Topology on locally finite metric spaces

TOPOLOGY ON LOCALL Y FINITE METRIC SP A CES V ALERIO CAPRARO Abstra ct. The necessity of a theory of General T op ology and, most of all, of Algebraic T op ology on locally ﬁnite metric spaces comes from man y areas of re search in both Applied and Pure Mathematics: Molecular Biology , Mathematical Chemistry , Computer Science, T op ologica l Graph Theory and Metric Geometry . In this pap er we prop ose the basic no- tions of such a theory and some app lications: w e replace th e classi cal notions of contin uous function, h omeomorphism and homotopic equiv alence with the notions of NPP-function, NPP-local-isomorphism and N PP-homotop y (N PP stand s for Nearest Poin t Preserving); w e also in tro duce the notion of N PP-isomorphism. W e construct th ree in v arian ts un- der N PP-isomorphisms and, in particular, we deﬁn e the fundamental group of a lo cally ﬁnite metric space. As ﬁrst applications, we propose th e follo wing: motiv ated by th e longstanding question whether there is a purely metric condition which extend s the no- tion of amenability of a group to any metric space, w e prop ose the p roperty SN (Small Neighborho od); motiv ated by some applicativ e problems in Comput er Science, w e prov e the analog of the Jordan curve theorem in Z 2 ; motiv ated by a question asked during a lecture at Lausanne, we extend to any locally ﬁn ite metric space a recent inequality of P .N.Jolissain t and V alette regarding the ℓ p -distortion. 1. Introduction 1.1. Motiv ations, applications and related literature . Th e necessit y of a theory of General T op ology and, m ost of all, of Algebraic T op ology on lo cally ﬁ nite metric spaces comes from sev eral areas of researc h: • Mole cular biologists, as Bon, V ernizzi, Orland and Z ee, started in 2008 a classiﬁ- cation of R NA structures lo oking at their genus (see [ ? ] and also [Bo-Or11] and [RHAPSN11] for dev elopmen ts). • Chemists, as Herges, started in 2006 to stud y molecules lo oking lik e a discr etizatio n of the M¨ obius strip or of the Klein b ottle (see [He06] and also the b o ok [Ro-Ki02]). • Computer scien tists found ed in the late 80s a new ﬁ eld or researc h, called Digital T op ology , motiv ate d by th e study of the top ology of the scree n of a compu ter, whic h is a lo cally ﬁn ite spaces whose p oin ts are pixels (see, f or instance, [Ko]). Although the motiv atio ns are mainly from App lied Mathematics, the theory has many applications also in Pure Mathematics. (1) The theory , in tro ducing a n otion of conn ectedness for lo cally ﬁ nite metric spaces, giv es the tools to bring up m any results from Gr ap h Theory to lo cally ﬁnite metric 2000 Mathematics Subje ct C lassiﬁc a tion. Primary 52A01; Secondary 46L36. Supp orted by Swiss S NF Sinergia pro ject CRS I22-13043 5. 1 2 V ALERIO CAPRARO spaces. In particular, motiv ated b y a question aske d d uring a lecture of Alain V ale tte, w e p ro v e a version f or lo cally ﬁ nite metric spaces of a recen t inequalit y pro v ed by P .N.Jolissain t and A.V alette regarding the ℓ p -distortion. (2) The th eory , int ro ducing a n otion of con tin uit y for lo cally ﬁn ite metric spaces, giv es the to ols to bring down man y results fr om, say , T op ology of Manifolds to lo cally ﬁnite metric spaces. In particular, motiv at ed b y a p roblem in Computer Science, w e p ro v e the an alogue of the J ordan curve th eorem in Z 2 . (3) The theory leads to the disco v ery of the p rop ert y SN, which is, as far as we kno w, the ﬁ rst kn o wn pur ely metric p r op ert y that r educes to amenabilit y for th e Ca yley graph of a ﬁ nitely generated grou p . Here is a little sur v ey ab out related ideas in literature. • Dig ital T op ology con tains many go o d ideas that h a v e led to the d isco v ery of Al- gebraic T opology on the so-called digital spaces (see [Kh87], [Ko89], [ADF Q00 ] and [Ha10]). Unfortunately , this theory , desp ite concerns only subsets of Z n (see [ADF Q00], Example 2.1), is q u ite tec hnical, f ar from b eing in tuitiv e and con tains w eird b eha viors, as the fact that the digital f undament al group do es n ot alwa ys v er- ify the multiplicati ve prop erty (see [Ha10] and r eference therein) and the S eifert-v an Kamp en theorem (see again [ADF Q00]). • T o p ological Graph Th eory is based on the follo wing identiﬁcat ion: every vertex is seen as a p oint; an y edge joinin g tw o verte x is seen as a cop y of [0 , 1] j oinin g the t w o extremal p oin ts. This theory has many applications that one can ﬁn d in an y of the b o oks on the topic, but in our opinion it d oes not capture the r eal top ologica l essence of a graph. W e will se e that there are tec hnical and natur al reasons to exp ect that 3-cycle and the 4-cycl e m ust ha v e trivial fundamental group. T he latter example also sho ws that Rips complexes are n ot enough to reac h our p urp ose 1 . • Man y authors h a v e stud ied coarse cohomology on lo cally ﬁnite metric sp aces (see [Ho72], [Ro93 ] and, for a sys tematic app r oac h, [Ro03 ]). Th is is of cour se an inte r- esting theory , with man y applications in K-theory , where one is inte rested only on large-scal e b ehaviors of a s p ace, but is u seless in our case where one is in terested in lo cal b ehavio rs and, more sp eciﬁcally , in ﬁnite spaces (molecules are ﬁ nite, th e screen of a computer is ﬁnite ...), that get trivial in coarse geometry . Hoping to hav e con vinced the r eader th at s omething new is really necessary , the very basic id ea th at we are going to follo w is a c hange of p ersp ectiv e ab out the notion of con tin uit y . Let X b e a top ologica l sp ace, a con tin uous path in X is classically seen as a con tin uous function f : [0 , 1] → X . Hence, the n otion of con tin uit y is, in some sense, imp osed from outside , taking as a u nit of reference th e in terv al [0 , 1]. This c hoice wo rks 1 T rying to deﬁ ne the fundamen tal group via Rips complexes leads to many d iﬃcu lties: ﬁ rst of all, since it dep ends on the radius δ , it would not b e clear whic h is the right fundamental group; moreov er, a non- trivial deﬁnition w ould imp ly that the fundamental group of the un it square Q in Z 2 with the Euclidean metric is Z , con tradicting, ﬁrst of all, the natural requirement of the v alidit y of th e Jordan curve theorem in Z 2 (indeed Z 2 \ Q w ould b e p ath- c onne cte d ), but also the intuitio n that an observer living on Q has no unit of measuremen t to ﬁnd a hole. TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 3 v ery w ell for m anifolds, b ecause ev ery op en set con tains copies of [0 , 1], b ut it do es n ot work for lo cally ﬁnite metric spaces. The idea is to change p ersp ectiv e lo oking at the notion of con tin uit y from inside . Contin uit y is to do steps as short as p ossible : a con tin uous path in a locally ﬁnite metric space will b e constructed, rou gh ly sp eaking, as follo ws: start fr om a p oin t x 0 , mo v e to one of the nearest p oin ts x 1 and so on. F rom a p hilosophical p oin t of view, this pap er is a corollary of this simple c hange of p ers p ectiv e. 1.2. Structure of the pap er. • In S ec. 2 w e dev elop the basics of Algebraic T op ology on lo cally ﬁnite metric spaces: w e introd uce the notion of fund amen tal group of a locally ﬁnite metric space, taking inspiration fr om the original d eﬁnition b y P oincar ` e, and w e p ro v e that the construc- tion is in dep endent on the b ase p oin t and giv es an NPP-inv arian t (see Th eorem 10). W e give some examples of fu ndamen tal group s , sho wing that this fund amen tal group b eha v es like a discr etization of the classical fu ndamen tal group: for instance, the fu ndamen tal group of the grid Z 2 is th e same as the classical fund amental group of R 2 , the f undament al group of Z 2 \ { (0 , 0) } is the same as the classical fu nda- men tal group of R 2 \ { (0 , 0) } . W e state some r esu lts ab out the fun d amen tal group that will b e prov ed in the companion piece of this p ap er [Ca-Go-Pi11]. W e in- tro duce the discrete analog of the notions of con tin uous fun ction, homeomorphism and homotopic equiv alence and we pro v e, in Theorem 21, that they in duce group homomorphisms/isomorph isms at the lev el of the fund amen tal group s. • In Sec. 3 w e p rop ose t wo applications of the fun d amen tal group : motiv ated by a problem in Comp uter Science, we prov e the analog in Z 2 of the J ordan cur v e theorem (see Th eorem 26); motiv ated b y a question ask ed du ring a lecture we extend to an y ﬁn ite metric space a r ecen t result by P .N. Jolissaint and A. V alette, whic h giv es a lo w er b ound for the ℓ p -distortion of a ﬁnite connected graph (see Theorem 32 and C orollary 33). • Motiv ated b y the theory dev elop ed in Sec. 2, in S ec. 4 we introdu ce a new notion of isomorphism b et w een lo cally ﬁnite metric spaces, which will b e called NPP- isomorphism. In S ec. 4.3 we collect s ome simple examples of NPP-isomorph isms, sho wing also that this NPP-w a y to em b ed a metric space into another is not equiv- alen t to an y of the other w ell-kno wn w a ys. I n particular, w e introd uce the notion of gr ap h-typ e metric sp ace and we prov e that any such a metric space is canonically NPP-isomorphic to a T r aveling Salesman graph (Th eorem 44). • In Sec. 5 we in tro duce the second NPP-inv arian t for lo cally ﬁn ite m etric s p aces (the fundamental group b eing the ﬁr st one), w hic h w e call isop erimetric c onstant (Deﬁnitions 47, 48 and Theorem 49), since the construction is similar (and reduces) to the isop erimetric constant of a lo cally ﬁnite, inﬁn ite, connected graph (that is not an NPP-inv arian t, by the wa y). • Motiv ated by the qu estion What is the pr op erty that c orr esp on ds to have isop eri- metric c on stant e qual to zer o , in Sec. 6 we introdu ce a pr op ert y for any m etric space, the Small Neigh b orho o d p rop ert y (pr op ert y SN), in Deﬁnition 52. At ﬁrst w e give some examples of sp aces with or without pr op ert y SN (Prop osition 55 ) and 4 V ALERIO CAPRARO then we stud y the relation b et w een the prop ert y S N and the amenabilit y: we prov e that the pr op ert y SN is equiv alen t to the amenability for inﬁ nite, lo cally ﬁnite, connected graph (Theorem 60 and Prop osition 67). So, p r op ert y SN is th e ﬁ r st kno wn purely metric p rop ert y that reduces to amenabilit y for (the Ca yley graph of ) a ﬁnitely generated group . Finally , we giv e a descrip tion of the prop er ty SN of a lo cally ﬁn ite space in terms of the isop erimetric constant (Theorem 69), whic h is a v ersion f or lo cally ﬁ n ite metric spaces of well-kno wn resu lts by Cecc herini- Silb erstein, Grigorc h uk, de la Harp e and Elek, S ´ os. I n particular, it turns out that the prop erty SN is an NPP -inv ariant. • In S ec. 7 w e in tro duce the third inv arian t of the theory , whic h we call zo om isop eri- metric c onstant , since it give s more precise information than the isop erimetric con- stan t, using a construction that, in some sense, increases the kn o wledge of the space, step b y step (Deﬁnition 71 and Theorem 72). W e p ro v e that the t w o isop erimetric constan ts are, in some s ense, ind ep enden t, meaning that in general there is no w a y to dedu ce one from th e other (Theorem 79). Using this new inv ariant, we prop ose a s tronger notion than amenab ility , the notion of lo c al ly amenable sp ac e , a space whic h is amenable w ith resp ect to any lo c al observation . W e prov e in Theorem 75 that in case of graph-t yp e sp aces, a single observ ation is enough to conclude if the space is completely amenable. This suggests (Sec. 7. 3 ) some wa y to extend the classical notion of expand er of graphs and the recen t notion of geometric p r op ert y (T) to any lo cally ﬁnite m etric sp ace. Before starting the tec hnical discussion, let us ﬁx some notation. Th roughout the p ap er: • Giv en a ﬁnite set A , w e d enote by | A | the num b er of elemen ts in A . • ( X , d ) will b e, a priori, any metric space. Giv en x ∈ X and r > 0, B ( x, r ) will denote the op en b all of r adius r ab out x . • Giv en a subset A of X , we d en ote b y A the closure of A . • Giv en A ⊆ X and α > 0 w e d enote – cN α ( A ) = { x ∈ X : d ( x, A ) ≤ α } – cN α ( A ) ◦ = { x ∈ X : d ( x, A ) < α } The diﬀeren ce cN α ( A ) \ cN α ( A ) ◦ is denoted by cB α ( A ). 1.3. Other p ossible applications. This subsection can b e skipp ed at ﬁ rst reading. It is little sp eculativ e and sho ws the picture that th e author had in min d writing this pap er. The (hop efu lly) in teresting part is the formulati on of a new class of problems that might b e of interest b oth f or Pur e and Applied Mathematics. Let ( X , d ) b e a metric sp ace and su pp ose that ther e is an observer that wan ts to mak e a map of X . The strong restriction, that often an observe r f aces in the real life, is that the distance b et w een tw o p oin ts is n ot kn own exactly . The observ er is only able to say , giv en t w o p oin ts y , z , whic h one is n earer to another p oin t x . One can construct d ozens of real examples wh ere suc h a situation ma y happ en; h ere are a couple of them: TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 5 • An astronomer wan ts to make a picture of the universe. He or she d oes not kno w the exact distance b et we en t w o ob jects, bu t (sometimes) he can d ecide ind irectly whic h one is nearer to a th ird ob ject. • An applied computer scientist wan ts to upload some data r elate d to a metric sp ace. Ma yb e in this case he kno ws mathematical ly the function d , but it could b e imp os- sible to u pload it exactly (mayb e just b ecause d ( x, y ) is an irrational num b er and (s)he is allo w ed to us e just rational num b ers with a ﬁ x ed num b er of d igits). His/her idea migh t b e to replace d with another function which preserv es the relation of nearness/farness b et w een p oin ts. • A trav e ling salesman who do es not kn o w the exact distance b etw een the cities that he has to visit, b ut kno ws, ma yb e b y mere observ atio n of a map, their relations of nearness/farness. W e w ill formalize later the in tuitiv e id ea that, abstractly , the pro cedure of making suc h a map is describ ed by an NPP-em b eddin g, namely an em b eddin g of ( X , d ) into another metric space ( X ′ , d ′ ) wh ich is Ne a r est P oints Pr eserving . The image of ( X, d ) und er this em b edding is called sym b olic map . T he th eory has clearly tw o general questions: • Is it p ossible (and, in case, ho w?) to mak e a go o d symbolic map? • What kind of information are enco ded in a symbolic map? These questions generate a new class of mathematical questions: • Ho w to ﬁ nd go od NPP-embedd ings? • What are th e in v arian ts u nder NPP -isomorphisms? The s econd qu estion will b e widely d iscussed in this pap er, w hose primary pur p ose is to ﬁn d three diﬀerent inv arian ts u nder NPP-isomorph isms. T he ﬁrst question will not b e discussed and so w e list b elo w some explicit questions related to it. Problem 1. Given a metric sp ac e, ﬁnd its b est NP P-emb e dding (Banach sp ac e? lowest dimensional Hilb ert sp ac e? c onstant curvatur e su rfac es? As already happ ened a couple of cen turies ago, also in this case Cartography helps to formulate interesting and d iﬃcult q u estions: the general problem in cartograph y is to represent some metric s pace into a map that w e can easily construct and bring with u s. It is clear that the b est map w ould b e a planar map , but w e can b e satisﬁed also of a su rface of constant curv ature. So an inte resting p r oblem for the application w ould b e Problem 2. Find hyp othesis that gu ar ante e the existenc e of an NPP- emb e dding into a surfac e of c onsta nt curvatur e and, in p articular, into the Euclide an plane. A classical problem in Applied Computer Science is that of u ploading some data in suc h a wa y to k eep the relations of farness and nearness among p oin ts. Since in this case w e can only u se a ﬁ nite n umber of digits, s ay N , th e most imp ortant p roblem seems to b e the follo wing: Problem 3. Find hyp otheses that guar ante e the existenc e of an NPP- emb e dding of a (ne c- essarily ﬁnite) metric sp a c e into some (( Q N ) n , d ) , wher e Q N is the set of r ational numb ers with N digits and d is a c omputable metric (for i nstanc e, the ℓ 1 -metric). 6 V ALERIO CAPRARO Another interesti ng p roblem r egards the p ossibilit y to ﬁn d the b est NPP-emb edding Problem 4. Find hyp otheses that guar an te e the existenc e of an N PP-emb e dding into some Z n , e quipp e d with some computable m etrics , as for instanc e the ℓ 1 -metric. In this latter case, it is likely that a theory of homology of a lo cally ﬁ nite metric space can h elp. 2. Basics of Algebraic Topology on lo call y finite sp aces This section is d ev oted to the basic concepts of General T op ology and Algebraic T op ology on a locally ﬁ nite metric space: we introd u ce the notion of fund amen tal group, we present some basic prop erties and we introd uce the discrete analog of the classical notions of con tin uous function, h omeomorphism and homotop y . 2.1. The fundamental group of a lo cally ﬁnit e space. In the spirit of the original deﬁnition of the f undament al group of a top ological space, w e constru ct a group whose r ole is to capture the holes of a lo cally ﬁnite m etric space. The b asic examp le to k eep in mind is the follo wing: Z 2 has no holes, Z 2 \ { (0 , 0) } h as a hole. T ry to attac h a fun damen tal group to a lo cally ﬁnite metric space is not a n ew id ea, at least when the metric comes from a lo cally ﬁ nite, connected graph . In this case the standard deﬁnition, u sing spann ing trees, is n ot th at interesting. V ery recentl y , Diestel and Spr ¨ ussel ha v e stu died a more in teresting n otion of fund amen tal group for lo cally ﬁ- nite connected graphs with end s (see [Di-Sp11]). O n the other hand, many authors h av e studied s ome coarse cohomogy (see [Ho72], [Ro93] and, f or a sy s tematic approac h, Ch apter 5 in [Ro03]). As we told in th e in tro duction, our motiv ating problems force to consider something new, something able to capture th e details of the space. W e recall th at our main int uition is to c hange p ersp ectiv e lo oking at the notion of contin uit y from inside . So, w e rep lace the usu al n otion of a con tin uous p ath with th e notion of a path made b y th e shortest admissible steps and th e notion of h omotopic equiv a lence w ith an equiv ale nce that puts in relation paths wh ic h diﬀer by the shortest ad m issible steps. Throughout th is section, let ( X , d ) b e a lo cally ﬁn ite metric space; i.e. a metric space whose b oun ded sets are ﬁnite. Deﬁnition 5. L et x ∈ X . The set dN 1 ( x ) , c a l le d discr ete 1-neighb orho o d of x , is c o nstructe d as fol lows: let r > 0 b e the smal lest r adius such that | B ( x, r ) | ≥ 2 , then dN 1 ( x ) = B ( x, r ) . Deﬁnition 6. L et ( X , d ) b e a lo c al ly ﬁnite metric sp ac es. A c ontinuous p ath in X i s a se quenc e of p oints x 0 x 1 . . . x n − 1 x n such that for al l i = 1 , 2 , . . . , n , one has x i ∈ dN 1 ( x i − 1 ) and x i − 1 ∈ dN 1 ( x i ) (1) Observe that to b e joine d by a c ontinuous p ath is an equiv ale nce relation. T he equiv alence classes are called path-connected comp onents . TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 7 Remark 7. A simple bu t imp orta nt r emark is that a c ontinuous p ath x 0 x 1 . . . x n − 1 x n is char acterize d b y the pr op erty: d ( x i , x i − 1 ) = d ( x i +1 , x i ) , for al l i ∈ { 1 , . . . , n − 1 } . So, the fol lowing set of p oints inside R 2 with the Euclide an distanc e is not p a th c onne cte d • ( − 1 , 1) • (1 , 1) • ( − 1 , 0) • (0 , 0) • (1 , − 1) and, in p articular, it has thr e e p ath c onne cte d c o mp onents: C 1 = { ( − 1 , 1) , ( − 1 , 0) , (0 , 0) } , C 2 = { (1 , 1) } and C 3 = { (1 , − 1) } . An e xample of p a th-c onne cte d subset X of R 2 (with the Euclide an distanc e) i s the fol lowing: • ( − 1 , 1) • (0 , 1) • (1 , 1) • ( − 1 , 0) • (1 , 0) • ( − 1 , − 1) • (0 , − 1) • (1 , − 1) This latter example shows also that a lo c al ly ﬁnite p ath-c onne cte d metric sp ac e might not lo ok like a gr aph: ther e is no way to put on X a gr a ph distanc e which i s c o her ent to the one inherite d by the Eu clide an metric. As for the classical fund amen tal group, let us restrict our atten tion to path-connected lo cally ﬁnite metric spaces 2 X and let x ∈ X b e a ﬁxed b ase p oin t. A con tin uous circuit is simply give n b y a con tin uous p ath x 0 x 1 . . . x n − 1 x n suc h that x 0 = x n = x . As u sual, we equip the set of circuits with the same base p oin t with the op eration of concatenation ( x 0 x 1 . . . x n − 1 x n )( y 0 y 1 . . . y m − 1 y m ) = x 0 . . . x n y 0 . . . y m with the computational rule x i x i = x i for all x i ∈ X (2) No w let us in tro duce a notion of homotopic equiv al ence. Consider t w o con tin uous circuits x 0 x 1 . . . x n − 1 x n and y 0 y 1 . . . y m − 1 y m . By adding some x ’s or some y ’s we can supp ose that n = m (thanks to th e ru le 2). 2 It is a triv ial fact, but imp ortant to keep in mind , that a path -connected metric space can b e very far from b eing a graph. Consider for instance Z 2 \ { (0 , 0) } , equipp ed with the Euclidean metric. 8 V ALERIO CAPRARO Deﬁnition 8. The two cir cuits x 0 x 1 . . . x n − 1 x n and y 0 y 1 . . . y n − 1 y n ar e said to b e homo- topic e quivalent i f for al l i = 1 , . . . , n − 1 , ther e is a ﬁnite se quenc e z 1 i , . . . , z k i ( k do es not dep end on i , a p osteriori thanks to the c omputation al rule in 2) such that (1) z 1 i = x i (2) z k i = y i (3) z h +1 i ∈ dN 1 ( z h i ) and z h i ∈ dN 1 ( z h +1 i ) , for al l h = 1 , . . . , k − 1 (4) x 0 z h 1 z h 2 . . . z h n − 1 x n is a c o ntinuous cir cuit for al l h = 1 , . . . k A comfortable wa y to lo ok (and also to deﬁne) an h omotopic equiv alence is b y b uilding the homotopy ma t rix       x 0 x 1 . . . x n − 1 x n x 0 z 2 1 . . . z 2 n − 1 x n . . . . . . . . . . . . . . . x 0 z k − 1 1 . . . z k − 1 n − 1 x n x 0 y 1 . . . y n − 1 x 0       with the condition that ev ery r o w and every column f orm a contin uous path in X . Example 9. L et Q = { (0 , 0) , (1 , 0) , (1 , 1) , (0 , 1) } as a su bset of R 2 with the Euclide an metric and c o nsider the c o ntinuous cir cuit γ with b ase p oint (0 , 0) as in the ﬁgur e: • (0 , 1)   • (1 , 1) o o • (0 , 0) / / • (1 , 0) O O Now, the deﬁnition, says that it is p ossible r eplac e the p o ints (0 , 1) and (1 , 1) with one of closest p oints, for instanc e (0 , 0) and (1 , 0) . Henc e, γ is homotopic e quivalent to the fol lowing cir cuit • (0 , 1) • (1 , 1) • (0 , 0) , , • (1 , 0) l l Now, we c an r eplac e (1 , 0) with one of the ne ar est p oint, for instanc e (0 , 0) . In this way, we obtain the c o nstant p ath in (0 , 0) . So, the initial cir cuit is homotopic e qu i valent to the c o nstant p ath in (0 , 0) . In terms of homotopic matrix we have   (0 , 0) (1 , 0) (1 , 1) (0 , 1) (0 , 0) (0 , 0) (1 , 0) (1 , 0) (0 , 0) (0 , 0) (0 , 0) (0 , 0) (0 , 0) (0 , 0) (0 , 0)   This is a rough p ro of that π 1 ( C 4 ) = { 0 } , wher e C 4 is the 4-cycle. Notic e that su ch a pr o c e dur e c annot b e adapte d for C n , with n ≥ 5 . TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 9 It s h ould b e clear that ev erything go es w ell. L et u s summarize all basics pr op erties with the f ollo wing Theorem 10. (1) The homotop ic e quivalenc e is an e q uivalenc e r ela tion on the set of cir cuits with b ase p oint x . The q u otient set is denote d by π 1 ( X, x ) . (2) The op er ation of c onc atenation is wel l deﬁne d on π 1 ( X, x ) , giving rise to the struc- tur e of gr oup. (3) π 1 ( X, x ) do es not dep end on x i n the fol lowing sense: give n another y ∈ X , ther e is a gr oup i somorphism b etwe e n π 1 ( X, x ) and π 1 ( X, y ) . Pr o of. (1) an d (2) are eviden t. (3) is the discr etization of the standard pro of: let c 0 c 1 . . . c n − 1 c n b e a con tin uous path connecting x w ith y , then th e m apping Φ : π 1 ( X, x ) → π 1 ( X, y ) de- ﬁned by Φ([ x 0 x 1 . . . x n − 1 n ]) = [ c n c n − 1 . . . c 1 c 0 x 0 x 1 . . . x n − 1 x n c 0 c 1 . . . c n − 1 c n ] is the desired isomorphism.  Let us giv e s ome basic examples, ju st to sh o w ho w the fu n damen tal group b eha v es: Example 11. • The fundamental gr oup of Z 2 is trivial and the fu ndament al g r oup of Z 2 \ { (0 , 0) } is e qual to Z (b o th with r esp e ct to the gr aph metric and the Euclide an one). These two examples also show that this way to do algebr aic top olo gy on lo c al ly ﬁnite sp ac es is not c o arsely invariant. This is natur al sinc e the details of the sp ac e ar e very imp o rtant for our motivating pr o blems. • L et C n b e the n - cycle gr aph, then π 1 ( C n ) =  { e } , if n ≤ 4 ; Z , if n ≥ 5 . The pr o of is b asic al ly the gene r alization of the ar gument in E xampl e 9. In some sense, thr e e of f our p oints ar e not enough to ﬁnd a hole and one ne e ds mor e. This is clariﬁe d in much mor e g ener ality in the fol lowing The or em 12. W e conclude this in tro ductory s ection stating some resu lts that will b e pro v ed in the companion p iece of this pap er [Ca-Go-Pi11 ]. Theorem 12. L et X b e a c omp act metrizable p ath-c onne cte d manifold and let T b e a ﬁne enough triangulation of X . E quip the 1-skeleton of T with the natur al structur e of ﬁnite c o nne cte d gr aph . Then π 1 ( T ) = π 1 ( X ) The ﬁrst application of this result is to mak e u s optimistic ab out th e theory: it is not ortho gonal to the classical one, but we obtain it as a limit the o ry . Another application is th at it allo ws to construct (ev en explicitly) ﬁnite graph s wh ose fun d amen tal group has torsion, whic h is a completely new p h enomenon in T op olog ical Graph Theory . Finally , this result gets very close to the construction of an algorithm for c omp uting the fu ndamen tal group of a m anifold, knowing a triangulation, which is an op en problem for n ≥ 4 (see [KJZLG08]). I t is clear that s uc h an algorithm will n ot w ork in general, b ecause the wo rd 10 V ALERIO CAPRARO problem on ﬁn itely present ed groups is not solv able by a celebrated result by Novik o v (see [No55]), b u t this is of cour se a interesting topic for fu r ther researc h. Theorem 13. L et X , Y b e two lo c al ly ﬁnite p ath-c onne cte d metric sp ac e i n normal form 3 . Then X × Y is p ath-c onne cte d with the ℓ 1 -metric. O ne has π 1 ( X × Y ) = π 1 ( X ) × π 1 ( Y ) Theorem 14. L et X b e a lo c al ly ﬁnite p ath -c onne cte d metric sp ac e and { U λ } b e a c ontin- uous c overing 4 of X . Then π 1 ( X ) is uniquely determine d b y the π 1 ( U λ ) ’s in Seifert-van Kamp en ’s sense. 2.2. Con tin uous functions, homeomorphisms and homotopic equiv alences b e- t w een lo cally ﬁnite metric spaces. In this section we wan t to in tro duce the discrete analog of the classical notions of con tin uous fun ction, homeomorphism and homotopic equiv a lence. In order to ﬁnd the discrete an alog of the classical notion of con tin uous function, it suﬃ ces to discr etize th e stand ard notion of contin uous function fr om a metric s pace to another. W e get Deﬁnition 15. A fu nc tion f : X → Y b etwe e n two lo c al ly ﬁnite metric sp ac es i s c al le d NPP-function if for al l x ∈ X one has f ( dN 1 ( x )) ⊆ dN 1 ( f ( x ) ) Observe that NPP-functions map con tin uous circuits to con tin uous circuits (exactly as con tin uous fun ctions do in T op ology of Manifolds). Hence, the image of a lo cally ﬁ n ite path-connected metric space is still so. I n particular, if X is a lo cally ﬁnite path-connected metric space and f : X → Z is an NPP-function. S upp ose that there are x 0 , x 1 ∈ X such that f ( x 0 ) f ( x 1 ) < 0, then ther e is x ∈ X su ch that f ( x ) = 0. This is a ﬁ rst very simp le example sh o wing ho w the discrete analog of classical results for cont inuous fun ctions hold for NPP-function. By the wa y , notice that suc h a result is not true for con tin uous fun ction in classical sense (jus t thin k of f : Z → Z deﬁn ed b y f ( x ) = x 2 − 2). An other s im p le remark ab out NPP-function is that comp osition of NPP-fun ction is still an NPP-function. This sim p le prop erty will b e often used in th e sequel with ou t furth er comment. Deﬁnition 16. An NPP- function f b etwe e n two lo c al ly ﬁnite metric sp ac es X and Y is c a l le d NPP-lo c al-isomor phism if it is b i je ctive and the inv e rse f − 1 is stil l an NPP- function. 3 Let X b e a locally ﬁnite path - connected metric space and x ∈ X . Let R x b e the smallest R > 0 such that | B ( x, R ) | ≥ 2. On e can easily pro ve that R x does n ot d epend on x and so it is a constant of the space, called step o f X . A lo cally ﬁn ite path-connected space is said to b e in normal form if the d istance is normalized in such a wa y that the step is equal to 1. Observe that one can pass from a general form to the normal form via a simple re-normalization, whic h do es n ot aﬀect the fundamental group (see Theorem 21). It follo ws th at the hypothesis in the Theorem is wi thout loss of gener al ity . 4 The notion of contin uous cov ering is the discrete counterpart of an op en cov ering, closed un der ﬁnite inters ections and made of path-connected subsets, which is the standard h yp othesis of Seifert-v an Kamp en’s theorem. TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 11 The adjectiv e lo c al will b e clariﬁed in Sec. 4. At the momen t, we can just sa y that it comes fr om the fact that s u c h functions look ju st at the smallest non -trivial neighborh o o d of a p oin t x . In order to study some prop erties w e will need s ome more global n otion. Let us observe explicitly that the requir emen t th at f − 1 is NPP is n ecessary . I n deed, consider X = { 0 , 1 , 2 } with the Eu clidean metric and let ∆ b e the r egular triangle in R 2 . An y bijection f : X → ∆ is an NPP-fu nction, but n one of th em is an NPP-lo cal-isomorphism. Deﬁnition 17. L et f , g : X → Y b e two NPP-fu nctions. They ar e said to b e homoto pic e quivalent if ther e is a se quenc e of N P P-functions f n : X → Y such that (1) f 1 ( x ) = f ( x ) , for al l x ∈ X (2) for al l x ∈ X , ther e is a p ositive inte ger k ( x ) such that f 1 ( x ) f 2 ( x ) . . . f k ( x ) ( x ) is a c o ntinuous p ath joining f ( x ) and g ( x ) . If f is homotop ic to g we write f ≃ g . Remark 18. N otic e that every p oint f ( x ) is tr ansp orte d c ontinuously in g ( x ) in ﬁnite time, but the se quenc e f n might b e inﬁnite. As an explicit example, we now show that the identity on Z is homotopic e quivalent to the c onsta nt p a th at the origin (henc e Z is c o ntr a ctible). Deﬁne f n : Z → Z to b e f n ( x ) =    0 , if | x | ≤ n x + n, if x < − n x − n, if x > n It is cle ar that e a ch f n is an NPP-fu nc tion and that the se quenc e f 0 ( n ) f 1 ( n ) . . . f n ( n ) is a c o ntinuous p ath c onn e cting 0 with n , as claime d . Deﬁnition 19. Two lo c al ly ﬁnite metric sp ac es ar e c al le d homotopic e quivalent if ther e ar e NPP -functions f : X → Y and g : Y → X such that f ◦ g ≃ I d Y and g ◦ f ≃ I d X . No w we wa nt to prov e the d iscrete analog of well-kno wn prop erties of Algebraic T op ol- ogy; namely , that con tin uous fu nctions indu ce group homomorphism s at the lev el of the fundamentals group and that homeomorphisms and homotopic equiv a lences ind uce isomor- phisms at th e leve l of the fund amen tal groups. T h e last result is a ﬁ rst example of how to discr etize an existing pr o of . In particular, we present th e discretization of th e pro of in [Ma], Ch apter I I, Section 8. Let us ﬁx some notation: let X, Y b e tw o lo cally ﬁnite path-connected spaces an d f , g : X → Y t wo homotopic equ iv alen t NPP-fu nctions. Let { f n } b e the s equ ence of NPP- functions describing the h omotopic equiv alence b et w een f and g . Fix a base p oin t x 0 ∈ X and let γ = f 1 ( x 0 ) f 2 ( x 0 ) . . . f n − 1 ( x 0 ) f n ( x 0 ) b e the contin uous path connecting f ( x 0 ) and g ( x 0 ) along the homotopic equiv ale nce. Deﬁne the isomorp hism (as in T heorem 10) u : π 1 ( Y , f ( x 0 )) → π 1 ( Y , g ( x 0 )) to b e u ( β ) = γ − 1 β γ . Lemma 20. The fol lowing diagr am is c ommutative 12 V ALERIO CAPRARO π 1 ( X, x 0 ) f ∗ / / g ∗ ' ' O O O O O O O O O O O π 1 ( Y , f ( x 0 )) u   π 1 ( Y , g ( x 0 )) Pr o of. Let α ∈ π 1 ( X, x 0 ) and let us p ro v e that g ∗ ( α ) = γ − 1 f ∗ ( α ) γ . Let α b e r epresen ted b y the con tin uous circuit x 0 x 1 . . . x n − 1 x n . Add ing,if necessary , some constan t paths, we can b uild the follo wing homotopic matrix       f ( x 0 ) f ( x 1 ) . . . f ( x n − 1 ) f ( x n ) f 2 ( x 0 ) f 2 ( x 1 ) . . . f 2 ( x n − 1 ) f 2 ( x n ) . . . . . . . . . . . . . . . f n − 1 ( x 0 ) f n − 1 ( x 1 ) . . . f n − 1 ( x n − 1 ) f n − 1 ( x n ) g ( x 0 ) g ( x 1 ) . . . g ( x n − 1 ) g ( x n )       where the terminology homotopic matrix means that ev ery row and eve ry column is a con tin uous path. Observe that if we read around th e b oundary of this matrix we h a v e exactly f ∗ ( α ) γ ( g ∗ ( α )) − 1 γ 1 . Th e fact that the m atrix is an h omotopic matrix implies that this circuits is homotopic equiv alen t to the constan t path (just eliminate eac h column starting from the righ t side and, ﬁ nally , collapse the ﬁrst column to the constan t path). Hence f ∗ ( α ) γ ( g ∗ ( α )) − 1 γ 1 = 1, as required.  Theorem 21. L et f : X → Y b e an NP P -function b etwe en two lo c al ly ﬁnite p ath -c onne cte d metric sp ac es. Consider the induc e d function f ∗ : π 1 ( X, x ) → π 1 ( Y , f ( x )) deﬁne d b y f ∗ [ x 0 x 1 . . . x n − 1 x n ] = [ f ( x 0 ) f ( x 1 ) . . . f ( x n − 1 ) f ( x n )] (1) f ∗ is always wel l-deﬁne d and it is always a gr oup homomorp hism. (2) If f is an NP P -lo c al-isomorp hism, then f ∗ is a gr oup i somorphism. (3) If f is an homotopic e quivalenc e, then f ∗ is a gr oup i somorphism. Pr o of. (1) f ∗ is we ll-deﬁned, since it maps circuits to circuits (by deﬁnition) and pre- serv es the h omotop y b et we en circuits (by deﬁnition, again, f m aps homotop y ma- trices to homotopy m atrices). At this p oin t it is clear that f ∗ is a group homomor- phism. (2) It is clear, by the pr evious item and the fact that f − 1 is an NPP -fu nction. (3) Sin ce g ◦ f ≃ I d X , by Lemma 20 w e ob tain the follo wing commuta tiv e diagram π 1 ( X, x ) f ∗ / / u ' ' O O O O O O O O O O O O π 1 ( Y , f ( x )) g ∗   π 1 ( X, ( g ( f ( x )) Since u is an isomorphism, it follo ws that f ∗ is a monomorphism and g ∗ is an epimorp hism. Applyin g the same argument to f ◦ g we obtain that g ∗ is a TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 13 monomorphism and then it is an isomorphism. No w, since g ∗ ◦ f ∗ = u and b oth g ∗ and u are isomorp hism, it follo ws that f ∗ is an isomorphism, as requir ed.  3. Tw o app lica tions The theory that w e are d evelo ping is, in some sense, b et w een Graph Theory and T op ol- ogy of Manifolds and this is why one can ﬁnd man y app lications bringing up r esults fr om Graph Th eory to lo cally ﬁnite metric spaces or bringi ng down results f rom T op ology of Manifolds. What we mean, more sp eciﬁcally , is the f ollo wing: supp ose that one needs a version for lo cally ﬁnite metric space of a th eorem for graphs. The id ea is to d ecom- p ose the metric space into its connected comp onen ts. They are not graphs, but many argumen ts can b e applied with little v aria tions (basically b ecause there is a notion of con- nectedness) and s o one can re-pro v e v ariatio ns of the claimed theorem for eac h of them and, then, pu t ev erything together bac k. This is the spirit b ehind our ve rsion for general metric sp aces of the r ecen t P .N.Jolissain t-V ale tte’s inequalit y . O n the other hand, since ev- ery path-connected comp onen t is a discr etization of the classica l notion of path-connected comp onen t in General T op ology , many arguments of T op ology of Manifolds can b e applied in eac h p ath-connected comp on ent (basically b ecause there is a notion of contin uit y), giv- ing versions of r esults for manif olds in lo cally ﬁn ite metric spaces. Said this, one can ha v e fun giving many applications of the theory . W e h av e chosen to p resen t t w o particular app licat ions for sp eciﬁc reasons that will b e explained at the b eginning of their o wn su bsection. 3.1. The Jordan curv e theorem in Z 2 . The classical Jordan curve theorem states that a simp le closed curv e γ in R 2 separates R 2 in t w o p ath-connecte d comp onents, one b oun ded and one unboun d ed and γ is the b oundary of eac h of these comp onen ts. In this section we w an t to pr o v e the analog ue result in Z 2 with the Euclidean distance. The reason b ehin d the c hoice of this application is that the Jord an curv e theorem in Z 2 is of interest in Digital T op ology (see, for instance, [Bo08] an d reference ther ein), but it seems that the version that we are going to pr esen t is not kn o wn. Ind eed, it is based on a particular deﬁ n ition of simplicity of a cur v e that seems to b e n ew. In fact, one is tempted to deﬁn e a simp le circuit in Z 2 as a cont inuous circuit x 0 x 1 . . . x n − 1 x n suc h that the x i ’s are p airwise distinct for i ∈ { 1 , . . . , n − 1 } . With this d eﬁnition th e Jord an cu rv e theorem is false: consider the follo wing con tin uous circuit: (0 , 0)(0 , − 1)(1 , − 1)(2 , − 1)(2 , 0)(2 , 1 )(1 , 1)(1 , 2)(0 , 2)( − 1 , 2)( − 1 , 1 )( − 1 , 0 )(0 , 0) it is simple in the previous sense; it h as fundamenta l group equal to Z , b ut it do es not separate the grid Z 2 in t w o path-connected comp onen ts. In some sense, this circuit b eha ve s lik e the 8-shap e cu r v e in R 2 , which is not simple. The discrete analog of simplicit y is indeed something d iﬀeren t. Let us ﬁx some notation • Let ( x 0 , y 0 ) ∈ Z 2 , we denote b y dB 1 (( x 0 , y 0 )) the set { ( x 0 , y 0 + 1) , ( x 0 , y 0 − 1) , ( x 0 − 1 , y 0 ) , ( x 0 + 1 , y 0 ) } 14 V ALERIO CAPRARO whic h is called discrete 1-b oundary of ( x 0 , y 0 ). • Let ( x 0 , y 0 ) ∈ Z 2 , we denote b y dB 2 (( x 0 , y 0 )) the set { ( x 0 + 1 , y 0 + 1) , ( x 0 − 1 , y 0 + 1) , ( x 0 − 1 , y 0 − 1) , ( x 0 + 1 , y 0 − 1) } whic h is called discrete 2-b oundary of ( x 0 , y 0 ). Deﬁnition 22. A simple curve in Z 2 is a c ontinuous p ath x 0 x 1 . . . x n − 1 x n such that • The x i ’s ar e p airw ise distinct for i ∈ { 1 , . . . , n − 1 } . • Whenever x i ∈ dB 2 ( x j ) , for j > i , then j = i + 2 and x i +1 ∈ dB 1 ( x i ) ∩ dB 1 ( x j ) This deﬁnition m igh t seem ad ho c , but it is indeed exactly how simplicit y b eha v es in a discrete setting. Ind eed, the main prop erty of the classical notion of simple curve γ in R 2 is not the injectivit y of the m apping t → γ ( t ), bu t the fact (implied by the in jectiv- it y), that γ can b e r e- c onstructe d as follo ws: let I n t ( γ ) b e the b ounded p ath-connected comp onen t of R 2 \ γ and ( x 0 , y 0 ) ∈ I nt ( γ ). Construct four p oints of γ as follo ws: ( x + 0 , y 0 ) is the ﬁrst p oin t of the half-line y = y 0 , with x > x 0 , hitting γ ; analogously , constru ct ( x − 0 , y 0 ) , ( x 0 , y + 0 ) , ( x 0 , y − 0 ). Then γ is un iquely determined b y the set of these p oin ts, when ( x, y ) runs ov er the path-connected comp onent of R 2 \ γ contai ning ( x 0 , y 0 ). Notic e that this re-construction prin ciple is false in R 2 for closed curve havi ng f undament al group equal to Z or for the 8-shap e cur v e. In the f ollo wing, w e shows that our d eﬁnition of simple curve in Z 2 giv es exactly the pr op ert y whic h allo w to make this re-construction pr ocedu re. Let γ b e a simple cur v e. Observe that, u p to h omotop y , w e can supp ose that γ do es not con tain any unit squar e ; n amely , it do es not con tain four p oin ts of the shap e ( x 0 , y 0 ) , ( x 0 + 1 , y 0 ) , ( x 0 + 1 , y 0 + 1) , ( x 0 + 1 , y 0 ). Let ( x, y ) ∈ Z 2 \ γ , we constru ct (at most) four p oin ts in the follo wing w a y: • ( x + 0 , y 0 ) is the ﬁrst p oin t (if exists) where the horizon tal half-line y = y 0 , for x > x 0 , hits γ ; • ( x − 0 , y 0 ) is the ﬁrst p oin t (if exists) where the horizon tal half-line y = y 0 , for x < x 0 , hits γ ; • ( x 0 , y + 0 ) is the ﬁ rst p oin t (if exists) where the v ertical half-line x = x 0 , for y > y 0 , hits γ ; • ( x 0 , y − 0 ) is the ﬁ rst p oin t (if exists) where the v ertical half-line x = x 0 , for y < y 0 , hits γ ; Deﬁnition 23. A p oint ( x 0 , y 0 ) is c a l le d quasi-internal to γ if al l four p oints deﬁne d ab ove exist. Deﬁnition 24. A subse t C of Z 2 is c al le d internal to γ i f • Every ( x, y ) ∈ C is quasi-internal to γ , • C is c al le d under the r econstruction p ro cedure ; namely, if ( x 0 , y 0 ) ∈ C , then any p oint ( x, y 0 ) , with x ∈ [ x − 0 , x + 0 ] , b elongs to C and also any p oint ( x 0 , y ) , with y ∈ [ y − 0 , y + 0 ] , b elongs to C . TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 15 In this c ontext, the p oints of the shap e ( x ± 0 , y 0 ) and ( x 0 , y ± 0 ) , with ( x 0 , y 0 ) ∈ C , ar e c al le d extr e mal p oints of C with r esp e c t to γ . The set of extr emal p o ints of C with r esp e ct to γ is denote d by E xtr γ ( C ) . It is clear that one of the crucial part of the pr oof of the Jordan curve theorem in Z 2 is the existence of an internal set. F or no w, let us supp ose th at it exists and prov e the re-construction p rinciple. Let C ⊆ Z 2 b e inte rnal to γ and su pp ose that f or any e ∈ E xtr γ ( C ) one of the follo wing prop erties hold: • | dB 1 ( e ) ∩ E xtr γ ( C ) | = 2 • | dN 1 ( e ) ∩ E xtr γ ( C ) | = 1 and | dB 2 ( e ) ∩ E xtr γ ( C ) | = 1 • | dN 1 ( e ) ∩ E xtr γ ( C ) | = 0 and | dB 2 ( e ) ∩ E xtr γ ( C ) | = 2 If e is a p oin t of the second shap e, w e denote by e ′ the unique p oin t in dB 2 ( e ) ∩ E xtr γ ( C ); if e is a p oint of the third shap e, we denote by e ′ , e ′′ the t w o p oints in dB 2 ( e ) ∩ E xtr γ ( C ). Under this hypothesis, we deﬁ ne th e γ -completion of C to b e the set C γ formed by the union b etw ee n C and all p oin ts e ′ and e ′ , e ′′ . No w we can pro v e the r e c onstruction principle in Z 2 . Lemma 25. L et γ b e a simple cir cuit in Z 2 which do es not c on tain unit squar es and let C b e a ﬁnite subset of Z 2 which i s internal to γ . Then γ = C γ Pr o of. Fix e = ( x 0 , y 0 ) ∈ E xtr γ ( C ). Since ( x 0 , y 0 ) is extremal, we kno w that there is at least one p oint in dN 1 ( x 0 , y 0 ) which b elongs to C . W e can s upp ose that it is ( x 0 + 1 , y 0 ). Since γ is con tin uous and ( x 0 , y 0 ) ∈ γ , we know that there are at least t w o p oints in dN 1 ( x 0 , y 0 ) b elonging to γ . Now, one has just seve ral p ossibilities to study . Let us go through a couple of them (the remaining will b e left to the reader). First case: ( x 0 , y 0 − 1) , ( x 0 , y 0 + 1) ∈ γ and ( x 0 + 1 , y 0 − 1) , ( x 0 + 1 , y 0 + 1) / ∈ γ . Since C is closed un der reconstruction, it follo ws that ( x 0 + 1 , y 0 − 1) , ( x 0 + 1 , y 0 + 1) ∈ C In this case it is evident that ( x 0 , y 0 − 1) and ( x 0 , y 0 + 1) are extremal and so w e b elong to the ﬁrst of th e previous three p ossibilities and there is no completion to do. Second case: ( x 0 + 1 , y 0 − 1) , ( x 0 + 1 , y 0 + 1) ∈ γ . Observe that these tw o p oin ts b elong also to E xtr γ ( C ) and so we b elong to th e third p ossibilit y and w e h a v e e ′ = ( x 0 , y 0 + 1) and e ′′ = ( x 0 , y 0 − 1) . Note that e ′ , e ′′ ∈ γ , sin ce γ is simple. Third case: ( x 0 , y 0 − 1) , ( x 0 + 1 , y 0 + 1) ∈ γ and ( x 0 + 1 , y 0 − 1) / ∈ γ . Since C is closed und er recon- struction, it follo ws that ( x 0 + 1 , y 0 − 1) ∈ C . So we are in the second p ossibilit y and so 16 V ALERIO CAPRARO e ′ = ( x 0 , y 0 + 1). Notice that e ′ ∈ γ , sin ce γ is simple. And so on tediously . No w, observe that w e h av e p r o v ed that for any e ∈ E xtr γ ( C ), w e can ﬁ nd a piece of γ in dB 1 ( e ) ∪ dB 2 ( e ) and this p iece of γ b elongs, by construction, to C γ . Sin ce C is ﬁn ite, this pro cedur e giv es rise a sub-circuit of γ , w h ic h has to coincide with γ , since the latter is simp le and d o es not contai n unit squ ares.  Theorem 26. L et γ b e a non-c onstan t simple cir cuit in Z 2 not c ontaining squar es 5 . Then Z 2 \ γ has exactly two p ath-c onne cte d c omp on ents, denote d by E xt ( γ ) and I nt ( γ ) and the fol lowing pr op erties ar e satisﬁe d : • I nt ( γ ) is ﬁnite; • E xt ( γ ) is inﬁnite; • γ = E xt r γ ( I nt ( γ )) γ Pr o of. F or the con v enience of the reader, we divide the pro of in sev eral steps. First st e p: deﬁnition of I n t ( δ ) and E xt ( δ ) . Let γ = c 0 c 1 . . . c n − 1 c n and Γ = { c 0 , . . . c n − 1 } . Fix the follo wing integ ers: x − = min { x : ( x, y ) ∈ Γ } x + = max { x : ( x, y ) ∈ Γ } y − = min { y : ( x, y ) ∈ Γ } y + = max { y : ( x, y ) ∈ Γ } No w, for an y in teger k ≥ 0 and for an y integ er y ∈ [ y − − k , y + + k ] consider the contin uous path γ y , k = ( x − − k, y )( x − − k + 1 , y ) . . . ( x + + k − 1 , y )( x + + k, y ) and d en ote b y Γ y , k the set { ( x − − k, y ) , ( x − − k + 1 , y ) , . . . , ( x + + k − 1 , y ) , ( x + + k, y ) } Let ( x 1 , y − )( x 2 , y − ) . . . ( x r , y − ) b e the ﬁrst contin uous path obtained by the inte rsection b et w een Γ and the horizontal line y = y − ; i.e. x 1 is th e min imal x suc h that ( x, y − ) ∈ Γ. Observe that r 6 = 1. Ind eed, if it was r = 1, then γ wo uld not ha v e b een simple. It follo ws that r ≥ 2 an d therefore Γ contai ns ( x 1 , y − ) , ( x 1 + 1 , y − ) and ( x 1 , y − + 1). Hence, ( x 1 + 1 , y − + 1) / ∈ Γ (since γ do es n ot conta in unit squares). W e d eﬁne I nt ( γ ) to b e the path-connected comp onen t of Z 2 \ γ con taining ( x 1 + 1 , y − + 1). O f course, we no w deﬁn e E xt ( γ ) to b e the complemen tary of I nt ( δ ) ∪ γ . Second ste p: I n t ( γ ) is ﬁnite and E xt ( γ ) is inﬁnite . By deﬁnition I nt ( γ ) is non-empt y . In order to pro v e that it is ﬁ nite and that E xt ( γ ) is inﬁnite, we p ro v e that I nt ( γ ) is con tained in the rectangle R = [ x − , x + ] × [ y − , y + ]. Let us denote by ( z 0 , w 0 ) the p oint ( x 1 + 1 , y − + 1) th at we hav e found in th e previous s tep. Let ( z 1 , w 1 ) ∈ dN 1 (( z 0 , y 0 )) ∩ I nt ( γ ). There are t w o p ossibilities: ( z 1 , w 1 ) = ( z 0 + 1 , w 0 ) or 5 This assumption is, in some sense, without loss of generalit y , because w e can alw ays assume it up to homotopic e quivalenc e . TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 17 ( z 1 , w 1 ) = ( z 0 , w 0 + 1). In eac h case, one has z 1 ≥ x − and w 1 ≥ y − . No w consider the ve rti- cal lin e z = z 1 and su pp ose, by cont radiction, that it inte rsects γ only in ( z 1 , w 1 − 1). Since γ is simp le, it necessarily follo ws that either γ ⊆ { ( x, y ) : x < z 1 } or γ ⊆ { ( x, y ) : x > z 1 } . But also in these cases, one contradicts the fact that γ is simple. It f ollo ws that the line x = z 1 in tersects γ in another p oint ( x 1 , y ) and then w 1 ≤ y ≤ y + . It follo ws that w 1 ≤ y + . Analogously , taking the h orizonta l line y = w 1 , one gets z 1 ≤ x + . Hence ( z 1 , w 1 ) ∈ R . No w rep eat the argument for an y ( z 2 , w 2 ) ∈ dN 1 (( z 1 , w 1 )) ∩ I nt ( γ ) and we get the same. Since I nt ( γ ) is, b y deﬁnition, the p ath-connecte d comp onen t con taining ( z 0 , w 0 ) we obtain indeed that I nt ( γ ) ⊆ R . Third step: I n t ( γ ) and E xt ( γ ) are path-connected. I n t ( γ ) is p ath-connecte d by deﬁnition. Let us prov e that also E xt ( γ ) is path-connected. Let ( x 0 , y 0 ) ∈ E xt ( γ ), it suﬃces to ﬁ nd a con tin uous path starting from ( x 0 , y 0 ) an d going outside R w ithout hitting γ . W e pro ceed by making cont inuously the r econstruction pro cedure: • W e run contin uously along the half-line x = x 0 , with y > y 0 . If w e go outside R without hitting δ , the pro of is o v er; otherw ise, let ( x 0 , y + 0 ) ∈ δ b e the ﬁ rst p oint where w e hit δ . A t the same wa y , w e construct ( x 0 , y − 0 ), ru nning cont inuously along the half-line x = x 0 , with y < y 0 ; then, we construct ( x − 0 , y 0 ) and ﬁ nally ( x + 0 , y 0 ). In this wa y we construct a con tin uous path δ 1 . • F or an y p oin t in δ 1 , w e rep eat the argument in the ﬁrst step, constructing a con- tin uous p ath δ 2 . • W e rep eat the argument, u ntil p ossible. No w, sup p ose th at the pro cedur e end s after a ﬁnite n umber of steps, we w an t to ob- tain a con tradiction by showing that ( x 0 , y 0 ) ∈ I nt ( γ ). I n deed, apply the r econstruction pro cedure ﬁ rst starting from ( x 0 , y 0 ) and then starting fr om ( z 0 , w 0 ) (it is the same p oin t as the pr evious s tep of the pr oof ). W e obtain tw o in ternal s ets C and C ′ whose extremal p oin ts deﬁn e, thanks to the reconstruction principle, the same circu it. In particular, it follo ws that C ∩ C ′ 6 = ∅ . Let ( x, y ) ∈ C ∩ C ′ and let C ′′ b e the set obtained b y making the reconstruction pr ocedu re starting from ( x, y ). It is clear that C = C ′′ = C ′ and, in particular, C = C ′ . I t follo ws that ( x 0 , y 0 ) is in the same path-connected comp onent of Z 2 \ γ con taining ( z 0 , w 0 ), which is indeed, by deﬁn ition, I nt ( γ ). Hence, the r econstruction pro cedure n ev er ends an d so it is alw a ys p ossible to add p oin ts. It follo ws that we ha v e a con tin uous path starting in ( x 0 , y 0 ) and ending arb itrarily far a w a y . F ourth step: γ = E xt r γ ( I nt ( γ )) γ This is no w clear, from th e p r oof of the previous steps and from the reconstruction princi- ple.  W e conclude this sub section with s ome remarks. 18 V ALERIO CAPRARO Remark 27. (1) It se ems to b e likely that π ( γ ) = Z , π 1 ( E xt ( γ )) = Z and I nt ( γ ) is c o ntr a ctible, i. e . homotopic e quivalent to one p oint. (2) It is likely that some we aker version of the Jor d an curve the or em holds for c ontinu- ous cir cuits, p ossibly not simple, having fundamental gr oup e qual to Z . In fact, it is wel l p ossible that such cu rves sep ar ate the grid Z 2 in a t le ast two p a th-c onne cte d c o mp o nents. 3.2. P .N.Jolissaint-V alett e ’s inequality for ﬁnite metric spaces. In a lecture at Lausanne, Alain V alette present ed a new and in teresting inequalit y for the ℓ p -distortion of a ﬁ nite connected graph , pr ov ed by himself and his co-author Pierre-Nicolas Jolissain t (see [Jo-V a11], Th eorem 1). At th at p oin t one of the au d itors came up w ith the question whether there is some version for general ﬁ n ite metric spaces of that inequalit y . Here we w an t to prop ose su c h a m ore general version. The author thanks Pierre-Nicolas Jolissaint for reading an earlier version of this section and su ggesting many impro v emen ts. Let ( X, d ) b e a ﬁnite metric s pace and let X 1 , . . . X n b e the partition of X in p ath-connecte d comp onen ts. The b asic id ea is clearly to app ly P .N.Jolissain t-V ale tte’s inequalit y on eac h of them, but unfortun ately this app licatio n is not straigh tforw ard, since a p ath-connecte d comp onen t migh t not look like a graph (thin k, f or instance, to the space [ − n, n ] 2 \ { ( 0 , 0) } equipp ed with the metric in d uced by the standard em b edding in to R 2 ). S o w e hav e to b e a bit careful to apply P .N.Jolissain t-V alette’s argumen t. Remark 28. Sinc e the ℓ p -distortion do es not dep end on r esc aling the metric and sinc e we ar e g oing to work on e ach p ath-c onne cte d c o mp o nent sep ar ately, we c an su pp ose that e ach X i is i n normal form. Since we are going to w ork on a ﬁxed path-connected comp onent , let us simplify the notation assuming d irectly that X is ﬁn ite path-connected metric space in normal form. A t the end of this section it w ill b e easy to pu t together all path-connected comp onents. Let x, y ∈ X , x 6 = y and let x 0 x 1 . . . x n − 1 x n b e a con tin uous path joining x and y of minimal length n . Denote by s the ﬂo or of d ( x, y ), i.e. s is the greatest p ositiv e in teger smaller than or equ al d ( x, y ). Notice that s ≥ 1, since X is in n orm al form. Denote by P ( x, y ) the set of co v erings P = { p 1 , . . . , p s } of the s et { 0 , 1 , . . . , n } su ch that • If a ∈ p i and b ∈ p i +1 , then a ≤ b , • the greatest elemen t of p i equals the smallest elemen t of p i +1 . W e denote by p − i and p + i resp ectiv ely the smallest an d the greatest elemen t of p i . No w we int ro duce th e follo wing s et E ( X ) =  ( e − , e + ) ∈ X × X : ∃ x, y ∈ X, p ∈ P ( x, y ) : e − = p − i , e + = p + i  (3) Remark 29. If X = ( V , E ) is a ﬁnite c onne cte d gr ap h e quipp e d with the shortest p ath metric, then E ( X ) = E . Inde e d i n this c ase s = n and so the only c overings b elonging to P ( x, y ) have the shap e p i = { x i − 1 , x i } , wher e the x i ’s ar e taken along a shortest p ath joining x and y . TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 19 W e deﬁne a metric analog of the p -sp ectral gap: for 1 ≤ p < ∞ , we set λ ( p ) 1 = inf ( P e ∈ E ( X ) | f ( e + ) − f ( e − ) | p inf α ∈ R P x ∈ X | f ( x ) − α | p ) (4) where the inﬁmum is take n o v er all functions f ∈ ℓ p ( X ) which are not constan t. Lemma 30. L et ( X, d ) b e a ﬁnite p ath -c onne cte d metric sp a c e in normal form. (1) F or any p e rmutation α ∈ S y m ( X ) and F : X → ℓ p ( N ) , one has X x ∈ X || F ( x ) − F ( α ( x )) || p p ≤ 2 p X x ∈ X || F ( x ) || p p (2) F or any bi-lipschitz e mb e d ding 6 F : X → ℓ p ( N ) , ther e is another bi- lipschitz emb e d- ding G : X → ℓ p ( N ) such that || F || Lip || F − 1 || Lip = || G || Lip || G − 1 || Lip and X x ∈ X || G ( x ) || p p ≤ 1 λ ( p ) 1 X e ∈ E ( X ) || G ( e + ) − G ( e − ) || p p Pr o of. (1) Th is p ro of is absolutely the same as the pro of of Lemma 1 in [Jo-V a11]. (2) Ob serv e that the construction of G made in [Gr-No10] is pu rely algebraic and so w e can apply it. The inequalit y ju st follo ws fr om the deﬁnition of λ ( p ) 1 .  No w we ha v e to p ro v e a v ersion for metric spaces of a lemma already prov e d by Lin ial and Magen for ﬁn ite connected graph (see [Li-Ma00], Claim 3.2). W e n eed to introd uce a n umber th at measures h o w far is the metric space to b e graph. W e set d ( X ) = max e ∈ E ( X ) d ( e − , e + ) (5) W e told that d ( X ) measures ho w far X is far from b eing a graph since it is clear that d ( X ) = 1 if and only if X is a ﬁnite conn ected graph, in the sense that the metric of X is exactly the length of the shortest p ath connected t w o p oints. Ind eed, in one sense, it is trivial: if X is a ﬁnite graph, then E ( X ) = E (b y Remark 29) and d ( X ) = 1, sin ce the 6 F or the conv enience of the reader we recall that a bi-lipschitz embed ding, in this context, is a mapping F : X → ℓ p such that t h ere are constants C 1 , C 2 such that for all x, y ∈ X one has C 1 d ( x, y ) ≤ d p ( F ( x ) , F ( y )) ≤ C 2 d ( x, y ) where d p stands for the ℓ p -distance. It is clear that F is injective and so w e can consider F − 1 : F ( X ) → X . So w e can deﬁ ne || F || Lip = sup x 6 = y d p ( F ( x ) , F ( y )) d ( x, y ) and || F − 1 || Lip = sup x 6 = y d ( x, y ) d p ( F ( x ) , F ( y )) The p roduct || F || Lip || F − 1 || Lip is called distortion of F and denoted by D ist ( F ). 20 V ALERIO CAPRARO space is su pp osed to b e in norm al form. Conv ersely , supp ose that d ( X ) = 1, c ho ose t w o distinct p oin ts x, y ∈ X an d let x 0 x 1 . . . x n − 1 x n b e a contin uous path of min imal length suc h that x 0 = x and x n = y . Of, cours e s ≤ d ( x, y ) ≤ n . So, it su ﬃces to pro v e that s = n . In order to do that, su pp ose that s < n and observe that eve ry p ∈ P ( x, y ) w ould con tain a p i with at least thr ee p oin ts p − i = x i − 1 , x i , x i +1 = p + i (w e supp ose th at they are exactly three, since the general case is similar. Since X is in n ormal form , it follo ws that d ( x i − 1 , x i ) = d ( x i , x i +1 ) = 1. No w, s u pp ose that d ( X ) = 1, it follo ws that also d ( x i − 1 , x i +1 ) = 1 and then the path x 0 . . . x i − 1 x i +1 . . . x n is still a contin uous path connecting x with y , contradict ing the minimalit y of the length of the previous p ath. Lemma 31. L et ( X , d ) b e a ﬁnite p ath-c onne cte d metric sp ac e in normal form and f : X → R . Then max x 6 = y | f ( x ) − f ( y ) | d ( x, y ) ≤ max e ∈ E ( X ) | f ( e + ) − f ( e − ) | ≤ d ( X ) m ax x 6 = y | f ( x ) − f ( y ) | d ( x, y ) (6) Pr o of. Let us pro v e only the ﬁrst inequalit y , since the second w ill b e tr ivial a p osteriori . Let x, y ∈ X w here the maxim um in the left hand side is attained and let x 0 x 1 . . . x n − 1 x n b e a conti nuous path of minimal length connecting x with y . Let p ∈ P ( x, y ) and let k b e an inte ger such that | f ( p + k ) − f ( p − k ) | ≥ | f ( p + i ) − f ( p − i ) | for all i . It f ollo ws | f ( p + k ) − f ( p − k ) | = s | f ( p + k ) − f ( p − k ) | s ≥ P s i =1 | f ( p + i ) − f ( p − i ) | s ≥ No w we use the fact th at the cov e ring p is made exactly by s sets. It follo ws that x and y b elong to the un ion of the p i ’s and w e can use the tr iangle in equ alit y and obtain ≥ | f ( x ) − f ( y ) | s ≥ | f ( x ) − f ( y ) | d ( x, y )  The follo wing r esu lt wa s already prov e d for ﬁnite connected graph in [Jo-V a1 1 ], Theorem 1 and Prop osition 3. Theorem 32. L et ( X , d ) b e a ﬁnite p at h-c onne cte d metric sp ac e in normal f orm. F or al l 1 ≤ p < ∞ , one has c p ( X ) ≥ D ( X ) 2 d ( X ) | X | | E ( X ) | λ ( p ) 1 ! 1 p (7) wher e D ( X ) = max α ∈ S y m ( X ) min x ∈ X d ( x, α ( x )) (8) and TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 21 c p ( X ) = inf  || F || Lip || F − 1 || Lip , F : X → ℓ p ( N ) bi-lipschitz emb e dding  (9) is the ℓ p -distortion of X . Pr o of. Let G b e a bi-lipsc hitz em b edding whic h ve riﬁes the second condition in Lemma 30 and let α b e a p erm utation of X without ﬁxed p oin ts. Let ρ ( α ) = min x ∈ X d ( x, α ( x )). One has 1 || G − 1 || p Lip = min x 6 = y || G ( x ) − G ( y ) || p p d ( x, y ) p ≤ min x ∈ X || G ( x ) − G ( α ( x )) || p p d ( x, α ( x )) p ≤ ≤ 1 ρ ( α ) p min x ∈ X || G ( x ) − G ( α ( x )) || p p ≤ ≤ 1 ρ ( α ) p | X | X x ∈ X || G ( x ) − G ( α ( x )) || p p ≤ No w app ly the ﬁ rst statemen t of Lemma 30: ≤ 2 p ρ ( α ) p | X | X x ∈ X || G ( x ) || p p ≤ No w app ly the s econd statemen t of Lemma 30: ≤ 2 p ρ ( α ) p | X | λ ( p ) 1 X e ∈ E ( X ) || G ( e + ) − G ( e − ) || p p ≤ ≤ 2 p | E ( X ) | ρ ( α ) p | X | λ ( p ) 1 max e ∈ E ( X ) || G ( e + ) − G ( e − ) || p p ≤ No w app ly Lemma 31: ≤ 2 p d ( X ) p | E ( X ) ||| G || p Lip ρ ( α ) p | X | λ ( p ) 1 No w recall th e deﬁnition in Equations 9 and 8 and just re-arrange the terms to get the desired in equ alit y .  Notice that d ( X ) ≥ 1 an d so th e inequalit y gets w orse when the metric space is not a graph. It suﬃces to write down some pictur es to und erstand that it is likel y that one can replace d ( X ) with a smaller constan t. F or the moment we are not in terested in ﬁ nding the b est inequalit y , but just in showing ho w to apply some notions that we ha v e intro- duced in ord er to extend resu lts from the setting of (lo cally) ﬁnite graph s to (locally) ﬁnite metric spaces. Analogously , now w e present a v ersion for general metric sp aces of P .N.Jolissain t-V alette’s in equalit y wh ic h is certainly n ot the b est p ossible, b ecause one can try to mak e it b etter studyin g how the diﬀeren t path-connected comp onents are r elate d among themselves. 22 V ALERIO CAPRARO Corollary 33. L et X b e a metric sp ac e such that ev ery p ath-c onne cte d c omp onent X i is ﬁnite. One has c p ( X ) ≥ su p i D ( X i ) 2 d ( X i ) | X i | | E ( X i ) | λ ( p ) 1 ! 1 p (10) Pr o of. Just observe that if X i is a partition of X th en c p ( X ) ≥ sup i c p ( X i ).  4. NPP-isomorphisms In Sec.2.2 we got very close to a new n otio n of isomorphism b et w een t w o lo cally ﬁ- nite metric spaces: the notion of NPP-lo cal-isomorphism is already a go o d n otion, but it is to o lo c al to captur e global b eh aviors. In this section we introd uce the n otion of NPP-isomorphism, whic h is just a global v ersion of NPP-lo cal isomorp hisms. If we w ere in terested only in lo cally ﬁnite metric spaces, th e deﬁnitions would ha v e b een m uc h easier. In ligh t of the p rop ert y S N, th at seems to b e of in terest for general m etric sp aces, in this section we w ork on general metric spaces. 4.1. The discrete neigh b orho od of a b ounded set. In this ﬁrst subsection we int ro- duce th e notion of discr ete k -neighb orho o d of a b oun ded set, as a generalization of the discrete 1-neigh b orho o d of a singleton, introdu ced in Deﬁnition 5. The basic idea is that w e wa nt a notion able to captur e at the same time information ab out con tin uity and dis- creteness on the neigh b orho o d of a b ounded sub set A of X . W e try do that int ro ducing a notion that gets tr ivial wh en th e metric space n ear A is intuitively con tin uous. Deﬁnition 34. L et P ⊆ [0 , ∞ ) , with 0 ∈ P . A c omp lete c hain in P is a ﬁnite subset p 1 , p 2 , . . . , p n such that (1) p 1 = 0 (2) p i < p j , f or al l i < j (3) if p ∈ P is such that p i ≤ p ≤ p i +1 , then either p = p i , or p = p i +1 . Deﬁnition 35. L et ( X, d ) b e any metric sp ac e, let A b e a b ound e d subset of X and let k b e a nonne gative inte ger. The discr ete k - neighb orho o d of A is the set of elements x ∈ X (r esp. x ∈ X \ A ) such that ther e is a ﬁnite se quenc e x 0 , x 1 , . . . , x l = x in X such that (1) x 0 ∈ A (2) l ≤ k (3) d ( x 0 , x 0 ) < d ( x 0 , x 1 ) < d ( x 0 , x 2 ) < . . . < d ( x 0 , x l ) is a c omp lete chain in P = { d ( y , A ) , y ∈ X } Notice that there are no relations of inclusion b et w een dN k and cN k − 1 (recall th at cN k ( A ) = { x ∈ X : d ( x, A ) ≤ k } . F or instance • T ak e A = (0 , 1) inside X = R equipp ed with the Euclidean metric. Then cN 1 ( A ) = ( − 1 , 2) and dN 2 ( A ) = [0 , 1]. • T ak e A = { 0 } inside X = { 0 , 2 } equipp ed with the Euclidean metric. Then cN 1 ( A ) = { 0 } and dN 2 ( A ) = { 0 , 2 } . TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 23 The ﬁrst example, in particular, s ho ws the meaning of our myste rious sentence : the discr ete b o undary is something that gets trivial when the sp ac e is intuitively c ontinuous . So let us deﬁne a n otion of global conti nuit y of a s p ace wh ic h will b e also usefu l in the sequel to giv e examples of spaces w ith prop erty S N (see Prop osition 55). Deﬁnition 36. The discr ete k - b oundary of a b ounde d subset A of X is dB k ( A ) = dN k ( A ) \ A . Deﬁnition 37. A metric sp ac e ( X, d ) is said to b e c ontinuous if for al l op en and b ound e d subsets A of X and for al l k ∈ N , one has dB k ( A ) = A \ A . 4.2. Boundary and NPP-isomorphisms. T his subsection is dev oted to t wo imp ortant deﬁnitions. Th e ﬁrst is th e deﬁnition of b oun dary that we are intereste d in. Since we wan t to capture at the same time con tin uit y and discreteness, it is natural to use tw o term s , one lo oking at the discrete p art, the other at the con tin uous part. Let us ﬁ x a p iece of notation. Let A ⊆ X and let k ≥ 1 b e an in teger. First of all, set M = max { d ( x, A ) , x ∈ dB k ( A ) } and then α ( k ) =  k , if M = 0; M , if M 6 = 0. Deﬁnition 38. L et k ≥ 1 b e inte ger and A ⊆ X . The k -b oundary of A is the set B k ( A ) = dB k ( A ) ∪ [ 0 ≤ α ≤ α ( k ) − 1 cB α ( A ) wher e α runs over the r e al numb ers (not j u st over the inte gers). Lemma 39. If ( X , d ) is lo c al ly ﬁnite and A i s ﬁnite and non empty, then B k ( A ) = dB k ( A ) Pr o of. By deﬁnition, there is no con tin uous con tribution; n amely , the con tin uous con tri- bution r ed uces to the discrete distribu tion.  Observe that this lemma w ould ha v e b een f alse, without in tro ducing the parameter α ( k ), since otherwise there migh t ha v e b een some extra-con tributions from the discrete p art. The second deﬁnition is th e notion of equ iv alence that is of our in terest. Deﬁnition 40. The metric sp ac e ( X, d X ) i s NP P-emb e ddable (r esp. isomorphic) into (r esp. to) the metric sp a c e ( Y , d Y ) if and only if ther e is an inje ctive (r esp. bije ctive) map f : X → Y such that if (r esp. and only i f ) x 1 , . . . , x n ∈ X give rise to a c ompl ete chain d X ( x 1 , x 1 ) < d X ( x 1 , x 2 ) < . . . < d X ( x 1 , x n ) inside P = { d X ( x 1 , x ) , x ∈ X } , then f ( x 1 ) , . . . f ( x n ) giv e rise to a c omp lete chain d Y ( f ( x 1 ) , f ( x 1 )) < . . . < d Y ( f ( x 1 ) , f ( x n )) inside P = { d Y ( f ( x 1 ) , y ) , y ∈ Y } 24 V ALERIO CAPRARO This condition b ecomes particularly s im p le in the case of lo cally ﬁn ite metric space, thanks to Lemma 39. Prop osition 41. A bije ctive map b etwe en lo c al ly ﬁnite metric sp ac es is a NPP -isomorph isms if and only for al l ﬁnite non-empty subset A ⊆ X , one has f ( dN k ( A )) = dN k ( f ( A )) This p rop osition make s clear wh y this n otion is the global v ersion of NPP -local-isomorphisms. 4.3. Examples of NPP-em b eddings. W e ha v e just introduced another w a y to embed a metric sp ace in to another metric sp ace. In deed, the theory of em b edd in g a (lo cally ﬁnite) metric s pace in to a (p ossibly well un d ersto o d ) metric space has a long and int eresting story o v er the 20th cen tury and still alive . At th e v ery b eginning of the story , p eople were in terested in isometric em b edd in gs, bu t at s ome p oin t it b ecame clear that it is ve ry rare to ﬁnd an isometry b et w een t w o metric spaces of int erest. A t that p oint p eople got in terested in v arious approxima te emb eddings: quasi-isometric em b eddin gs, bi-lipschitz embedd ings, coarse em b eddings, uniform em b eddings and so on. In particular, the theory b ecame v ery p opular after tw o breakthrough pap ers , whic h s h o w ed un exp ected correlations w ith other b r anc hes of Mathematics or, ev en, other s ciences: Linial, L ondon and R ab in o vic h found a link b etw ee n th is theory and some problems in theoretical computer science (see [Li-Lo-Ra95]); Guoliang Y u sh o w ed a link b et w een this theory , geometric group theory and K-theory of C ∗ -algebras (see [Y u00]). The common paradigm of all th ese embedd ings is to forget, more or less b r utely , the lo cal structure of the m etric s pace in order to try to understand the global structur e. As we said at the ve ry b eginning of this pap er, b ecause of our motiv ating problems, we are intereste d in the lo cal s tr ucture of a metric space and this is why w e cannot use one of those em b eddings. Remark 42. L et us observe explicitly that the notion of NPP -emb e dding is diﬀer ent f r om al l other notions use d to study metric sp ac es. • It is cle ar that an isometry i s also an NPP- emb e dding. The c onverse do es not hold: c onsider, inside the r e al plane with the Euclide an metric, the subsp ac e X = (0 , 1) ∪ { ( n, 0) , n ∈ N , n ≥ 2 } . It is cle ar that X is NPP- isomorp hic to Y = { ( n, 0) , n ∈ N , n 6 = 1 } , but ther e is no isometry fr om X to any subsp ac e of Z (simply b e c ause of the irr a tional distanc es). • T ake two p ar al lel c opies of Z inside the Euclide a n plane, for instanc e: X = { (0 , x ) , x ∈ Z } ∪ { (1 , x ) , x ∈ Z } . Then X is quasi-isometric 7 and c o arsely e quivalent 8 to a single c o py of Z . But it is cle ar that ther e ar e no NPP-i somorphisms b etwe en X and a 7 Recall that tw o metric spaces ( X , d X ) and ( Y , d Y ) are called quasi-isometric if t here is a map f : X → Y and tw o constants λ ≥ 1 and C ≥ 0 such that (1) λ − 1 d ( x 1 , x 2 ) − C ≤ d ( f ( x 1 ) , f ( x 2 )) ≤ λd ( x 1 , x 2 ) + C , for all x 1 , x 2 ∈ X (2) d ( y , f ( X )) ≤ C , for all y ∈ Y 8 Let ( X , d X ) an d ( Y , d Y ) b e tw o metric spaces. A map f : X → Y is called coarse embe dding if th ere are tw o n on-decreasing functions ρ ± : R + → R + such that ρ − ( d X ( x, y )) ≤ d Y ( f ( x ) , f ( y )) ≤ ρ + ( d X ( x, y )) TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 25 single c o py of Z . Inde e d, the discr ete 1-b oundary of a singleton in X c on tains thr e e p oints and the discr ete 1-b oundary of a singleton in Z c ontains two p oints . No w we wan t to prop ose a general example of NPP -isomorphism, which generalizes the situation when ( X , d ) is a connected grap h equipp ed w ith the shortest-path distance. Indeed, giv en a metric space, w e can alw a ys repr esen t it as a graph, joining any pair of p oin ts ( x, y ) with an edge lab eled with the num b er d ( x, y ). This soun ds particularly go o d if th e sp ace is a connected graph , b ecause d ( x, y ) ∈ N , bu t in general w e are not allo w ed to do such a constru ction, since the nature of the problem exclude the use of d ( x, y ) (see the p roblems discussed in S ec. 1.3). so the aim of what follo ws is to ﬁnd conditions th at guaran tee th e existence of a canonical NPP-isomorph ism b et w een a lo cally ﬁnite metric space ( X , d ) and a T S-graph 9 whose distances are natural num b ers. The pro of is based on the ﬁ rst intuitiv e pro cedure to mak e a m ap : start fr om a b ase p oin t x 0 ∈ X , lo ok at the n earest p oin ts (i.e., lo ok at dN 1 ( { x 0 } )) and draw a p oin t for eac h of them. W e obtain a symb olic representa tion of some region R 1 of X . No w, increase the knowle dge ab out the sp ace lo oking at dN 2 ( { x 0 } ) and d r a w a p oint for any n ew p oint that we ﬁ n d; and so on. Un fortunately , without some assu mptions, this pro cedure faces some diﬃculties. The simplest assump tions we can ﬁnd sa ys basically that the discrete neigh b orho o ds b ehav e lik e the balls. It is lik ely that one can ﬁn d more general results, but, at the same time, it is likely that in general ther e is no NPP-isomorphism b et w een a lo cally ﬁn ite m etric space an d a TS-graph w hose distances are natur al n umbers. Deﬁnition 43. A lo c al ly ﬁnite metric sp ac e is c al le d gr aph-typ e if (1) x ∈ dB k ( y ) \ dB k − 1 ( y ) ⇔ y ∈ dB k ( x ) \ dB k − 1 ( x ) (2) x ∈ dN h ( y ) and z ∈ dN k ( y ) , imply x ∈ dN h + k ( z ) Theorem 44. Any gr aph -typ e sp ac e X is NPP- isomorp hic to a c anon ic al TS-gr aph , which is c al le d symb olic gr aph of X . Pr o of. Fix a base p oin t 10 x 0 ∈ X and construct a we igh ted graph as follo ws: • F or any p oint y ∈ dB 1 ( x 0 ), dr a w an edge { x, y } lab eled with the num b er 1. • – F or any p oin t x ∈ dB 1 ( x 0 ), consider any y ∈ dB 1 ( x ). I f there is no edge joining x and y , d ra w an edge { x, y } lab eled with the num b er 1. The spaces X and Y are called coarsely equiv alent if th ere are tw o coarse embeddings f : X → Y and g : Y → X and a constan t C ≥ 0 su c h th at (1) d X ( x, g ( f ( x ))) ≤ C , for all x ∈ X (2) d Y ( y , f ( g ( y )) ) ≤ C , for all y ∈ Y 9 A weigh ted graph X = ( V , V 2 ) (t h en any pair of vertex is joined by an edge) is called trav eling salesman graph (TS graph) , if any edge { x, y } is lab eled by a non-negative intege r d ( x, y ), whic h gives rise to a metric on V . 10 It will b e clear at th e en d of the construction th at it do es not dep end on the base p oint (it will follo w by the deﬁn ition of graph-type metric spaces). 26 V ALERIO CAPRARO – F or an y p oin t x ∈ dB 1 ( x 0 ), join x to any p oint y ∈ dB 2 ( x ) and lab el this edge with the n umber 2. • – F or any p oin t x ∈ dB 2 ( x 0 ), consider any y ∈ dB 1 ( x ). I f there is no edge joining x and y , d ra w an edge { x, y } lab eled with the num b er 1. – F or an y p oin t x ∈ dB 2 ( x 0 ), consider any y ∈ dB 2 ( x ). I f there is no edge joining x and y , d ra w an edge { x, y } lab eled with the num b er 2. – F or an y p oin t x ∈ dB 2 ( x 0 ), j oin x to an y p oin t y ∈ dB 3 ( x ), w ith an edge lab eled with th e num b er 3. • and so on. It is clear that this construction p ro duces a graph with v ertex set X . Let us d enote by S y mb ( X ) this graph. W e hav e to c hec k that th e lab els giv e rise to a m etric on S y mb ( X ), that we denote by S y mb ( d ). Obvio usly , this metric is deﬁned as follo ws: S y mb ( d )( x, y ) =  0 , if x = y ; ℓ ( x, y ) , otherw ise. where ℓ ( x, y ) is the lab el of the (unique by constru ction) edge joinin g x and y . The construction implies th at ℓ ( x, y ) is the uniqu e integ er h ≥ 1 su c h that y ∈ dB h ( x ) \ dB h − 1 ( x ). F rom the prop erties in Deﬁnition 43, immediately f ollo ws that the symbolic metric is indeed a metric on the symb olic graph. It remains to pro v e that the identit y map is an NPP-isomorphism b et w een the t w o metrics. L et x 0 , x 1 , . . . , x n ∈ X giving rise to a complete chain d ( x 0 , x 0 ) < d ( x 0 , x 1 ) < . . . in P = { d ( x 0 , x ) , x ∈ X ). It follo ws that x j ∈ dB j ( x 0 ) \ dB j − 1 ( x 0 ), for all j = 1 , . . . , n . Hence S y mb ( d )( x 0 , x j ) = j whic h is a complete c hain in { S y mb ( d )( x 0 , x ) , x ∈ X } . Analo- gously , one gets the con v erse.  Theorem 44 is, in some sense, a starting th eorem. It u ses s ome natural conditions in order to guaran tee the (almost) b est result: to mak e a symb olic map using only intege r n umbers . In Sec. 1.3 w e ha v e already discussed that th er e are basically t wo diﬀerent wa ys to dev elop this result: relax the h yp othesis in ord er to h a v e a symbolic map whic h uses only rational num b ers with a ﬁxed n um b er of digits; strengthen/c hange the h yp othesis in ord er to hav e a ﬁnite-dimensional s ym b olic map. The p oin t that make s the p r oblem sometimes diﬃcult is that, in the real-life problems, one needs b oth th e condition. 5. The isoperimet ric cons t ant of a locall y finite sp a ce No w that w e ha v e a new notion of isomorphism b etw ee n lo cally ﬁ nite m etric s paces, we w an t to construct some inv ariants. W e already ha v e an NPP-in v aria nt; namely the fund a- men tal group, w hic h comes from th e c ontinuous world of manifolds thanks to a pr o c e dur e of discr etization of c ontinuity . I n this section we w an t to int ro duce another inv arian t, which comes from the very discr ete world of c onne cte d gr aphs thanks to a pr o c e dur e of c ontinuiza- tion of discr eteness . The main application of th is in v ariant is to lead to the disco v ery of th e prop erty S N, wh ic h is the ﬁrs t kn o wn pur ely metric prop erty that reduces to amenability for ﬁn itely generated group s. TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 27 5.1. Deﬁnition of the isop erimetric constan t. Let X = ( V , E ) b e an in ﬁnite, lo cally ﬁnite, connected graph. Th ere are m an y diﬀeren t deﬁnitions of isop erimetric constan t. W e will start fr om the one that seems to u s easier to b e generalized. W e will deﬁn e lo cals isop erimetric constan ts ι k and a global isop erimetric constan t ι . W e will observe that the deﬁnition of ι would b e th e same, starting from one of the other deﬁnitions of isop erimetric constan t accessible in literature. The follo wing deﬁnition is p robably due to McMullen (see [McMu89]) and it also app ears in [Be-Sc97], in [Ce-Gr-Ha99] and [El-So05]. Deﬁnition 45. L et X = ( V , E ) b e an inﬁnite, lo c al ly ﬁnite, c onne cte d gr ap h and let V e quipp e d with the shortest path metric. The isop erimetric c o nstant of X is ι 1 ( X ) = inf  | dB 1 ( A ) | | A | , A ⊆ V is ﬁnite and non-empty  (11) Generalizing Deﬁnition 45, Elek and S` os p rop osed in [El-So05] the follo wing Deﬁnition 46. L et X = ( V , E ) b e an inﬁnite, lo c al ly ﬁnite, c onne cte d gr ap h and let V e quipp e d with the s h ortest path metric. F or any inte ger k ≥ 1 , deﬁne the k -isop erimetric c o nstant of X to b e ι k ( X ) = inf  | dB k ( A ) | | A | , A ⊆ V is ﬁnite and non-empty  (12) An observ ation is needed at this p oin t. It is clear th at th ese deﬁn itions we re diﬀerent , since they d id n ot know the discrete b oundary . Indeed they used cN k ( A ) \ A instead of dN k ( A ), which is the same, wh en the graph is connected and the metric is the shortest path metric (for a pro of, u s e Lemm a 59, w h ic h holds also f or lo cally ﬁnite connected graphs). Seen in this w a y , these deﬁn itions do not us e the graph stru cture, but only the metric structure. So, theoretically , w e could hav e u sed them as deﬁn itions ev en in our case. Th e p oin t is that they would not hav e b een an NPP-inv arian t. Since NPP isomorph isms of lo cally ﬁn ite m etric space p reserv es th e discrete b oundary , the tric k is ind eed to us e it. Deﬁnition 47. L et ( X, d ) b e a lo c al ly ﬁnite metric sp ac e. The k -isop erimetric c onstant of X is ι k ( X ) = inf  | dB k ( A ) | | A | , A ⊆ X ﬁnite and non-empty  (13) Notice th at the n otion of k -isop erimetric constant is by d eﬁnition a lo cal notion, since it lo oks at the discrete k -neighborh oo d of the sets. S ometimes, it is b etter to hav e a global notion. Deﬁnition 48. L et ( X, d ) b e a lo c al ly ﬁnite metric sp ac e. The isop erimetric c onstant of X is ι ( X ) = sup k ≥ 0 inf  | dB k ( A ) | | A | , A ⊆ X ﬁnite and non-empty  (14) 28 V ALERIO CAPRARO The follo wing resu lt is n o w obvio us by deﬁnition (and by Pr op osition 41), bu t we state it as a theorem b ecause of the intrinsic imp ortance to ha v e an in v aria nt. Theorem 49. The k -isop erimetric c on stant ι k is an NPP - invariant. Conse quently, also the isop erimetric c onstant ι is an NPP -invariant. Remark 50. A s we told at the b e g i nning of this se ctio n, ther e ar e other notions of isop eri- metric c onstant. Using these other notions, pr ob ably we would have obtaine d other symb olic k -isop erimetric c onstant, but the same glob a l symb olic isop erimetric c onstant. Inde e d al l these deﬁnitions diﬀer fr om e ach other lo c al ly. • The deﬁnition that we have u se d makes a c om p a rison b etwe en A and d 1 ( A ) , namely, in the language of Gr aph The ory, the set of vertic es outside A which ar e c onne cte d by an e d ge to some element of A . • Another deﬁnition, as in [Do84] and [Co-Sa93] , makes u se of the v ertic es inside A . • Ornstein-Wei ss’ deﬁnition makes use b oth of the vertic es b oth inside and outside A (se e [Or-W e87] ) Al l these diﬀer enc es vanish making use of the p ar ameter k . No w, it is we ll-kno wn by a result of Cecc herini-Silb erstein, Grigorc h uk and de la Harp e (for ﬁ nitely generated grou p ) and then by a r esult of E lek and S´ os (for lo cally ﬁnite graphs), that the condition ι ( X ) = 0 is equiv alent to th e notion of amenab ility . The natural question is then: What metric pr op erty c orr esp ond s to have ι ( X ) = 0 , wher e X is a lo c al ly ﬁnite metric sp ac e? This question leads to the ﬁrst kno wn pu rely metric prop erty whic h redu ces to amenability for ﬁnitely generated groups. W e will discuss this pr op ert y , called prop erty SN, in the next section. 5.2. Some observ ation ab out the Mark o v op erator and the Lapla cian. A t this p oin t one is tempted to deﬁne th e Mark o v op erator, the L aplacia n, the Gradien t and so on in suc h a wa y to ha v e some isop erimetric inequalities. Let us recall the d eﬁnition of the Mark o v op erator f or a connected graph of b ounded degree equipp ed with the sh ortest path distance. Let H b e the Hilb ert space of functions h : X → C su c h that P x ∈ X deg ( x ) | h ( x ) | 2 < ∞ . Deﬁnition 51. The Markov op e r ator is the self-adjoint and b ounde d op er ator T on H deﬁne d by ( T h )( x ) = 1 deg ( x ) X y ∈ B 1 ( x ) h ( y ) (15) wher e B 1 ( x ) = { y ∈ X : d ( x, y ) = 1 } . The p oin t is that it is n ot clear h o w to r eplace deg ( x ) and B 1 ( x ), in order to h a v e some isop erimetric inequalities. In deed there is some am biguit y d ue to the follo wing fact: the isop erimetric constan t lo oks, in some sens e, outside the sets. Namely , giv en a ﬁ nite s et F , it looks at the b ehavior of | dB 1 ( F ) | | F | . But the p oin ts in dB 1 ( F ) are not the nearest p oints to some p oint in F , but th ey are the p oints that are nearest to F wh en an observ er sitting on F looks outside F . Now, th e am biguit y should b e clear. In ord er to ha v e a Marko v op erator TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 29 whic h agrees w ith the isop erimetric constant, one should rep lace B 1 ( x ) with some set w hic h tak es in to accoun t also the n earest p oin ts outside ... outside wh at? Our idea is that in this case one should ﬁx a base p oin t x 0 ∈ X , consider the increasing and co v ering sequence of sets { x 0 } ⊆ dB 1 ( { x 0 } ) ⊆ dB 2 ( { x 0 } ) ⊆ . . . and d eﬁne the Mark o v op erator T x 0 , replacing B 1 ( x ) with th e union b etw een dB 1 ( { x } ) and the set of nearest p oin t to x ou tsid e the ﬁrst dB k ( { x 0 } ) con taining x . This pro cedur e seems to ha v e a su rprising philosophical aﬃnity with T essera’s viewp oints tec hnicalit y us ed to deﬁne analogo us notions in any metric space (see [T e08]). Unfortu nately , we cannot adopt word by word T essera’s tec hnique, that use classical b alls, if we wan t to k eep some relation with the isop erimetric constan t, wh ic h use balls in the sens e of discrete b oundary . Ind eed, as obser ved more or less explicitly man y times, the discrete b oundary of a set can b e lo cally very diﬀerent from the classical b oundary deﬁn ed using the balls. In many cases, it is well p ossible that the t w o b oundaries diﬀer v ery little at large scale. In th is case it is exp ected that, at lar ge sc ale , our Marko v op erator, our Laplacian and our gradient should b e almost the same as the ones deﬁ ned by T essera. F or instance, it is philosophically likel y that the large scale S ob olev’s inequalities (see [Co96] and [T e08]) are th e same. This is of course an inte resting ﬁeld of researc h. 6. The Small Ne ighborhood proper ty In this section we constru ct a pur ely metric prop erty (meaning th at the d eﬁnition can b e give n for an y metric s p ace) that corresp onds , for lo cally ﬁnite metric spaces, to hav e isop erimetric constan t equ al to zero. T his p r op ert y will b e called Sm all Neigh b orho o d prop erty (p rop ert y SN). Indeed, the geometric int uition b ehind this prop ert y is that c ertain big sets ha v e ne gligible b oundary . This is the same in tuition used to d escrib e the amenabilit y of a ﬁ nitely generated group, through the celebrated Følner condition (see [F o5 5 ]). W e will see, in fact, that there is a precise corresp ondence b et w een pr op ert y SN and amenability: the t w o are equiv alen t f or Ca yley graph s of ﬁnitely generated groups. Hence, as far as w e kno w, the p rop ert y S N tur ns out to b e th e ﬁrst known p urely metric d escription of amenabilit y , solving, in part 11 , the longstanding question whether or not there is a pur ely metric descrip tion of amenabilit y . 6.1. Deﬁnition of Small Neigh b orho o d prop ert y. As already said, the idea b eh in d the p r op ert y SN is quite intuitiv e: w e w an t th at c erta in su b sets of a metric sp ace ha v e ne gligible b oundary . T rying to formalize this p rop ert y , one faces some trou b les. In deed • It is not clear whic h c ert ain sets w e should consider. Of course not an y . Indeed, whatev er th is p r op ert y is, we w ant that the Euclidean metric on th e real line has this pr op ert y . So we need to exclude those sets, as the set of rational num b ers, that are intrinsic al ly to o big . • W e w an t a notion able to describ e th e amenabilit y of a f.g. group; namely , w e w an t that a f .g. group is amenable if and only if its Cayley graph h as the prop erty S N. No w the C a yley grap h of a f.g. group is alwa ys a lo cally ﬁn ite metric space and then the (top ological) b oundary of a b ounded set is alw a ys empt y , h ence n eglig ible 11 In p art means that it is n ot clear what hap p ens for non ﬁnitely generated groups. 30 V ALERIO CAPRARO in every sense. Th is means that w e cannot adopt the usual deﬁn ition of b oundary , otherwise ev ery (Ca yley graph of a) f .g. group would ha v e the prop er ty S N, against our aim. • A general metric space do es not hav e a n atural measur e to d eﬁne the n otion of negligible set. F ortunately , w e are help ed by the w ork done in the previous sections: w e hav e a n otion of b ound ary able to tak e in accoun t, at th e same time, discrete and con tin uous b ehavio rs; on the other h and, w e in terpret ne g ligibility sa ying that the set is top olo gic a l ly bigger than its k -b oun dary; n amely , th e k -b oundary do es n ot con tain any cop y of the set. Finally , as w e said, we h a v e to lo ok just at those sets whic h are not intrinsicall y to o big; namely , op en and b ounded sets whose complementa ry con tains an op en non-empty set. A t this p oin t we can make the id ea completely formal ve ry easily . Let u s ﬁx some notation • giv en a su bset A of the metric space X and a non n egati ve inte ger k . Let us denote b y F ( A, k ) the set of con tin uous, op en and injectiv e maps f : X → X such that f ( A ) ⊆ B k ( A ). • let F d enote the class of all s u bsets A of the m etric space X whic h are op en, b ounded and suc h that X \ A con tains an op en non-empty set. Deﬁnition 52. A metric sp ac e ( X , d ) is said to have the Sm al l Neighb or ho o d pr op erty if for al l nonne gative inte gers k ther e exists A ∈ F such that F ( A, k ) = ∅ . 6.2. Examples of spaces wit h or wit hout the prop erty SN. In this subsection w e w an t to collect some examples of s paces w ith or without the p rop ert y S N. I t is an o ccasion to introdu ce a new class of sp aces, the pitless spaces. Let x ∈ X and r > 0. Denote b y B ( x, r ) = { y ∈ X : d ( x, y ) < r } and N ( x, r ) = { y ∈ X : d ( x, y ) ≤ r } . It is alw a ys B ( x, r ) ⊆ N ( x, r ), bu t it is well kn o wn that the inclusion could b e d ramatica lly prop er. Th e prop er ty N ( x, r ) = B ( x, r ), for all x ∈ X and r > 0 is kno wn to b e equiv alen t to th e prop erty th at for an y x ∈ X , the m apping y ∈ X → d ( x, y ) has a u nique lo cal prop er minimum at x . T his means that we can mo v e from a p oin t y to a p oint x without going b ack : in some sense, the space has n o pits . Deﬁnition 53. A p it on a metric sp ac e ( X , d ) is a p oint y for which ther e exists another p oint x ∈ X such that the mapping z ∈ X → d ( z , x ) has a lo c al pr op er minimum at y . A metric sp ac e is c al le d p itless if it has no pits. Remark 54. It is cle ar that pitlessness implies c ontinuity (se e Deﬁnition 37). Inde e d, if X is not c ontinuous, let A b e an op en and b ounde d set for which ther e i s a c omplete chain d ( x 0 , A ) < d ( x 1 , A ) inside { d ( x, A ) , x ∈ X } . Consider cN d ( x 1 ,A ) ( A ) ◦ , which c oincides with A and then i ts closur e c annot b e e qual to cN d ( x 1 ,A ) . The c onverse is not true. T o have an example of a c ontinuous sp a c e (also with pr op erty SN) and pits it suﬃc es to c onsider suitable sub se ts of R 2 ; for instanc e, the sp ac e X c onstructe d as fol lows: • let R the r e ctangle deﬁne d by the p oints ( − 1 , 0) , ( − 1 , 2) , (0 , 2) , (0 , 0) . TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 31 • let X b e the interse ction b etwe en R and the epigr aph of a r e gular function f : [ − 1 , 0] → R such that f ( − 1) = f (0) = 0 , f ( x ) > √ 1 − x 2 , in a neighb orho o d of − 1 , f ( x ) < √ 1 − x 2 in a neighb orho o d of 0 and f ( x ) = √ 1 − x 2 only in a single p oint. It is cle ar that X is a c ontinuous sp ac e with pr op erty SN and tips. Prop osition 55. (1) Ev ery pitless sp ac e has the pr op erty SN. (2) The metric sp ac e ( R , d ) , with d ( x, y ) = min { 1 , | x − y |} do es not have the pr op erty SN. Pr o of. (1) As already said, pitlessness is equiv a len t to the prop erty: N ( x, r ) = B ( x, r ), for all x ∈ X and r > 0. This latter p rop ert y clearly implies that cB α ( A ) \ cB α ( A ) ◦ = ∅ , for all α . Usin g con tin uit y (see Deﬁnition 37) we get B k ( A ) = A \ A , that clearly cannot conta in any op en set. (2) Let k = 2 and A ∈ F . The sp ace under consideration is clearly con tin uous, and so dB k ( A ) = A \ A . On the other hand, a d irect computation, s h o ws that [ 0 ≤ α ≤ k − 1 ( cN α ( A ) \ cN α ( A ) ◦ ) = R \ A It f ollo ws that B k ( A ) = R \ A , w hic h con tains op en sets. It is clear that A can b e mapp ed in suc h op en s ets, since the top ology is the same as the Euclidean top ology .  Remark 56. (1) Obse rvi ng that Banach sp a c es ar e pitless, we g e t that any Banach sp ac e has the pr op erty SN. (2) O bserve that ( R , d ) , with d ( x, y ) = min {| x − y | , 1 } , is c ontinuous. So pitlessness is in this c ase a c ruc ial pr op erties, much str onger than c o ntinuity. (3) P r op o sition 55 shows also that the pr op e rty SN is not invariant under quasi-isometries. Inde e d the metric sp ac e ( R , d ) with d ( x, y ) = min { 1 , | x − y |} do es not have the pr o p- erty SN, but it is b ounde d and then quasi- isometric to a singleton, which has the pr op erty SN . 6.3. Corresp ondence b etw een amenability and prop erty SN. As w e told, the idea b ehind the prop erty S N is p h ilosophically the s ame as the idea b ehind th e amenabilit y of a ﬁnitely generated group . So the pur p ose of this section is to clarify this relation. In particular, w e pr ov e that a ﬁn itely generated group is amenable if and only if its Ca yley graph, equipp ed with its natural metric, has the prop erty S N. F or the sak e of completeness, let us recall the deﬁnition of the C a yley graph of a group. Giv en a ﬁ nitely generated group G , with a generating set S that we can su pp ose to b e symmetric an d not con taining the id en tit y , its C a yley graph Γ is deﬁn ed as follo ws: • ev ery elemen t g ∈ G is a v ertex of Γ, • t wo ve rtex g and h are joined by an edge if and only if there is s ∈ S suc h that h = g s . The natur al shortest p ath d istance on Γ is called word metric on G with resp ect to the generating set S and it is denoted by d S . Since S is ﬁn ite, th e metric space ( G, d S ) is 32 V ALERIO CAPRARO lo cally ﬁnite and this is fun damen tal to introd uce the doubling condition in Deﬁnition 57. In the sequel, S w ill b e alw a ys a ﬁ xed symmetric generating subset of G non-con taining the id en tit y . Finally we recall the notion of amenabilit y of a f.g. group through an equiv alent con- dition prop osed by Cecc herini-Silb erstein, Grigorc h uk and de la Harp e (see [Ce-Gr-Ha99], Theorem 32). Deﬁnition 57. A f.g. gr oup G is amenable if and only if the metric sp ac e ( G, d S ) do es not verify the doubling c ondition 12 . Lemma 58. L et A b e a ﬁnite non-empty subset of G , g ∈ G \ A and denote d ( g, A ) = l . F or any l ′ ∈ { 0 , . . . , l } , ther e is g ′ ∈ G such that d ( g ′ , A ) = l ′ Pr o of. Since A is ﬁnite, there is a ∈ A and s 1 , . . . , s l ∈ S suc h that a = s 1 · . . . · s l and we can assu m e that this is already a reduced w ord. It suﬃces to tak e g ′ = as 1 · . . . · s l ′ .  Lemma 59. L et A b e a ﬁnite subset of G . F or al l k ≥ 1 , one has dB k ( A ) = cB k − 1 ( A ) Pr o of. Let x ∈ cN k − 1 ( A ) and a ∈ A realizing the minim um of d ( x, A ). Let us sa y d ( x, a ) = l ≤ k − 1. Using lemma 58 w e can ﬁnd a sequence a = x 0 , x 1 , . . . , x l = x su c h that d ( A, x i ) = i . S ince the metric d attains only in teger v al ues, it follo ws that the c hain d ( x 0 , A ) < . . . < d ( x l , A ) is complete and it has length equal to l + 1 ≤ k . Hence x = x l ∈ dB k ( A ). C onv ersely , let x ∈ dB k ( A ). T h erefore, there are x 0 , x 1 , . . . , x l = x , with l ≤ k − 1, suc h that the chai n d ( x 0 , A ) < d ( x 1 , A ) < . . . < d ( x l , A ) is complete. Completeness and Lemma 58 imply that d ( x i , A ) = i for all i and in p articular d ( x l , A ) = l ≤ k − 1. It follo ws that x = x l ∈ cB k − 1 ( A ).  Theorem 60. A f.g. gr oup G is amenable if and only if the metric sp ac e ( G, d S ) has the pr op erty SN . Pr o of. The pr oof is based on Lemma 59. Indeed, ﬁrst of all it implies that α ( k ) = k and then [ 0 ≤ α ≤ k − 1  cN α ( A ) \ cN α ( A ) ◦  = cB k ( A ) (17) It follo ws, applying Lemma 59, that B k ( A ) = dB k ( A ) ∪ [ 0 ≤ α ≤ k − 1  cN α ( A ) \ cN α ( A ) ◦  = cB k ( A ) (18) No w we can prov e the theorem. L et us su pp ose that G is amenable. Fix k ≥ 1 and let A b e a ﬁnite subset of G such that | cN k ( A ) | < 2 | A | . So th ere is no injective mappin g from A to 12 A locally ﬁnite metric space satisﬁes the doubling condition if there exists a constant k > 0 such that | cN k ( A ) | ≥ 2 | A | (16) for an y non-empty ﬁnite subset A of G . TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 33 cN k ( A ) and, from the observ ati on ab ov e, F ( A, k ) = ∅ . C on v ersely , assume that ( G, d A ) has the p rop ert y SN. Let k ≥ 0 and assume that for all A , one has | N k ( A ) | ≥ 2 | A | . T h erefore there is an injectiv e mapping f : A → N k ( A ) \ A , w hic h is automatica lly con tin uous and op en, since the top ology is discrete. By the initial observ ation w e get f ∈ F ( A, k ), con tradicting th e prop er ty SN.  Remark 61. A longstanding q u estion asks whether ther e is a pur ely metric c o ndition for the amenability of a lo c al ly c omp act gr oup, not ne c essa rily ﬁnitely gener a te d. A p a rt authors’ opinion that amenability, in gener al , is a me asur e-the or etic al notion that b e c omes metric in some p a rticular c ases - namely, i n the lo c al ly ﬁnite c ase - let us observe e xplicitly that the pr op erty SN is not able to c aptur e amenability in gener al . Inde e d, the lo c al ly c o mp a ct amenable gr oup ( R , d ) , with d ( x, y ) = min {| x − y | , 1 } , do es not have the pr op erty SN. L o oking at Pr op osition 55, this happ ens b e c ause we have for c e d the sp ac e to have pits. This is why, it would b e nic e to have a b ett er understanding of the fol lowing que stion Problem 62. Given a lo c al ly c omp act amenable g r oup which is metrizable by a c ontinuous metric. Is i t always p ossible to ﬁnd an e q uivalent pitless metric? Conversely, it is not cle ar whether or not ther e exists a non-amenable metrizable gr oup with the pr op erty SN. 6.4. Corresp ondence b etw een prop ert y SN and the other notions of amenability . In the previous section we hav e pro v ed that the p rop ert y S N is a purely metric prop ert y (meaning that it can b e formula ted for any metric space) whic h generalizes amenabilit y of a ﬁnitely generated group. The idea that such a purely metric condition has to exist is not new and it relies, in fact, in Følner’s c haracterizatio n of amenability [F o55], wh ic h can b e intuitiv ely explained sa ying that a ﬁn itely generated group is amenable if and only if, lo oking at its Ca yley graph, the b ou n dary of arbitrarily b ig set is small. Th ere h a v e b een, indeed, m an y tent ativ e to ﬁnd this purely metric condition, but all of them unsatisfactory , since applicable just to lo cally ﬁ nite m etric sp aces (see [Ce-Gr-Ha99]) or to metric sp aces with c o arse b ounde d ge ometr y (see [Bl-W e92 ]). F rom another p oin t of view, Laczko vich generalized the n otion of p arado xical d ecomposition to any metric space ([La90 ], [La01] and also [De-Si-So95 ]). I t tu r ns out that R is p arado xical and this mak es sense thinking of translation in v arian t σ -add itiv e measures deﬁned on the p o w er set of R , but it do es not tak e in to account the existence of translation in v arian t ﬁn itely add itiv e measures deﬁned on a nic e algebra of sub sets. T h e latter is indeed ho w amenabilit y b ehav e s in the c ontinuous case. In this section we w an t to mak e a comparison among the prop erty S N and the other t w o n otions of amenability that h a v e b een prop osed in the past for metric spaces that are more general than the ones arising as the Ca yley graph of a group. The ﬁ r st deﬁnition is take n from [Ce-Gr-Ha99]. In their pap er, it is present ed in a diﬀeren t w a y , but the reader can easily reconstruct the follo wing d eﬁ nition using their Deﬁn ition 5 and Theorem 32. In particular, their Th eorem 32 giv es man y other equiv alen t deﬁnitions. Deﬁnition 63. A lo c al ly ﬁnite metric sp ac e is said to b e amenable if the doubling c o ndition is not satisﬁe d . 34 V ALERIO CAPRARO The second notion of amenabilit y , prop osed by Block and W ein b erger (see [Bl-W e9 2 ]) is a bit more tec hnical. Deﬁnition 64. A metric sp ac e ( X , d ) is said to have c o arse b ounde d ge ometry if it admits a quasi-lattic e; namely, ther e is Γ ⊆ X such that (1) Ther e exists α > 0 such that cN α (Γ) = X , (2) F or al l r > 0 ther e exists K r > 0 such that, for al l x ∈ X , | Γ ∩ B r ( x ) | ≤ K r (19) wher e B r ( x ) is the op en b al l with r a dius r ab out x . Deﬁnition 65. A metric sp ac e ( X, d ) with c o arse b ounde d ge ometry, given by the quasi- lattic e Γ , is said to b e amenable if for al l r, δ > 0 , ther e exists a ﬁnite subset U of Γ such that | ∂ r U | | U | < δ (20) wher e ∂ r U = { x ∈ Γ : d ( x, U ) < r , d ( x, Γ \ U ) < r } . The natural question is ab out the p ossible logic implications among these tw o notions of amenabilit y and the p rop ert y S N. T h e follo wing d iscussion give s an almost complete summary of the situation. Let u s start from comparing Deﬁnitions 63 and 65. The m ysterious Remark 42 in [Ce-Gr-Ha99] conta ins some ideas that w e w an t to make formal. First of all, the fact that there are lo cally ﬁnite metric spaces that do not ha v e coarse b ounded geometry 13 and, con v ersely , there are spaces with b ounded geometry that are n ot locally ﬁnite, forces to consider th e problem in the restricted class of lo cally ﬁn ite spaces with coarse b ounded geometry . W e ha v e the follo wing r esult: Prop osition 66. L et ( X , d ) b e a lo c al ly ﬁnite metric sp ac e with c o arse b ounde d ge om etry. It is amenable i n the sense of Deﬁnition 63 if and only if i t is amenable in the sense of Deﬁnition 65. Pr o of. Let Γ ⊆ X b e a quasi-lattice. By deﬁn ition of quasi-lattice, X is quasi-isometric to Γ. Since amenabilit y in the sense of Deﬁnition 63 is in v arian t un der quasi-isometries (see [Ce-Gr-Ha99], P rop osition 38), w e can r estrict our attenti on on Γ. The r esult f ollo ws by the obs erv ation that the Følner condition on Γ in E q u ation 20 is equiv alen t to the negation of the doubling condition on Γ in Equation 16.  Ab out the relation b etw een Prop ert y SN and amenabilit y in the sense of Deﬁnition 63 w e can give the follo wing Prop osition 67. If ( X , d ) c omes fr om a lo c al ly ﬁnite c onne cte d gr ap h, then it has the pr op erty SN if and only if it i s amenability in the sense of Deﬁnition 63. 13 F or instance, attach ov er any natural num b er n a set A n with n elemen ts and deﬁne the metric to b e 1 b etw een p oints in the same A n and n + m b etw een a p oin t in A n and one in A m , with n 6 = m . TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 35 Pr o of. It suﬃces to observ e that Lemma 59 holds in this case and th at the pr o of of T heorem 60 do es n ot really d ep end on the fact that X is th e Cayley graph of a group .  Remark 68. In gener al , it is cle ar that ther e ar e lo c al ly ﬁnite metric sp ac es with pr o p erty SN which ar e non amenable in the sense of Deﬁnition 63. A simple instanc e is given by the sp ac e X c onstructe d as fol lows: let F 2 = < a, b > b e the fr e e gr o up on two gene r ators and c onsider the usu al Cayley g r aph with r esp e ct to the gener ating set { a, a − 1 , b, b − 1 } . Now get rid of al l b al ls of r ad ius which is not a p ow er of 2 . The sp ac e obtaine d is amenable i n the sense of De ﬁnition 63, but i t do es not have the pr op erty SN. On the other hand, i t is not cle a r at the moment if the other implic ation c an b e true: ar e ther e non amenable lo c al ly ﬁnite sp a c e with the pr op erty SN? Ab out th e relation b etw een pr op ert y SN and amenabilit y in the s ense of Deﬁnition 65 w e can sa y right no w that the t w o are, in general, distin ct prop erties. In deed the space ( R , d ) w ith d ( x, y ) = min { 1 , | x − y |} do es n ot ha v e the pr op ert y S N, but it is amenable in the s en se of Blo c k and W ein b erger, taking Γ = { 0 } . On the other han d , if w e consid er the additional hyp othesis that the space is lo cally ﬁnite, then Deﬁn itions 63 an d 65 are basically th e same, by Prop osition 66, and s o Prop osition 67 and Remark 68 apply . 6.5. Description of prop erty SN using the sym b olic isop erimetric constant. In this ﬁnal subsection, we giv e the pr omised c haracterization of pr op ert y S N in terms of the isop erimetric constant. This c haracterizat ion is the equiv alent of resu lts already pro ve d for connected lo cally ﬁ n ite graphs by Cecc herini-Silb erstein, Grigorc h uk and d e la Harp e (see [Ce-Gr-Ha99], Th eorem 51) and by Elek and S ´ os (see [El-So05], Prop osition 4.3). Theorem 69. A lo c al ly ﬁnite metric sp a c e ( X , d ) has the pr op erty SN if and only if ι ( X ) = 0 . Pr o of. ι ( X ) > 0 if and only if there is k such that inf  | dB k ( A ) | | A | , A ⊆ X ﬁnite and non-empty  > 0 (21) It is no w a standard argumen t to p ro v e that th ere is (another) k suc h that | dB k ( A ) | ≥ | A | for all A ⊆ X ﬁnite and n on-empt y (see f or ins tance [Ce-Gr-Ha99 ], just b elo w Deﬁnition 30). No w, since X is lo cally ﬁn ite, B k ( A ) has no cont inuous part, namely B k ( A ) = dB k ( A ) (see L emma 39). It follo ws th at ι ( X ) > 0 if and only if there is a p ositiv e integ er k such that | B k ( A ) | ≥ | A | for all A ⊆ X and this is the negation of the prop erty SN, as in the pro of of T heorem 60.  F rom this result we can d educe a ﬁ rst application of the NPP-inv ariance of the isop eri- metric constant. Corollary 70. F or lo c al ly ﬁnite metric sp ac es, the pr op erty SN is invariant under NPP- isomorph isms. 36 V ALERIO CAPRARO Observe that amenability in the sense of Deﬁn itions 63 and 65 is not inv arian t und er NPP-isomorphisms. This do es not mean that our pr op ert y SN is b etter, as a generalizatio n of amenabilit y , than the others, bu t jus t that those generalizatio ns d o not catc h the essen ce of the problems that we are considering and then w e n eed something else. 7. The zoom isoperimetric const ants In this section we w an t to in tro duce another NPP -in v ariant, wh ic h is, in some sen s e, more p recise than the isop erimetric constant . In fact, the isop erimetric constan t lo oks at the b est b eha vior inside the m etric space and so the information enco ded in it could b e v ery far from w hat a lo cal observ er really sees. In ord er to mak e this obscure sente nce clearer, let u s consider the f ollo wing example. Con s ider a regular tree of degree d ≥ 3 and attac h ov e r a v ertex a copy of the in tegers. W e get a connected graph of b ounded d egree with isop erimetric constan t equal to zero. Th is graph is then amenable in ev ery sense: it has prop erty S N and it is amenable in the sense of Deﬁnitions 63 and 65. This h app ens b ecause the isop erimetric constan t forgets wh at happ ens in the non-amenable p art. On the other hand, consid er an observer living in the v ertex x . His/her w a y to observ e the unive rse w here (s)h e liv es is to incr e ase his/her know le dge step by step , considering at the ﬁrst step th e set A 0 = { x } , then A 1 = dN 1 ( { x } ) and so on: his/her p oin t of view on the u niv erse is giv en b y the sequence A n = dN n ( { x } ). I t is clear that, wherev er x is, the sequence A n is even tually doubling, in the sense that | A n | ≥ 2 | A n − 1 | . So, the isop erimetric constan t s ays that the un iv erse is amenable, but ev ery local observ er would sa y that the unive rse is non-amenable. This lac k of agreemen t is the reason why in this section w e try to introdu ce a new in v arian t wh ic h tak es in to account all these lo cal observ ations. By the w a y , th is inv arian t turn s out to b e new ev en for lo cally ﬁnite graphs. 7.1. Deﬁnition of zo om isop erimetric constan ts. Let us ﬁx some n otatio n. Let ( X, d ) b e a lo cally ﬁnite metric space, x ∈ X and k , n b e tw o p ositiv e in tegers. W e deﬁn e ζ k ( x ) = inf  | dN nk ( x ) | | dN ( n − 1) k ( x ) | , n ≥ 1  (22) Then deﬁn e ζ ( x ) = sup { ζ k ( x ) , k ≥ 1 } (23) Deﬁnition 71. The upp er zo om isop erimetric c o nstant of X is ζ + ( X ) = su p { ζ ( x ) , x ∈ X } (24) The low er zo om isop erimetric c onstant of X is ζ − ( X ) = inf { ζ ( x ) , x ∈ X } (25) Also in this case, the follo wing result is obvio us by construction and Prop osition 41. Theorem 72. The zo om i sop erimetric c onstants ar e invariant u nder NP P-isomorphism s. TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 37 Let us introd uce the parameter λ ( X ) = ζ + ( X ) − ζ − ( X ). I n our op in ion, the case λ ( X ) = 0 is particularly interesting, since it describ es the s ituation when the observ ati ons made by t w o arbitrary observ ers agree. Unfortunately , it can happ en that λ ( X ) 6 = 0, as sho wn in the next example (wh ich is a v ariatio n of an example suggested by Ily a Bogdano v). In the next subsection we will sho w a seemingly in teresting example when λ ( X ) = 0. Example 73. Consider a binary tr e e with a r o ot v ertex a , that is, a has de g r e e 1 and e ach other vertex has de gr e e 3 . F or some inte ger of the shap e n 1 ! lar ge enough, delete some vertic es (and the subtr e e hanging fr om them) fr o m the n 1 ! -th layer in such a way that | dB n 1 ! − 1 ( { a } ) | | dN n 1 ! − 2 ( { a } ) | ≈ 1 1000 and observe that | dB n 1 ! ( { a } ) | | dN n 1 ! − 1 ( { a } ) | ≥ 1 800 Now, for an inte ger of the shap e n 2 ! much lar ger than n 1 ! , delete vertic es fr om the n 2 ! -layer in the same manner, and so on. F or any k ∈ N , we get | dB n k ! − 1 ( { a } ) | | dN n k ! − 2 ( { a } ) | ≈ 1 1000 (26) and | dB n k ! ( { a } ) | | dN n k ! − 1 ( { a } ) | ≥ 1 800 (27) Final ly, g lu e two c opies of this gr aph at the ve rtex a . Now, fr om the formula in 26, we get ζ k ( a ) ≤ 1 + 1 1000 , for al l k (sinc e the se quenc e of the multiple of k eventual ly interse cts the se quenc e n k ! ). On the other hand, if b is one of the neig hb ors of a , the formula in 27 gives ζ k ( b ) > 1 + 1 1000 . 7.2. Lo cally amenable metric spaces. It is clear that 0 ≤ ι ( X ) ≤ ζ − ( X ) − 1 ≤ ζ + ( X ) − 1 and then the condition ζ + ( X ) = 1, implies the pr op ert y SN. On the other hand, th e condition ζ + ( X ) = 1, m eans ζ ( x ) = 1, for all x ∈ X and this is qu ite a strong condition. It means that the space is seen to b e amenable by any lo cal observ er. Deﬁnition 74. A lo c al ly ﬁnite metric sp ac e is c al le d lo c al ly amenable if ζ + ( X ) = 1 . The follo wing result is basic, b ut it has a nice in terpretation: in some cases, it suﬃces that an observer sees the s p ace amenable, to conclude that the s pace is amenable for any observ er. Theorem 75. L et ( X, d ) b e a gr aph-typ e lo c al ly ﬁnite metric sp ac e. Supp ose ther e e xists x ∈ X such that ζ ( x ) = 1 , then ζ + ( X ) = 1 (and, obviously, λ ( X ) = 0 ) Pr o of. Let y ∈ X , w e h a v e to pro v e that also ζ ( y ) = 1. L et k b e ﬁ xed and let n 0 suc h that x = A h 0 ( { x } ) ∈ A k n 0 − 1 ( y ) (notice that h is s till undetermined, since A h 0 ( { x } ) = x holds 38 V ALERIO CAPRARO for ev ery h ). Let h suc h that A h 1 ( { x } ) ⊇ A k n 0 ( y ). No w, since X is of graph -t yp e, then th e discrete n eighb orh o o d s b eha v e like the b alls. It follo ws that for all m ∈ N , one has  A h m − 1 ( { x } ) ⊆ A k n 0 + m − 1 ( y ) A h m ( { x } ) ⊇ A k n 0 + m ( y ) whic h clearly imp lies that | A k n 0 + m ( y ) | | A k n 0 + m − 1 ( y ) | ≤ | A h m ( x ) | | A h m − 1 ( x ) | (28) No w, the left hand side is ≥ 1 and, by hypothesis, the right hand side has inﬁmum equal to 1, indep endently on h . It follo ws that also the left hand side has inﬁm um equal to 1, as required.  No w w e wan t giv e a ﬁ rst pr op osition ab out the relation b etw ee n the gro wth rate of a ﬁnitely generated group and the lo cal amenabilit y of its Ca yley graph . Let G b e a ﬁn itely generated group, ﬁx a ﬁ n ite symm etric generating set S and consid er th e lo cally ﬁn ite metric space ( G, d S ). Denote b y B ( n ) the ball of radius n ab ou t the identit y . W e recall the f ollo wing deﬁn ition: • A group is said to hav e p olynomial gro wth if there is a p ositiv e integ er k 0 suc h that | B ( n ) | ≤ n k 0 , for all n . • A group is said to hav e exp onential gro wth if there is a > 1, su c h that | B ( n ) | ≥ a n , for all n . A group wh ic h d oes not hav e exp onen tial growth is said to h a v e sub-exp onentia l growth. It is w ell-kno wn that th ere is n o p erfect corresp ondence b et w een amenabilit y and gro wth rate. Indeed, groups of su b-exp onen tial growth are amenable, but there are amenable groups generated by m elemen t wh ose gro wth appr oac h the one of the fr ee grou p on m generators (see [Ar-Gu-Gu05 ]). Here w e wan t to try to obtain a corresp onden ce b et w een gro wth rate and lo cal amenabilit y . Prop osition 76. (1) If G has p olyno mial gr owth, then ( G, d S ) is lo c al ly amenable. (2) If ( G, d S ) is lo c al ly amenable, then G has sub- e xp onential gr o wth. Pr o of. If G has p olynomial growth, then there exists a p ositive int eger k 0 suc h that B ( n ) ∼ n k 0 (this b asicall y follo ws by Gromo v’s theorem 14 ). Then, for all p ositiv e inte ger k , one has | B (( n + 1) k ) | | B ( n k ) | ∼ (( n + 1) k ) k 0 ( nk ) k 0 → 1 hence ζ ( e ) = 1. By Th eorem 72, it follo ws that ζ ( G ) = 1, hence G is lo cally amenable. 14 Gromo v’s theorem[Gr81] states that a group of polynomial growth contains a nilp otent group of ﬁ nite index and then , a p osteriori, the gro wth rate of the original group is not just b ounded ab ov e by a p olynomial, but it is exactly p olynomial. TOPOLOGY ON LO CALL Y FINI TE METRIC SP ACES 39 Supp ose that ( G, d S ) is lo cally amenable and sup p ose, by cont radiction, that it has ex- p onen tial gro wth. Then the limit of the sequence n p | B ( n ) | , wh ic h alw a ys exists (see, for instance, [Ha00], VI.C, Prop osition 56), is equal to some a > 1 (this is just another w a y to deﬁne groups with sub-exp onential growth. S ee, for ins tance, the in tro duction of [Ar-Gu-Gu05]). It f ollo ws that | B ( n ) | ∼ a n and then inf  | B ( n + 1) | | B ( n ) | , n ∈ N  > 1 (29) It f ollo ws that ζ ( e ) > 1 and, by Theorem 72, getting the con tradiction th at ( G, d S ) is not lo cally amenable.  Problem 77. Do e s ther e exist a ﬁnitely gener ate d gr oup with interme dia te gr o wth such that ( G, d S ) is lo c al ly amenable? 7.3. Expanders and geometric prop ert y ( T). In this subsection w e w an t to describ e another connection b et w een our theorizatio n and one of the recen t ﬁelds of researc h in Pure and Applied Mathematics, wh ic h is th e theory of expand ers and the geometric pr op ert y (T). In the previous section, w e ha v e prov ed that for grap h -t yp e m etric s paces, the lo cal condition ζ ( x ) = 1, for a particular p oint x , is actually a global condition, holding for an y p oin t. Con v ersely , also the lo cal condition ζ ( x ) > 1 is actually global. W e can see this in an other wa y: start from the p oint x and construct a sequence of graph G 0 = { x } , G 1 = dN 1 ( { x } ) and so on. If ζ ( x ) > 1, th en, apart the hyp othesis on the degree, we obtain an expander (see [Lu11] f or an in tro duction to expanders fr om b oth p ure and app lied p oin t of view). S ince ζ ( y ) > 1 for all y , it means that w e can sa y that the sp ace is expander, without am biguit y . Analog ously , w e think that w e can also generalize Willett- Y u’s geometric prop erty (T) to an y graph-t yp e metric spaces (for the deﬁnition of geometric prop erty (T), see [Wi-Y u10b]. F or basics and motiv a tion, see [Wi-Y u10a] and [Oy-Y u09]). In ord er to d o that, we n eed a go o d d eﬁnition of the Laplacian. Find such a deﬁn ition is probably the most imp ortan t basic pr oblem that we left op en and certainly is one of th e purp ose of the s econd article on this sub ject. 7.4. Indep endence b et w een ι ( X ) and ζ − ( X ) . Un til no w we h a v e introd uced tw o isop eri- metric constants for lo cally ﬁnite metric spaces. The relation ι ( X ) ≤ ζ − ( X ) − 1, says that the t w o inv arian ts are not completely indep endent, meaning th at when ζ − ( X ) = 1, we can deduce the v alue of ι ( X ). In th is section, w e wa nt to observ e explicitly th at th is is basically the uniqu e case: in general, there is no w a y to d educe a b ound for the v alue of ζ − ( X ), kno wing ι ( X ). It seems that ther e is no n otion of indep en d ence b et we en inv arian ts in literature. So let us p rop ose a deﬁn ition that lo oks quite n atural and easy to v erify . Deﬁnition 78. L et I 1 , I 2 b e two r e al-value d invariants (of some c ate gory). We say that I 2 is upp er indep endent on I 1 if ther e is no r e al value d fu nc tion f 2 such that I 2 ( X ) ≤ f 2 ( I 1 ( X )) (30) 40 V ALERIO CAPRARO for al l obje cts X in the c ate gory. Ana lo gously one c a n deﬁne lower indep endenc e and c ompletely indep endenc e. Theorem 79. ζ − is u pp er indep endent on ι . Pr o of. Consider a regular tree of degree n and attac h a cop y of Z o v er a v ertex. W e obtain an inﬁn ite, lo cally ﬁnite, connected graph. Let ( X, d ) b e this graph equipp ed with the shortest path metric. In this case ι ( X ) = 0, ind ep enden tly on n , sin ce it is enough to tak e ﬁnite sets concen trated in th e cop y of Z . On the other hand it is clear that ζ − ( X ) can b e made arbitrarily big with n .  Referen ces [ADFQ96] Ayala R., Dom ´ ınguez E., F ranc ´ es A.R., Qu in tero A. Determining the c omp one nts of the c om- plement of a digi tal ( n − 1) -manifold in Z n , Discrete Geometry for Computer Imagery; Lecture N otes on Computer S cience, vol. 11 76/1996 (1996), 163-176. [ADFQ00] Ayala R., D om ´ ınguez E., F ranc´ es A.R., Q uin tero A. Homotopy in digital sp ac es , Lecture notes in Computer S cience 1953 (2000), 3-14. 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