Twisted conjugacy classes in R. Thompsons group F

TWISTED CONJUGA CY CLASSES IN R. THOMPSON’S GR OUP F COLLIN BLEAK, ALEXANDER FEL’SHTYN, A ND DA CIBER G L. GON C ¸ AL VES Abstra ct. In this short article, w e prov e that an y automorphism of the R. Thompson’s group F h as infinitely many twisted conjugacy classes. The result foll o ws from the work of Brin [2], together with a standard facts on R. Thompson’s group F , and elemen tary properties of the Rei- demeister num bers. 1. Introduction Let φ : G → G b e an automorphism of a group G . A class of equiv alence x ∼ g xφ ( g − 1 ) is called the R eidemeister class or φ - c onjugacy class or twiste d c onjugacy class of φ . The num b er R ( φ ) of Reidemeister classes is called the R eidemeister numb er of φ . The in terest in t wisted conjugacy relations has its origins, in particular, in the Nielsen-Reidemeister fixed p oint theory (see, e.g. [16, 4]), in Selb er g theory (see, eg. [19, 1]), and Algebraic Geometry (see, e.g. [15]). A curr ent imp ortan t problem of the fi eld concerns obtaining a t wisted analogue of the celebrated Bur nside-F rob eniu s theorem [7, 4, 10, 11, 23, 9 , 8 ], that is, to sho w the coincidence of the Reidemeiste r num b er of φ and the n um b er of fixed p oin ts of the in d uced homeomorphism of an appropriate dual ob ject. One step in th is p ro cess is to describ e th e class of groups G , suc h that R ( φ ) = ∞ f or an y automorphism φ : G → G . The wo rk of disco ve ring whic h g roups b elong to the m entioned class of groups was b egun by F el’sht yn and Hill in [7 ]. Later, it w as s h o w n by v arious authors that the follo wing groups b elong to th is class: (1) non-element ary Gromo v h yp erb olic groups [5, 18], (2) Baums lag-Solit ar groups B S ( m, n ) = Date : Octob er 29, 2018. 1991 Mathematics Subj e ct Cl assific ation. 20E45;3 7C25; 55M20. Key wor ds and phr ases. Reidemeister num b er, twisted conjugacy classes, R. Thomp- son’s group F , R ∞ prop erty . 1 2 COLLIN BLEAK, ALEXANDE R FEL’SHTYN, AND D ACIBER G L. GONC ¸ AL VES h a, b | ba m b − 1 = a n i except for B S (1 , 1) [6], (3) generalized Baumslag-Solitar groups, that is, finitely generated groups which act on a tr ee with all edge and vertex stabilizers infinite cyclic [17], (4) lamplighter groups Z n ≀ Z iff 2 | n or 3 | n [14], (5) the solv ab le generaliza tion Γ of B S (1 , n ) give n b y the short exact sequence 1 → Z [ 1 n ] → Γ → Z k → 1 as w ell as any group quasi- isometric to Γ [21], grou p s which are quasi-isometric to B S (1 , n ) [20] (while this prop erty is not a quasi-isometry in v arian t), (6) saturated weakly b ranc h groups (including the Grigorch uk group and the Gupta-Sidki group) [12]. The p ap er [21] su ggests a terminology for this prop ert y , w hic h we w ould lik e to follo w. Namely , a group G has pr op erty R ∞ if all of its automor- phisms φ ha v e R ( φ ) = ∞ . F or the immediate consequences of the R ∞ prop erty in topological fixed p oint theory see, e.g., [20]. In the present note w e pro v e that R. Thompson’s group F has the R ∞ prop erty . W e do n ot know if this is the case for injectiv e homomorph isms. The R. T hompson g roup F is a finitely-presen ted group whic h has exp o- nen tial growth, do es not con tain a fr ee nonab elian subgroup; F is not a residually finite group and is not an elementa ry amenable group [3]. It is still unkn o wn wh ether or not F is amenable. Ric h ard Th ompson introduced th e groups F ≤ T ≤ V [22] in connection with his studies in log ic. Thompson’s groups hav e sin ce app eared in a v ariet y of m athematica l topics: the w orld problem, infinite s im p le group s, homotop y and shap e th eory , group cohomology , dyn amical systems and analysis. W e are in terested n ot only in Thompson’s group F , but also in the groups T and V , and the generaliza tions F n ≤ T n ≤ V n (Note that Higman introdu ces th e groups V n and solves the conjugacy problem f or th ese group s in [25], while Bro w n carries this generalization out f or T n and V n in [24]). Our wo rk here only discusses the group F . The r esu lts of the present pap er demonstr ate that the furth er study of Reidemeister theory for R. Th omp son’s group F has to go along the lines sp ecific for the infi nite case. On the other hand, this result shrin ks the class of groups for whic h the t wisted Burn side-F rob enius conjecture [7, 10, 11, 23, 9, 8] has ye t to b e verified. W e wo uld lik e to complete the introduction with the follo win g question TWISTED CONJUGACY CLASSES IN R . THOMPSON’S GROUP F 3 Question 1.1. Do the generalised T hompson’s groups F n ha v e the R ∞ prop erty? Ac knowledgmen ts : Th e first author wo uld lik e to thank Professor Brin for helpful conv ersations and discu ssion of h is w ork in [2], up on which this note strongly relies. The second author would lik e to thank V. Guba, R. Grigorc huk, M. K ap o vic h and M. Sapir, for stim ulating discussions and commen ts. 2. R. Thompson’s group F : de f initions and st andard f a cts W e will b e w orkin g with R. Thomps on’s grou p F . F ollo wing the definition in [2], F consists of a restricted class of h omeomorphisms of the r eal line under the op eration of comp osition. A h omeomorph ism α : R → R is in F if and only if α (1) is piecewise-linea r (admitting fin itely man y breaks in slop e), (2) is orien tation-preserving, (3) has all slop es of affine p ortions of its graph in the set { 2 k | k ∈ Z } , (4) has all breaks in slop e o ccurin g o v er the dyadic rationals Z [1 / 2], (5) maps Z [1 / 2] in to Z [1 / 2], and (6) has fi rst and last affine compon ents equal to pure translations b y (p oten tially d istinct) in tegers α l and α r . Note that the fifth cond ition is actually equiv alent to sa yin g that α maps Z [1 / 2] bijectiv ely onto Z [1 / 2] in an ord er preserving fashion, giv en the other axioms. Ev ery normal subgroup of F con tains the comm utator subgroup F ′ = [ F , F ] of F (see Theorem 4.3 in the standard introdu ctory su rv ey [3] on F ). F urther, in our realization of F , the comm u tator sub group F ′ of F consists of those elements of F whic h are the iden tit y function near ± ∞ . In particular, there is the stand ard on to h omomorphism Ab : F → Z × Z , with k ernel F ′ , w hic h is defined b y the rule Ab ( f ) = ( f l , f r ) (this is Theorem 4.1 of [3]). Here, f l is the translational part of f near −∞ , a nd f r is the translational part of f n ear ∞ , as indicated by the similar notation in the definition ab ov e for F . In fact, giv en any k ∈ F , let us fix the notation k l 4 COLLIN BLEAK, ALEXANDE R FEL’SHTYN, AND D ACIBER G L. GONC ¸ AL VES and k r as the translational parts of k (near −∞ and ∞ resp ectiv ely) for the rest of this sh ort pap er. Finally , we will need a deep r esult of Matthew Brin from [2]. T o state this resu lt, let us first define Rev : F → F to b e the a utomorphism pro- duced by conju gating an elemen t of F b y th e real homeomorphism x 7→ − x . W e also need to d efine the set of eventual ly T - lik e piecewise linear self- homeomorphisms (admitting an infin ite discrete set of “breaks” in the do- main, where the fi rst deriv ativ e is n ot defined) of R as follo ws. A homeomorphism α : R → R is eventual ly T - like if and only if α (1) is piecewise-linea r (admitting a discrete (p ossib ly infinite) collection of breaks in slop e), (2) is orien tation-preserving, (3) has all slop es of affine p ortions of its graph in the set { 2 k | k ∈ Z } , (4) has all breaks in slop e o ccurin g o v er the dyadic rationals Z [1 / 2], (5) maps Z [1 / 2] in to Z [1 / 2], and (6) has minimal domain v alue R α ≥ 0 and m aximal domain v alue L α ≤ 0 so that α ( x + 1) = α ( x ) + 1 for all x ∈ R \ ( L α − 1 , R α ). The final condition says that to the far left and the far r igh t, any suc h α (whic h could b e in F in ev ery other w a y) pro j ects to circl e maps, as in those lo cations it satisfies the p erio dicit y equation α ( x + 1) = α ( x ) + 1. I f α satisfied the p erio dicit y equation across all of R , th en it would represent a lift of an eleme nt of R. Thompson’s group T . Note that elemen ts of F are ev entually T -lik e. F or any even tually T -lik e elemen t α , we will use the notation L α and R α as in the ab o ve definition. In [2 ], Brin sh ows that the inner automorph ism of H omeo ( R ) defin ed b y conjugating elemen ts of H omeo ( R ) b y an y sp ecific ev en tually T -lik e elemen t restricts to an automorph ism of F . W e will call such an automorph ism of F an eventual ly T -like c onjugation of F . The follo win g is a restatemen t of a p ortion of Th eorem 1 in [2] us in g our language. Theorem 2.1 . (Brin) The automorphism gr oup of F is gener ate d by Rev and the ev entual ly T -like c onjugations of F . W e w ill measure the impact of an automorphism in Z × Z , the ab elian- ization of F . In ord er to do th is, w e will us e th e f ollo wing lemma. TWISTED CONJUGACY CLASSES IN R . THOMPSON’S GROUP F 5 Lemma 2.2. L et f ∈ F , and let g : R → R b e eventual ly T -like. If k = f g , then f l = k l and f r = k r . Th at is, eventual ly T -like c onjugation do es not change the tr ansla tional p arts of f ne ar plus or minu s i nfinity. Pr o of. Let f ∈ F , and g : R → R b e an eve nt ually T -like homeomorph ism of R . Let R g and L g b e the lo cations so that for all x ∈ R \ ( L g − 1 , R g ) w e ha v e that g ( x + 1) = g ( x ) + 1. Define v ariation functions V ar f , V ar g : R → R by the rules V ar f ( x ) = f ( x ) − x , V ar g ( x ) = g ( x ) − x . Define V R = V ar ( R g ). F or x > R g , we ha ve that | V ar g ( x ) − V R | < 1. T h at is, on a global scale, homeomorphisms satis- fying the p erio dicity equation are close to p ure translations. In particular, the graph of g is not only p erio dic near ∞ , it is also alw ays within a distance of 1 to the pu re translation b y the constan t V R . Note also that f or x > R f , V ar f ( x ) = f r . Let M = | R g | + | R f | + | V R | + | f r | + 1, and sup p ose x ∈ [ M , ∞ ). W e see that the follo wing equations are true. g ( f ( g − 1 ( x )) = g ( g − 1 ( x ) + f r ) = g ( g − 1 ( x )) + f r = x + f r = f ( x ) (In the thir d to last equalit y , if f r is negativ e, w e are actually using the fact that g ( y ) = g ( y + 1) − 1, w here y > R g .) The argument near min us infinity is similar.  Let H 1 = H Gp 1 b e the first in tegral h omology functor from group s to ab elian groups. The ab o v e work pro duces this corollary . Corollary 2.3. If φ : F → F is an eventual ly T -like automorph ism of F then the induc e d automorphism on the ab elianization H 1 ( F ) ∼ = Z × Z of F is the i dentity. In p articular, if H 1 ( Rev ) is the in duced h omomorphism on the ab elian- ization H 1 ( F ) of F we h a ve the follo wing. Corollary 2.4. The image of Aut ( F ) in Aut ( H 1 ( F ) = Z × Z ) is a cyclic gr oup Z 2 having H 1 ( Rev ) as a gener ator . 6 COLLIN BLEAK, ALEXANDE R FEL’SHTYN, AND D ACIBER G L. GONC ¸ AL VES 3. Simple f acts about Reidem e ister classe s an d the Main theorem 3.1. Reidemeister classes and inner automorphisms. Let u s denote b y τ g : G → G the automorphism τ g ( x ) = g x g − 1 for g ∈ G . Its restriction on a normal subgroup w e will denote by τ g as w ell. W e will need the follo wing statemen ts. Lemma 3.1. { g } φ k = { g k } τ k − 1 ◦ φ . Pr o of. Let g ′ = f g φ ( f − 1 ) b e φ -conjugate to g . Th en g ′ k = f g φ ( f − 1 ) k = f g k k − 1 φ ( f − 1 ) k = f ( g k ) ( τ k − 1 ◦ φ )( f − 1 ) . Con v ersely , if g ′ is τ k − 1 ◦ φ -conjugate to g , then g ′ k − 1 = f g ( τ k − 1 ◦ φ )( f − 1 ) k − 1 = f g k − 1 φ ( f − 1 ) . Hence a shift maps φ -conjugacy classes on to classes r elated to another au- tomorphism.  Corollary 3.2. R ( φ ) = R ( τ g ◦ φ ). In particular R (Id) = R ( τ g ) is the num b er of the usu al conjugacy classes in G . Consider a group extension resp ecting homomorph ism φ : (3.1) 0 / / H i / / φ ′   G p / / φ   G/H φ   / / 0 0 / / H i / / G p / / G/H / / 0 , where H is a n ormal subgroup of G . First, notice that the Reidemeister classes of φ in G are mapp ed epimorphically on to classes of φ in G/H . Indeed, (3.2) p ( e g ) p ( g ) φ ( p ( e g − 1 )) = p ( e g g φ ( e g − 1 ) . Supp ose that the Reidemeister num b er R ( φ ) is infinite, the previous remark then implies that the Reidemeister n umber R ( φ ) is infinite. See [13 ] for a ge neralization of this elemen tary fact to homomorphisms of sh ort exact sequences. An endomorp hism φ : G → G is said to b e ev entually comm utativ e if there exists a n atural num b er n suc h that the subgroup φ n ( G ) is commuta tiv e. TWISTED CONJUGACY CLASSES IN R . THOMPSON’S GROUP F 7 W e are n ow ready to compare the Reidemeister n umber of an endomor- phism φ with the Reidemeister n um b er of H 1 ( φ ) : H 1 ( G ) → H 1 ( G ). Theorem 3.3 ([16]) . The c omp osition η ◦ θ , G θ − → H 1 ( G ) η − → Coker  H 1 ( G ) 1 − H 1 ( φ ) − → H 1 ( G )  , wher e θ is ab elianization and η is the natur al pr oje ction, sends every φ - c onjugacy class to a single element. Mor e over, any gr oup homomor phism ζ : G → Γ which se nds every φ -c onjugacy class to a single element, factors thr ough η ◦ θ . If φ : G → G is e v entual ly c ommutative (for example, i f the gr oup G is ab elian) then R ( φ ) = R ( H 1 ( φ )) = #C oker (1 − H 1 ( φ )) . The first p art of this theorem is trivial. If α ′ = γ αφ ( γ − 1 ) , then θ ( α ′ ) = θ ( γ ) + θ ( α ) + θ ( φ ( γ − 1 )) = = θ ( γ ) + θ ( α ) − H 1 ( φ )( θ ( γ )) = θ ( α ) + (1 − ( H 1 ( φ )) θ ( γ ) , hence η ◦ θ ( α ) = η ◦ θ ( α ′ ). This Theorem sh o w s the imp ortance of th e group Cok er (1 − ( H 1 ( φ )) Corollary 3.4. ([16 ], p.33) Let G b e a finitely generated free Ab elian group. If det ( I − φ ) = 0 then R ( φ ) = # C ok er (1 − φ ) = ∞ . W e can n o w state the main r esu lt. Theorem 3.5. F or any automorphism φ of R.Thompson ’s gr oup F the R ei - demeister numb er R ( φ ) is infinite. Pr o of. W e w ill calculate the matrix M of the automorphism H 1 ( Rev ). Let g , k ∈ F with R ev ( g ) = k . Then k r = − g l and k l = − g r . S o, the matrix of an automorphism H 1 ( Rev ) is the matrix M = 0 − 1 − 1 0 ! . Hence det ( I − M ) = 0 . F rom corollary 3.4 it follo ws that the num b er of Reidemeister classes of H 1 ( Rev ) is infinite. The same is true for all p o w er s M k , since for k ev en M k = id and M k = M for k o dd. Hence, by C orollary 2.3 and 2.4, the Reidemeister n um b er R ( H 1 ( φ )) is also infinite. F rom (3.1)- (3.2) it follo w s that the Reidemeister n um b er R ( φ ) is in finite as well .  8 COLLIN BLEAK, ALEXANDE R FEL’SHTYN, AND D ACIBER G L. GONC ¸ AL VES In a more concrete fashion, and for th e curious reader, it is not difficult to generate an infinitude of twisted conjugacy classes in Z × Z for H 1 ( Rev ). Consider the set Γ of pairs { (0 , a ) | a ∈ Z } , one may chec k directly that no t wo elemen ts of Γ are t wisted-conjugate equiv alen t. Referen ces 1. J. A rt hur and L. Clozel, Simple algebr as, b ase change, and the advanc e d the ory of the tr ac e formula , Princeton Un iversit y Press, Princeton, NJ, 1989. MR90m:22041 2. Brin, Matthew G., The chamele on groups of Richard J. Thompson: automorphisms and dyn amics, Inst. Hautes ´ Etudes Sci. Publ. Math. , 84 ( 1997), 5–33. 3. 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