Groups with the same cohomology as their profinite completions

Groups with the same cohomolog y as their profinite completions Karl Lorensen Mathematics Departmen t P ennsylv ania State Univ ersit y , Alto ona College 3000 Ivyside Park Alto ona, P A 16601- 3760 USA e-mail: kql3@psu. edu Octob er 30, 2018 Abstract F or any p ositiv e integer n , A n is the class of all groups G such that, for 0 ≤ i ≤ n , H i ( ˆ G, A ) ∼ = H i ( G, A ) for ev ery finite discrete ˆ G -mo dule A . W e describe certain t yp es of free p rodu cts with amalgam and HNN extensions that are in some of the classes A n . In addition, w e investig ate th e residually finite groups in th e class A 2 . Keywor ds : Profinite completions; Goo d groups; HNN extensions; Ascending HNN extensions; F ree pro ducts with amalgamatio n; R esidually finite groups; Right-angled Artin groups. 1 In tro duction If G is a group, let ˆ G denote the profinite co mpletion of G a nd c G : G → ˆ G the completio n map. In this pap er we study groups G such that, for any finite discrete ˆ G -mo dule A , the map induced by c G from the contin uous cohomology group H i ( ˆ G, A ) to the discre te cohomolog y gr oup H i ( G, A ) is an isomorphism for a range o f dimens io ns i . When this o ccurs for 0 ≤ i ≤ n , we iden tify the gr oup a s lying in the class A n . T his definition makes bo th A 0 and A 1 equal to the class of all gro ups. How ev er, for n ≥ 2 mem ber ship in A n bec omes mor e restr ic ted: Q / Z and PGL 2 ( C ), for exa mple, a re b oth no t in A 2 . Nevertheless, a n umber of importa nt types of groups are kno wn to b elong to all the classes A n ; these include p olycyc lic groups, f ree g roups and, o f course, finite groups. The classes A n hav e received intermitten t attention in the literature since at least the early 1960s. The first ment ion of these groups appea r s to hav e b een b y J-P . Ser r e [ 13 , Ex- ercises 1 , 2, Chapter 2], who establishe d most o f their basic pro p er ties. Later, H. Schneebeli [ 13 ] examined the clas s A 2 by employing the vir tual c o homology of a group. Moreov er, more recently , in his w ork o n Grothendiec k’s question o n Brauer gr oups, S. Sc hr¨ oer [ 14 ] inv oked groups that a re in A n for all n ≥ 0. 1 F rom t he p er sp ective of a group theorist the class A 2 is par ticularly interesting. As observed in [ 15 ], this cla s s consists precisely of those groups G such that, for every group extension N ι ֌ E ǫ ։ G with N finitely generated, the sequence of pro finite completions 1 → ˆ N ˆ ι → ˆ E ˆ ǫ → ˆ G → 1 is exact. This is equiv alen t to the asser tion that the pro finite top ology on E induces the full profinite top olog y on N ; in other words, for every H ≤ N with [ N : H ] < ∞ , there exists K ≤ E with [ E : K ] < ∞ such that K ∩ N ≤ H . As shown in Sectio n 2, the ab ov e prop erty of gr oups in A 2 reveals that the r esidually finite gr oups in that cla ss ar e o f a remark able sp ecies: a gro up G is b oth residually finite and in A 2 if and o nly if every extension of a finitely generated r esidually finite gr oup by G is again residually finite. Suc h gro ups are the the s ub ject of J. Corso n and T. Ratkovitc h’s article [ 3 ], where they ar e lab elled “sup er re sidually finite.” Mo difying Corson and Ratkovitc h’s ter m slightly , we will refer to g roups that are b oth residually finite and in A 2 as highly r esi dual ly finite . The hig hly res idually finite groups include all polycyclic groups a nd f ree groups; moreov er, the clas s of finitely g enerated highly r esidually finite gr oups is closed under the formation of extensions. Although the clas s es A n for n ≥ 2 are e ndowed with app ea ling prop er ties, it is in gener al very difficult to determine whether a given group b elongs to A n . The goal of the present article is to remedy this situatio n by identif ying so me new types of gro ups in the clas ses A n , particularly ones that are highly residually finite. Our main focus is on free pro ducts with amalgama tio n and HNN extensions formed fro m gr oups already known to b elong to A n . In Section 3 we employ the Mayer-Vietoris seq uences to ascerta in conditions under which such constructions yield groups that are aga in in A n . F o r instance, one consequence o f our r esults is that a ny free pro duct o f tw o vir tually p olycyclic g roups with cyc lic amalgam is in A n for all n ≥ 0. In addition, w e deduce that an y right-angled Artin group is in A n for all n ≥ 0 . The last sec tio n, Section 4, is devoted to proving that certain as c ending HNN extensions are highly r e sidually finite. Ascending HNN extensions ar e HNN extensions that are for med from a monomorphis m of the bas e g r oup. More precisely , if φ : G → G is a monomor phis m, then the a scending HNN extension of G with respect to φ , deno ted G φ , is g iven by G φ = h G, t | t − 1 g t = φ ( g ) for a ll g ∈ G i . These constructio ns hav e already been s tudied extensively w ith regar d to their r esidual finiteness in [ 1 , 8 , 7 ]. F or instance, it is known that ha ving a base group that is either po lycyclic or fr e e o f finite rank ensures that a n ascending HNN ex tension m ust b e re s idually finite. Genera lizing these results, w e prov e that any a s cending HNN extension with a base group of one of these tw o types is, in fact, highly residually finite. In the pro cess, we furnish a new pro of, sig nificantly s ho rter than those already present in the liter ature, that an ascending HNN extension of a po lycyclic group is residually finite. Finally , in the following theorem, we establish our most genera l result concerning the highly residual finiteness of ascending HNN extensio ns. Theorem. L et 1 = G 1  G 2  G 3  · · · G r − 1  G r = G 2 b e a n ormal series in a gr oup G such that e ach factor gr oup G i /G i − 1 is either finite, fr e e of finite r a nk or p olycyclic. If φ : G → G is a monomo rphism such that φ ( G i ) ≤ G i for 1 ≤ i ≤ r , then G φ is high ly r esidual ly fi nite. Notation. W e write H ≤ f G (r esp ectively H  f N ) when H is a subg roup (resp ectively normal subg r oup) of finite index in the g r oup G . Since we will b e working with b oth contin uous and discrete cohomolo gy , it will b e ad- v an tageous to distinguish b et ween the tw o types in our notatio n. Henceforth we will employ H ∗ ( , ) for dis c r ete cohomolo gy and H ∗ c ( , ) for co ntin uous cohomo logy . 2 Elemen t ary prop erties of the classes A n In this section w e discuss the elementary prop erties o f the classes A n . W e be g in with the following lemma, due to Serre [ 13 , Exercise 1a, Chapter 2]. Lemma 2.1 . (Serr e) Assume n is a nonne gative inte ger. F or a gr oup G the fol lowing statements ar e e quivalent. (i) F or any finite discr ete ˆ G -mo dule A , the map H i c ( ˆ G, A ) → H i ( G, A ) induc e d by c G is bije ct ive for 0 ≤ i ≤ n and inje ctive for i = n + 1 . (ii) F or any finite discr ete ˆ G -mo dule A , the map H i c ( ˆ G, A ) → H i ( G, A ) induc e d by c G is surje ctive for 0 ≤ i ≤ n . Remark. F o r the conv enience of those readers who wish to w or k through the details of Serre’s pr o of in [ 13 , Exercise 1a, Chapter 2], w e p oint out that there is an erro r in his definition of the gro up-theoretic proper ty he iden tifies as C n : the mo dule M ′ should be in the class C ′ K ; that is, it m ust b e finite. This mispr int also a pp ea rs in the F rench editions of [ 15 ]. Serre’s lemma forms the basis for his definition of the class es A n . Definition. Let n be a nonnega tive in teger. The c lass A n is the class o f all gr oups G satisfying the t wo equiv alent pro pe rties in L e mma 2.1. W e can a t once mak e the follo wing assertion ab out the cla s ses A 0 and A 1 . Prop ositi o n 2.2. (Serre [ 13 , Ex ercise 1b, Chapter 2]) The cla sses A 0 and A 1 ar e e ach e qual to the cl ass of al l gr oups. In addition, we ca n immediately cite some examples of gro ups tha t lie in a ll of the classes A n . Prop ositi o n 2.3. Finite gr oups and fr e e gr oups ar e in the class A n for all n ≥ 0 . 3 Pr o of. That finite gro ups are in A n is obvious. Moreov er, if G is fre e , then, for any finite discrete ˆ G -mo dule A , b oth H i ( G, A ) and H i c ( ˆ G, A ) are trivial for i ≥ 2. Thus every free group is in A n . Serre [ 15 ] observes that the class A 2 has an intriguing proper ty . In or der to appreciate this pro p er ty , we will require the follo wing notion. Definition. If H ≤ G , we say that H is top olo gic al ly emb e dde d in G if the subspa ce top ology induced on H b y the profinite top ology on G co incides with the full profinite topo logy o n H . If H is a topologic a lly embedded subgroup in G , we write H ≤ t G ; if, in addition, H is normal, we write H  t G . Note that the follo wing three assertions concerning a subgroup H of a group G a re equiv alent: (i) H ≤ t G ; (ii) for ea ch K ≤ f H , ther e exists L ≤ f G such that L ∩ H ≤ K ; (iii) ˆ H e mbeds in ˆ G . As we shall see b elow, g roup extensions with quotients in the class A 2 behave particularly nicely with resp ect to pr ofinite co mpletion. In ge ner al, profinite completion is a right exact functor; in other words, for any extension N ι ֌ G ǫ ։ Q, the sequence ˆ N ˆ ι → ˆ G ˆ ǫ → ˆ Q → 1 is exa ct. Ho wev er , if Q b elong s to A 2 and N is finitely generated, then 1 → ˆ N ˆ ι → ˆ G ˆ ǫ → ˆ Q → 1 is exact; that is, Im ι  t G . This is the con tent o f the follo wing propos itio n, part of whos e pro of is sk etched in [ 13 , Exercise 2 , Chapter 2]. Prop ositi o n 2.4. The fol lowing statements ar e e quivalent for a gr oup G . (i) The gr oup G b elongs to A 2 . (ii) F or every gr oup ex t ension 1 → N ι → E ǫ → G → 1 with N fin itely gener ate d, the map ˆ ι : ˆ N → ˆ E is an inje ction. Pr o of. The direction (( i)= ⇒ (ii)) is proved in [ 15 ]. ((ii)= ⇒ (i)) W e prov e that the map H 2 c ( ˆ G, A ) → H 2 ( G, A ) is s urjective for an y finite discrete ˆ G -mo dule A . T o establish this, let ξ ∈ H 2 ( G, A ) and take A ֌ G ։ Q to b e a group extensio n corr esp onding to the cohomolo gy class ξ . F or ming the pr ofinite completions , we o btain a n extensio n of profinite g roups A ֌ ˆ G ։ ˆ Q . The cohomolog y cla ss in H 2 c ( ˆ G, A ) of this extension, then, is mapp ed to ξ b y the map H 2 c ( ˆ G, A ) → H 2 ( G, A ). The ab ov e prop osition enables us to establish that e a ch class A n is closed under t he formation of certain extensions. 4 Theorem 2 . 5. L et n b e a p ositive inte ger and N ֌ G ։ Q a gr oup extension. Assu me N and Q ar e b oth in A n and N is of typ e FP n − 1 . Then G is in A n . Recall that a g roup G is of type FP n if Z a dmits a pr o jective reso lution as a trivial G - mo dule that is finitely gener ated in the first n dimensions. Finite groups, p olycyclic groups and finitely generated free groups are a ll of t yp e FP n for all n ≥ 0. Moreov er, for each nonnegative in teger n , the class of groups of t yp e FP n is clos ed under the formation of extensions. The key ingredient in the pro of of Theorem 2 .5 is the following le mma on sp ectr al se- quences. Lemma 2.6. A ssume { E pq r } and { ¯ E pq r } ar e first quadr ant c ohomolo gy sp e ctr al se quenc es, and let { φ pq r } b e a map fr om { E pq r } to { ¯ E pq r } . Supp ose, further, that φ pq 2 : E pq 2 → ¯ E pq 2 is bije ct ive for 0 ≤ p + q ≤ n and inje ctive for p + q = n + 1 . Then t he map φ pq ∞ : E pq ∞ → ¯ E pq ∞ is bije ctive for 0 ≤ p + q ≤ n and inje ct ive for p + q = n + 1 . Pr o of. W e prove by induction on r that, for r ≥ 2, the map φ pq r : E pq r → ¯ E pq r is bijective fo r 0 ≤ p + q ≤ n and injective for p + q = n + 1. Assume r > 2. Consider the co mmutative diagram E p − r +1 ,q + r − 2 r − 1 − − − − → E pq r − 1 − − − − → E p + r − 1 ,q − r +2 r − 1   y   y   y ¯ E p − r +1 ,q + r − 2 r − 1 − − − − → ¯ E pq r − 1 − − − − → ¯ E p + r − 1 ,q − r +2 r − 1 . If 0 ≤ p + q ≤ n , then the fir st and second vertical maps in this diagra m are bijectiv e, whereas the third is injective. This means that, in this case, the map E r pq → ¯ E r pq is bijective. Now consider the cas e when p + q = n + 1 . Here we hav e that the s e c ond vertical map is injectiv e and the fir st bijective. Therefore, the map E pq r → ¯ E pq r is injective. In addition, we require the following elementary fact concerning one- dimens ional coho- mology , who se pro of we leav e to the reader. Lemma 2. 7. If G i s a gr oup and A a top olo gic al ˆ G -mo dule, then the map H 1 c ( ˆ G, A ) → H 1 ( G, A ) induc e d by c G : G → ˆ G is m onic. Armed with these tw o lemmas, we pro ce e d with the pr o of of Theor em 2.5. Pr o of. Assume n ≥ 2. By Prop o s ition 2.4, the sequence ˆ N ֌ ˆ G ։ ˆ Q is a profinite gro up extension. W e will employ the Lyndon- Ho chsc hild-Ser re cohomology spe ctral sequence s in ordinary and co ntin uous cohomolo g y , resp ectively , a sso ciated with the extensions N ֌ G ։ Q and ˆ N ֌ ˆ G ։ ˆ Q . In view of Lemma 2.6, the co nclusion will follow if it holds that the map H p c ( ˆ Q, H q c ( ˆ N , A )) → H p ( Q, H q ( N , A )) induced by c G is bijective for 0 ≤ p + q ≤ n and injectiv e for p + q = n + 1. First we observe that, s inc e N is of t yp e FP n − 1 , H q ( N , A ) is finite fo r 0 ≤ q ≤ n − 1. Consequently , since Q and N are both in A n , w e can immediately conclude that the map H p c ( ˆ Q, H q c ( ˆ N , A )) → H p ( Q, H q ( N , A )) has the desir ed prop erties provided q 6 = n, n + 1. In the las t t wo cases, ho wever, H q ( N , A ) may no t be finite, so they need to b e examined s eparately . The homomo rphism H n c ( ˆ N , A ) ˆ Q → H n ( N , A ) Q is readily seen to be bijective since the map H n c ( ˆ N , A ) → H n ( N , A ) is an isomorphism. Similarly , that the map H n +1 c ( ˆ N , A ) → H n +1 ( N , A ) is an injection ensur es that the ho momorphism H n +1 c ( ˆ N , A ) ˆ Q → H n +1 ( N , A ) Q is also mo nic. All that remains, then, is to verify that the 5 map H 1 c ( ˆ Q, H n c ( ˆ N , A )) → H 1 ( Q, H n ( N , A )) is injective; how ever, this follows immediately from Lemma 2.7. Therefo r e, the maps H p c ( ˆ Q, H q c ( ˆ N , A )) → H p ( Q, H q ( N , A )) all hav e the required pr o p erties for 0 ≤ p + q ≤ n + 1. The case of Theorem 2.5 when n = 2 w ar rants s pec ia l mention. Corollary 2. 8. (Sc hneeb eli [ 11 , Theor em 1 ]) If N ֌ G ։ Q is a gr oup extension in which Q is in A 2 and N is a finitely gener ate d gr oup in A 2 , then G b elongs to A 2 . Theorem 2.5 als o p ermits the following o bserv atio n. Corollary 2.9. Any gr oup that is virtual ly p oly-(finitely gener ate d f r e e) is i n A n for al l n ≥ 0 . In particular, finitely genera ted v irtually free groups and virtually polycy c lic groups lie in A n for all n ≥ 0. Not all finitely g enerated solv able g roups, how ever, lie in A 2 , as the following exa mple illustrates. Example. Let U b e the group of a ll upp er triangula r 3x3 matrices over the dyadic ra tionals with diago nal (1 , 2 k , 1), wher e k ∈ Z . F o r 1 ≤ i, j ≤ 3 , let E ij be the 3x3 matrix with a 1 in the ( i, j ) po sition and zer o s everywhere e lse. The group U is a solv able group ge ner ated by three elemen ts: the diagonal matrix with diagonal (1 , 2 , 1), the matrix 1 + E 12 and the matrix 1 + E 23 . Let φ : U → U b e the automorphism   1 a b 0 2 k c 0 0 1   7→   1 a 2 b 0 2 k 2 c 0 0 1   . Let A b e the matrix 1 + E 13 in U . Then φ induces an isomor phism U / h A i → U / h A 2 i . Hence, setting G = U / h A 2 i , we hav e a group e xtension Z / 2 ֌ G ։ G. If G were in A 2 , this w ould yield a profinite g roup extensio n Z / 2 ֌ ˆ G ։ ˆ G. How ever, this is an impossibility since every co nt inuous s urjective endomor phism o f a top o- logically finitely gener ated profinite g roup is a n auto morphism (see [ 14 , Prop os ition 4.2.2]). Therefore, G is not in A 2 . The group G w a s , incident ally , the first exa mple of a finitely g enerated so lv able g roup that fails to b e hopfian, due to P . Hall [ 4 ]. The residually finite groups in A 2 hav e a particularly striking prop erty , as describ ed in the following theorem. Theorem 2 . 10. The fo l lowing statements ar e e quivalent for a gr oup G . (i) The gr oup G is r esidual ly finite and li es in the class A 2 . (ii) F or any gr oup extension N ֌ E ։ G wi th N finitely gener ate d and r esidual ly finite, E is r esidual ly fin ite. (iii) F or any gr oup extension F ֌ E ։ G with F finite, E is r esidual ly finite. 6 Pr o of. ((i)= ⇒ (ii)) Let N ֌ E ։ G be a group extension with N finitely gene r ated and residually finite. Then w e ha ve a commutativ e diagr am 1 − − − − → N − − − − → E − − − − → G − − − − → 1   y c N   y c E   y c G 1 − − − − → ˆ N − − − − → ˆ E − − − − → ˆ G − − − − → 1 with exact r ows. Since the maps c N and c G are injective, c E is als o injective. ((ii)= ⇒ (iii)) is trivia l. ((iii)= ⇒ (i)) That G is r esidually finite is immediate. In order to show that G lies in A 2 , let N ֌ E ։ G b e a group extension suc h that N is finitely generated. W e will establish that ˆ N ֌ ˆ E ։ ˆ G is exact by demonstrating that N  t G . T o show this, let H ≤ f N . Then, since N is finitely generated, H co ntains a subgro up M  G such that [ N : M ] < ∞ . It follows, then, fr o m (iii) that E / M is residua lly finite. This implies that there exists K ≤ f G containing M suc h that K / M ∩ N / M is the trivia l subgro up of E / M . Thus K ∩ N = M ≤ H . Therefore, N  t G . As stated in the introduction, w e will r e fer to groups that s atisfy the three equiv alent conditions o f Theor em 2.10 as highly r esidual ly fin ite . In view of their remark a ble prop er ties, these ar e among the mo st interesting gro ups in the classes A n ; hence they are a c c orded a considerable amount of a tten tion in the s equel. F or the moment, how ever, we remain conten t merely to observe that finite gro ups, polycyc lic groups a nd free groups are all highly residually finite; moreov er, as follo ws fro m the cor o llary b elow, the class of finitely generated highly residually finite groups is closed under the formation o f extensions. Corollary 2.11. If N ֌ G ։ Q is a gr oup extension su ch that N and Q ar e b oth highly r esidual ly finite and N is finitely gener ate d, then G is also highl y r esidual ly finite . Pr o of. By Theor em 2 .1 0, G is residually finite, and, b y Corollary 2.8 , G is in A 2 . It needs to b e emphasized that not a ll residua lly finite gro ups are hig hly r esidually finite. F or exa mple, SL 3 ( Z ) is residua lly finite, but, in view of [ 5 , Theorem I(ii)], it is no t highly residually finite. F urther more, there are groups in A 2 that ar e not r esidually finite; the following exa mple is one suc h group. Example. Let G b e G. Higman’s [ 6 ] example of a n infinite group with four gener ators and four r elators without any pr op er subgroups of finite index. Since the deficiency of Higman’s presentation is 0 and G ab = 1, we have that H 2 G = 0. This means, by the universal co efficient theor em, that H 2 ( G, A ) = 0 for a ny finite G -module A . T her efore, since ˆ G = 1 , G lies in A 2 . 3 Ma y er-Vietoris sequences and the classes A n In this sectio n we prov e that certain free pr o ducts with ama lgamation a nd HNN extensions formed from gro ups in A n yield groups that also lie in A n . The main to ols that we emplo y are 7 the Mayer-Vietoris sequences for free pro ducts with amalgama tion a nd for HNN extensions, in b oth their o rdinary and pr o finite incarnations . W e b egin our discussion by recalling so me facts ab out pr ofinite free pro ducts with amal- gamation fro m [ 9 ]. If Γ 1 and Γ 2 are profinite g roups with a common clos ed subgr oup ∆, then we ca n a lwa ys form the pro finite fre e pro duct of Γ 1 and Γ 2 with amalga mated subgr oup ∆; this is the pushout of Γ 1 and Γ 2 ov er ∆ in the catego ry of pro finite g roups. If Γ 1 , Γ 2 and ∆ are all embedded in this pusho ut, which is b y no mea ns a lways the case, then the latter is referred to as a pr op er pr ofinite free pr o duct w ith a malgamation. A sso cia ted to prop er profinite free pro ducts with amalgamation are Ma yer-Vietoris sequence s that relate the coho mo logies o f the v arious gr oups to one a nother. W e are interested in the sp ecial s ituation when Γ 1 = ˆ G 1 , Γ 2 = ˆ G 2 and ∆ = ˆ H , where G 1 and G 2 are discr ete g roups with a shared top ologic ally embedded subgroup H . In this case, the profinite completion of G = G 1 ∗ H G 2 is the pr o finite free pro duct of ˆ G 1 and ˆ G 2 with amalgama ted s ubg roup ˆ H . Mor e ov er , this profinite free product with amalg am is proper if and only if b oth G 1 and G 2 are top olog ically e m b edded in G . In this case, we hav e a May er -Vietoris sequence for ˆ G rela ting the cohomolo gies of ˆ G 1 , ˆ G 2 and ˆ H . This se quence is describ ed in the following theore m, which als o illuminates the connection to the discre te May er -Vietoris s equence for G . Theorem 3.1. L et G 1 and G 2 b e gr oups with a c ommon sub gr oup H that is top olo gic al ly emb e dde d in b oth gr oups, and let G = G 1 ∗ H G 2 . Assume, further, that G 1 and G 2 ar e b oth top olo gic al ly emb e dde d in G . Then, for e ach discr ete ˆ G -mo dule A and p ositive inte ger i , we have a c ommutative diagr am H i − 1 c ( ˆ H , A ) − − − − → H i c ( ˆ G, A ) − − − − → H i c ( ˆ G 1 , A ) ⊕ H i c ( ˆ G 2 , A ) − − − − → H i c ( ˆ H , A )   y   y   y   y H i − 1 ( H, A ) − − − − → H i ( G, A ) − − − − → H i ( G 1 , A ) ⊕ H i ( G 2 , A ) − − − − → H i ( H, A ) , (3.1) in which the r ows ar e exact and the vertic al maps ar e induc e d by the c ompletion maps for G 1 , G 2 , H and G . Theorem 3 .1 a llows us to deduce the follo wing s et o f criteria for de ter mining if a free pro duct with amalgam is in A n . Corollary 3.2. L et G 1 and G 2 b e gr oups with a c ommon sub gr oup H that is top olo gic al ly em- b e dde d in b oth gr oups. Assu m e G 1 and G 2 ar e e ach top olo gic al ly emb e dde d in G = G 1 ∗ H G 2 . If G 1 and G 2 ar e b oth in A n and H is in A n − 1 , then G b elongs to A n . Pr o of. Let A b e a finite discrete ˆ G -mo dule. F or 0 ≤ i ≤ n the fir st and third maps in diagram (3.1) are bijections, whereas the fourth is an injection. Therefore, the second map is sur jective, placing G in A n . W e now pro cee d to ascer tain when the t wo factors in a f ree pro duct with amalga m are top ologically embedded. Prop ositi o n 3.3. L et G 1 and G 2 b e gr oups with a c ommon sub gr oup H . As s ume that, fo r e ach p air { N 1 , N 2 } with N i  f G i , ther e exists a p air { P 1 , P 2 } such that P i  f G i , P i ≤ N i and P 1 ∩ H = P 2 ∩ H . Then G 1 and G 2 ar e top olo gic al ly emb e dde d in G 1 ∗ H G 2 . 8 Pr o of. Let G = G 1 ∗ H G 2 . As s ume N 1  f G 1 and N 2  f G 2 . T he n t here exis ts a pair { P 1 , P 2 } suc h that P i  f G i , P i ≤ N i and P 1 ∩ H = P 2 ∩ H . Since P 1 ∩ H = P 2 ∩ H , P 1 H/ P 1 ∼ = P 2 H/ P 2 . W e can then identif y these t wo gr oups via this isomorphism a nd f orm the free product with amalgamation ¯ G = G 1 /P 1 ∗ P 1 H/P 1 G 2 /P 2 . As a free product of t wo finite groups with amalgamation, the group ¯ G is virtually free; moreov er, there is a homomorphism θ : G → ¯ G that maps G 1 and G 2 canonically onto G 1 /P 1 and G 2 /P 2 , resp ectively . Let U b e the inv erse image with respect to θ of a free subgroup of finite index in ¯ G . Then U ≤ f G and U ∩ G i ≤ P i ≤ N i . Therefore, G i ≤ t G for i = 1 , 2 . Corollar y 3.2 and Pr op osition 3 .3 provide us with a wa y to prove that a free pro duct with amalgam is in the cla ss A n . The only difficult y is that it is not e a sy to recog nize when the conditions stipulated in Pr op osition 3.3 might b e sa tisfied. Nevertheless, we will discer n tw o impo rtant cases where these conditions are fulfilled. The fir st inv olves a no rmal a malgam and is tr eated in the follo wing theor e m, inspired by [ 3 , Pro po sition 6.1]. Theorem 3 .4. Assume G 1 and G 2 ar e gr oups in A n with a shar e d finitely gener ate d normal sub gr oup N in A n − 1 . If G 1 / N and G 2 / N b oth b elong to A 2 , then G 1 ∗ N G 2 is in A n . Pr o of. Our plan is to show that, for each pair { N 1 , N 2 } with N i  f G i , there exists a pair { P 1 , P 2 } such that P i  f G i , P i ≤ N i and P 1 ∩ N = P 2 ∩ N . It will then follow by P rop osition 3.3 that G 1 and G 2 are b oth top o logically embedded in G 1 ∗ N G 2 , yielding the co nclusion at once by v irtue of Cor ollary 3.2. T o determine the gro ups P i , we let U b e a normal subgr oup of G contained in N ∩ N 1 ∩ N 2 such that [ N : U ] < ∞ – that such a subg r oup ex ists is a consequence of th e fact that N is finitely generated. Since N ≤ t G i , w e can find M i  f G i such that M i ≤ N i and M i ∩ N ≤ U . No w we let P i = U M i . Then P i ∩ N = U ( M i ∩ N ) = U ; moreov er, P i  f G i and P i ≤ N i . Thus we have constructed the desired pair { P 1 , P 2 } . The seco nd s itua tion where the h yp o theses of Pro p o siton 3 .3 are satisfied is when the amalgama ted s ubgroup is cyclic and the gr o ups ar e qua s ipo tent , a proper ty defined as follo ws. Definition. A gr oup G is qu asip otent if, for each g ∈ G , there exists k ∈ Z + and a sequence { N n } of normal finite index subgroups indexed b y Z + such that h g i ∩ N n = h g nk i for eac h n ∈ Z + . If, for each g ∈ G , such a s equence { N n } can b e chosen so that N n is a characteristic subgroup o f G for all n ≥ 1, then G is cha r acteristic al ly quasip otent . Finite groups are, trivially , exa mples of characteris tica lly quasip otent gr o ups. In [ 16 ] and [ 10 ], resp ectively , it is proven that fr ee gr oups and p oly cyclic gr oups are a lso in this class. Moreov er, from [ 2 , Theor em 5.1 ] one can glean the following prop osition, whic h ma y be used to construct more ex a mples of groups that are quasip otent and characteris tically quasip otent. Prop ositi o n 3 .5. (Burillo , Martino) L et N ֌ G ։ Q b e a gr oup extension w ith Q a quasip otent gr oup in A 2 and N a finitely gener ate d char acteristic al ly quasip otent gr oup. Then G is qu asip ot en t . If, in addition, Q is finitely gener ate d and N is a char acteristic sub gr oup of G , then G is char acteristic al ly qu asip otent. 9 Prop ositio n 3.5 yields, for example, that finitely generated virtually free groups and virtually p olycyclic gro ups are all c harac teristically qua s ipo tent . Another consequence of this prop osition is that (finitely g enerated free)-by-po lycyclic gro ups are quasip otent. Additional examples of groups that are qua sip otent ar e pr ovided by [ 2 , Theorems 3.6, 3.7, 3.8]. F or our purp oses the following pro pe rty o f quasip o tent g roups will be impo rtant; it can be deduced at once from the definition. Lemma 3.6. If G is a quasip otent gr oup, then every cyc lic sub gr oup of G is top olo gic al ly emb e dde d in G . Now we examine fre e pro ducts of qua sip otent gro ups with cy clic a ma lgam, showing that they satis fy the hypo thes e s of P rop osition 3.3. Lemma 3.7. (L. Ribes and P . Zalessk ii [ 11 ]) If G 1 and G 2 ar e quasip otent gr oups with a c ommon cyclic sub gr oup A , then G 1 and G 2 ar e b oth top olo gic al ly emb e dde d in G 1 ∗ A G 2 . Pr o of. W e need to show that, for eac h pair { N 1 , N 2 } with N i  f G i , t here exis ts a pair { P 1 , P 2 } such that P i  f G i , P i ≤ N i and P 1 ∩ A = P 2 ∩ A . T o determine the g roups P i , we b egin b y letting a b e a generator of A . Since G i is quasipotent, ther e exis ts a sequence { i M n } n ∈ Z + of no rmal finite-index subgroups of G i such that i M n ∩ A = h a nk i i for so me k i ∈ Z + . Mo reov er, by choosing subsequences of { i M n } , we ca n make k i larger than any fixed v a lue. Hence we c an select the sequence { i M n } so that i M n ∩ A = h a nk i i ≤ A ∩ N 1 ∩ N 2 . Now, if we let P 1 = 1 M k 2 ∩ N 1 and P 2 = 2 M k 1 ∩ N 2 , then the pair { P 1 , P 2 } has the desired prop erties. In conjunction with Coro llary 3 .2 , the pre ceding tw o lemmas immedia tely yield the fol- lowing theor em. Theorem 3. 8. A s sume G 1 and G 2 ar e quasip oten t gr oups with a c ommon cyclic sub gr oup A . If G 1 and G 2 ar e b oth in A n , then G 1 ∗ A G 2 also b elongs to A n . One imp ortant sp ecia l cas e of the ab ov e theor em is provided below. Corollary 3.9. Assum e G 1 and G 2 ar e gr oups that ar e e ach either virtual ly fr e e of finite r ank or virtual ly p olycyclic. If G 1 and G 2 have a shar e d cyclic sub gr oup A , then G 1 ∗ A G 2 is in A n for al l n ≥ 0 . The r emainder o f the sectio n is devoted to HNN extensio ns . W e will employ the following notation f or thes e constructions: given a discrete g r oup G a nd an iso morphism φ : H → K , where bo th H and K are subgroups of G , the HNN extension of G with resp ect to φ is denoted by G φ . In other words, G φ = h G, t | t − 1 ht = φ ( h ) for all h ∈ H i . In addition to HNN extensions of discre te groups, we will refer to profinite HNN ex- tensions. As describ ed in [ 9 ], from any pr ofinite group Γ a nd a ny continous isomor phism θ : ∆ → Λ, where ∆ and Λ are bo th c losed s ubgroups o f Γ, we can form the pro finite HNN extension of Γ with resp ect to θ . If Γ, ∆ and Λ a re each embedded in the pr ofinite HNN extension, w e refer to the latter as a pr op er profinite HN N extension. Any prop er profinite 10 HNN extensio n gives rise to a May er -Vietoris sequence that relates the co homology of the extension to that o f the gr oup Γ and the subg roup ∆. Our interest is in the case when Γ = ˆ G , ∆ = ˆ H , Λ = ˆ K a nd θ = ˆ φ , whe r e G is a discrete group with top ologica lly embedded subgroups H , K and φ : H → K is an is omorphism. In this case , ˆ G φ is the pro finite HNN extension of ˆ G with resp ect to ˆ φ . Mor eov er, if G is top ologically embedded in G φ , then this pro finite HNN extension is pro p e r a nd, therefore, gives rise to a Mayer-Vietoris sequence. This s equence and its r e la tionship to the discrete May er -Vietoris s equence for G φ are descr ibe d in the follo wing theorem. Theorem 3. 10. L et G b e a gr oup with isomorphic, top olo gic al ly emb e dde d s u b gr oups H and K . Assume φ : H → K is an isomorphism and G ≤ t G φ . Then, for e ach d iscr ete ˆ G -mo dule A and p ositive inte ger i , we have a c ommutative diagr am H i − 1 c ( ˆ H , A ) − − − − → H i c ( ˆ G φ , A ) − − − − → H i c ( ˆ G, A ) − − − − → H i c ( ˆ H , A )   y   y   y   y H i − 1 ( H, A ) − − − − → H i ( G φ , A ) − − − − → H i ( G, A ) − − − − → H i ( H, A ) , (3.2) in which t he ro ws ar e exact and the vertic al maps ar e induc e d by the c ompletion maps for G , H and G φ . Theorem 3.10 provides us with the following conditions under which a n HNN extension is in the cla ss A n . Corollary 3.11 . L et φ : H → K b e an isomorph ism, wher e H and K ar e top olo gic al ly emb e dde d sub gr oups of a gr oup G . A ssume, further, that G ≤ t G φ . If G is in A n and H is in A n − 1 , then G φ b elongs to A n . Pr o of. Let A b e a finite discrete ˆ G -mo dule, and consider diagra m (3.2) for 0 ≤ i ≤ n . Then the fir st a nd third v ertical maps are bijectiv e, and the fourth is injective. Therefore, the second map m ust b e surjectiv e, so that G φ is in A n . Analogous to Prop o sition 3 .3, w e have the following conditions that gua rantee that the base gro up is top olog ically embedded in an HNN extension. Prop ositi o n 3 .12. L et φ : H → K b e an isomorphism, whe r e H and K ar e sub gr oups of a gr oup G . A ssume that, for every N  f G , ther e exists P  f G such t hat P ≤ N and P ∩ K = φ ( P ∩ H ) . Then G ≤ t G φ . Pr o of. Let N  f G . Then there is a subgr oup P satisfying the three co nditions stated in the h yp o thesis. Since P ∩ K = φ ( P ∩ H ), φ induces an isomorphism ¯ φ : P H /P → P K /P . Hence w e can form the HNN extension ¯ G o f G/P with respec t to ¯ φ . Moreov er , we hav e a n epimorphism θ : G φ → ¯ G that maps G canonically onto G/P . As an HNN extension o f a finite gro up, ¯ G is virtually free. Let U b e the inv er se image under θ of a free subgro up of ¯ G with finite index. Then [ G φ : H ] < ∞ and U ∩ G ≤ P ≤ N . Therefore, G ≤ t G φ . Corollar y 3.11 a nd Prop osition 3.12 allow us to pr ov e the follo w ing result. Theorem 3 . 13. L et G b e a gr oup and H ≤ t G . Defin e t he gr oup Γ by Γ = h G, t | [ t, h ] = 1 for al l h ∈ H i . If G and H b elong to A n and A n − 1 , r esp e ctively, then Γ is in A n . 11 Pr o of. Applying Pr op osition 3.12 , G is rea dily seen to b e top o logically embedded in Γ . The conclusion then follo ws by Cor ollary 3.11. As a consequence of this theor em, we can deduce that any right-angled Artin gro up is contained in A n for n ≥ 0. These g roups are defined as follows. Definition. A right-angle d Artin gr oup is any group with a finite genera ting set X and a presentation o f the form h X | [ x , y ] = 1 for all ( x, y ) ∈ Σ i for some s ubs et Σ of the C a rtesian pro duct X × X . Our approa ch to proving that rig ht -angled Artin gro ups are in A n for all n ≥ 0 is similar to that emplo yed in the pro of o f [ 2 , Theorem 3.8 ]. Before pro ceeding with the pro of, w e require the follo wing lemma. Lemma 3. 14. If G is a ri ght-angle d A rtin gr oup wi th gener ating set X , then, for e very X ′ ⊆ X , h X ′ i ≤ t G. Pr o of. The pr o of is b y induction on the cardinality of X , the cas e | X | = 1 b eing trivial. Assume | X | > 1. If X ′ = X , then the co nclusion fo llows at once. Assume X ′ 6 = X , and let x ∈ X − X ′ . Define H to be the group ge nerated by X − { x } with all of the same relators as G except those inv olving x . F ur thermore, let Y b e the set o f all elements in X − { x } that commute with x in G . Then G = h H , x | [ x, y ] = 1 for a ll y ∈ Y i . Now let U ≤ f h X ′ i . By the inductiv e h y po thesis, h X ′ i ≤ t H , which means tha t H contains a normal subgroup N o f finite index such tha t N ∩ h X ′ i ≤ U . Next co nsider the group ¯ G = h H / N , ¯ x | [ ¯ x, N y ] = 1 for all y ∈ Y i . As an HNN extens ion of a finite gro up, ¯ G is virtually free. Mo reov er, there is a map θ : G → ¯ G that maps H canonically onto H/ N a nd x to ¯ x . Let V be the inv erse image under θ of a free subgroup of finite index in ¯ G . Then V ≤ f G a nd V ∩ H ≤ N . Th us V ∩ h X ′ i ≤ U . It follows, then, that h X ′ i ≤ t G . Theorem 3 . 15. Every righ t-angle d Artin gr oup is in A n for al l n ≥ 0 . Pr o of. The proo f is by induction on the nu mber of generators, the case of o ne generator being trivial. Let G b e a rig ht-angled Artin group with generating set X with | X | > 1, a nd assume that every r ight-angled Artin group with few er generators than G lies in A n for all n ≥ 0. If G has no relato r s, then it is free and thus in A n for all n ≥ 0. Supp ose G has at least one relator , say [ x 0 , y 0 ]. Let [ x 0 , y 0 ] , [ x 1 , y 0 ] , · · · , [ x l , y 0 ] b e a list of all the relato r s that in volve y 0 . N ow define H to be the gro up generated by X − { y 0 } with all of the same relators as G exc e pt tho se involving y 0 . In view of the inductive hypothesis, H must b elong to A n for all n ≥ 0. More over, G = h H , y 0 | [ x 0 , y 0 ] = [ x 1 , y 0 ] = · · · = [ x l , y 0 ] = 1 i . By Lemma 3.14 , the subgro up o f H g enerated by x 0 , · · · , x l is to po logically embedded in H . In addition, as a right-angled Artin gro up with fewer g enerator s tha n G , this subgroup m ust belo ng to A n for a ll n ≥ 0. Therefore, a ppe aling to Theorem 3.13, we can conclude that G is in the cla ss A n for all n ≥ 0 . 12 4 Ascending HNN extensions that are highly residually finite In this section we examine a sp ecia l t yp e o f HNN extens io n, known as an ascending HNN extension and defined as follows. Definition. Let φ : G → G be a gro up monomorphism. The asc ending HNN ex tension of G with r esp ect to φ , denoted G φ , is giv en by the presen tation G φ = h G, t | t − 1 g t = φ ( g ) for a ll g ∈ G i . Ascending HNN extensions hav e the following elementary prop erty , which is an immediate consequence o f the normal form for elemen ts of HNN extensions. Lemma 4.1. L et φ : G → G φ b e a gr oup monomorphism. Th en every ele ment of G φ c an b e expr esse d in the form t k g t − l , wher e g ∈ G and k , l ∈ Z + ∪ { 0 } . An a s cending HNN extens io n of a residually finite g r oup may not be re s idually finite; see, for instance, the examples described in [ 12 ] as well a s [ 8 , Theorem 1.2]. Ho wev e r, as established in [ 1 , 8 , 7 ], a scending HNN extensio ns o f s ome important sp ecial t yp es of residually finite groups are alwa ys residually finite. Among these sp ecial types are finitely generated free gro ups, whose ascending HNN ex tensions are in vestigated in [ 1 ] using metho ds from a lgebraic geometry . Theorem 4.2. (A. Borisov, M. Sapir) If G is a fi nitely gener ate d fr e e gr oup and φ : G → G a monomorp hism, then G φ is r esidual ly fin ite. Our ob jective in this sectio n is to strengthen the ab ov e res ult by showing that asce nding HNN extensions of finitely generated free groups are actually highly residually finite. In addition, w e c o nsider ascending HNN extensions of p olycyclic gro ups, proving that they , to o, ar e highly residually finite. Finally , we generaliz e th ese t wo results by establishing the highly re sidual finiteness of certain asc ending HNN extensions of gr o ups p oss e s sing a normal series whose factor groups are eac h either free, p olycyclic or finite. The tec hniques we emplo y in this section eschew an y explicit reference to co homology; instead we rely on the third c ha r acterizatio n of hig hly residually finite gr oups g iven in Theorem 2.10 . W e b eg in with the following elementary pr op erty o f subgr oups of finite index in a finitely generated g roup. Lemma 4. 3. L et G b e a finitely gener ate d gr oup, N  f G and φ : G → G a homomorphi sm. Then N c ont ains a su b gr oup M  f G such tha t φ ( M ) ≤ M . Pr o of. F or each nonnegative integer i , let φ − i ( N ) = { x ∈ G : φ i ( N ) ∈ N } , where φ 0 is understo o d to b e the identit y map from G to G . Now set M = T ∞ i =0 φ − i ( N ). It is easy to see that M ≤ N , M  G a nd φ ( M ) ≤ M . All that remains to b e shown, then, is that [ G : M ] < ∞ . T o establish this, we first observe that, for each i ≥ 0, [ G : φ − i ( N )] ≤ [ G : N ] . Since G p ossesses only finitely many subgro ups with index ≤ [ G : N ], it fo llows that there are only finitely many subgroups o f the form φ − i ( N ) for i ≥ 0. Therefore, M , a s the intersection of finitely man y subgroups with finite index , has finite index. 13 Our analysis of as cending HNN extensions is bas ed on the f ollowing pro p erty of a group endomorphism. Definition. A gr oup endomorphism φ : G → G ha s prop erty P if, for every g ∈ G , there exists N  f G such that, for i = 0 , 1 , · · · , φ i ( g ) ∈ N ⇔ φ i ( g ) = 1 . F or injective endomorphisms the abov e prop erty ca n be expressed as follows. Lemma 4.4. A gr oup monomorphism φ : G → G has pr op erty P if and only if, for every nontrivial element g in G , ther e exists N  f G such tha t φ i ( g ) / ∈ N for al l i ≥ 0 . The sig nificance of prop erty P for the study o f ascending HNN extensio ns is revealed by the following prop osition. Prop ositi o n 4.5. L et G b e a finitely gener ate d gr oup and φ : G → G a monomo rphism. The gr oup G φ is r esidual ly fin ite i f and only if φ has pr op erty P . Pr o of. (= ⇒ ) Assume g ∈ G with g 6 = 1. Let M  f G φ such that g / ∈ M . Set N = M ∩ G . If φ i ( g ) ∈ N for some i ≥ 0, then t − i g t i = φ i ( g ) ∈ N ≤ M , which implies that g ∈ N , a contradiction. Hence φ i ( g ) / ∈ N f or a ll i ≥ 0 . Therefore, φ ha s the desired pr op erty . ( ⇐ = ) Let x ∈ G φ with x 6 = 1. W e require a subgroup of finite index in G φ that misses x . By Lemma 4.1, x = t k g t − l for some g ∈ G , w he r e k and l are nonneg ative integers. First we disp ose of the c ase when g = 1, i.e., x = t n for n ∈ Z − { 0 } . Let ǫ : G φ → Z b e the epimorphism that maps G to { 0 } and t to 1. Then, if A ≤ f Z such that n / ∈ A , we can take ǫ − 1 ( A ) as the desired subgroup. Next w e treat the really serio us case, namely , when g 6 = 1. By Lemma s 4.4 a nd 4.3, we can find N  f G such that φ ( N ) ≤ N and φ i ( g ) / ∈ N for i ≥ 0 . Let M = S ∞ i =0 φ − i ( N ), where φ − i ( N ) is as defined abov e in the pro of of Lemma 4.3. T he n g / ∈ M , M  f G and φ ( M ) ≤ M . In a ddition, the map G/ M → G/ M induced by φ is an isomorphism. Hence we can f orm the s emidirect product G/ M ⋊ h ¯ t i in which ¯ t has infinite order a nd ¯ t − 1 ( M y ) ¯ t = M φ ( y ) for all y ∈ G . Moreo ver, we hav e a map θ : G φ → G/ M ⋊ h ¯ t i that maps G ca nonically o n to G/ M and t to ¯ t . N ow let H = θ − 1 ( h ¯ t i ). Then H ≤ f G φ and g / ∈ H . Also, t ∈ H , so that x / ∈ H , th us allowing H to serve as the requir ed subgro up. In order to detect the ab ov e g roup-theor etic pr op erty , it is desirable to understand when this pr op erty is inherited by extensio ns. In this r esp ect the following lemma will prov e useful. Lemma 4.6. Assu me N ֌ G ։ Q is a gr oup ex tension su ch that N is finitely gener ate d and r esidual ly finite and Q is hig hly re sidual ly finite. L et φ : G → G b e a homomorphism such that φ ( N ) ≤ N . If the homomorphisms Q → Q and N → N induc e d by φ b oth p ossess pr op erty P , then φ must also have pr op erty P . Pr o of. Let g ∈ G with g 6 = 1. W e requir e a normal subgr oup of finite index that misses all the nontrivial ele ments of the form φ i ( g ) for i ≥ 0 . If φ k ( g ) = 1 for some k > 0, then φ i ( g ) = 1 for all i ≥ k , which means that the ex istence of suc h a subgroup follows immediately fr om the r esidual finiteness of G . Thus we only need consider the cas e when φ i ( g ) 6 = 1 for all i ≥ 0. First suppos e that φ i ( g ) ∈ N for some i ≥ 0, and let k be the smalles t nonnegative int eger suc h that φ k ( g ) ∈ N . Then, because φ induces a map with pro per ty P on N , there exists M  f N such that φ i ( g ) / ∈ M for i ≥ k . Moreov er, since Q lies in A 2 , we can find 14 U 1  f G s uch that U 1 ∩ N ≤ M . Therefor e, φ i ( g ) / ∈ U 1 for all i ≥ k . If k = 0 , then we are done. F or k > 0 we can in voke the r esidual finiteness of G to obtain U 2  f G s uch that φ i ( g ) / ∈ U 2 for 0 ≤ i < k . Then, setting U = U 1 ∩ U 2 , we have U  f G and φ i ( g ) / ∈ U f or a ll i ≥ 0. Finally , in the ca se where φ i ( g ) / ∈ N for a ll i ≥ 0, w e can a pply the h yp othesis that the map Q → Q induced by φ has pr op erty P in order to pro duce a subgr oup of G with the desired pro p er ties. Lemma 4 .6 can be used to prove a result analog ous to Pr op osition 4 .5 that provides a wa y to determine when an a scending HNN extension of a highly r e sidually finite g roup is itself highly residually finite. Prop ositi o n 4.7. L et G b e a finitely gener ate d highly r esidual ly fi nite gr oup. If φ : G → G is a monomorp hism with pr op erty P , then G φ is high ly r esidual ly finite. Pr o of. Assume F ֌ E ǫ ։ G φ is a group ex tension with F finite. Set H = ǫ − 1 ( G ), and let u ∈ E such that ǫ ( u ) = t . Conjugatio n by u induces a mono morphism ψ : H → H such that ψ ( F ) = F . Mo reov er, we hav e E ∼ = H ψ . Also, since ψ induces φ : G → G , ψ has prop erty P by Lemma 4.6, so that E is residually finite. Therefore, G φ is highly residually finite. The above prop osition, combined with Theorem 4.2 and Pr op osition 4.5, yields the fol- lowing res ult. Theorem 4.8. If G is a fr e e gr oup of finite r ank and φ : G → G a monomo rphism, then G φ is high ly r esidual ly fi nite. Now we pro cee d to iden tify s ome classes of gr o ups for which every endo morphism has prop erty P . The first is the class of finite g roups, for which every endomorphism sa tisfies the prop er t y trivia lly . Lemma 4. 9. If G is a finite gr oup, then ev ery endomorph ism of G has pr op ert y P . Next we co nsider endomor phisms o f finitely generated ab elian groups. Lemma 4.10 . If A is a finitely gener ate d ab elian gr oup, t hen any homomorphism φ : A → A has pr op erty P . Pr o of. First w e consider the case when φ is a monomorphism. W e may deco mp os e A as A = F ⊕ ¯ A such that ¯ A is the direct sum of finitely many copies of Z a nd F is finite. Let ¯ φ : ¯ A → ¯ A b e the mono morphism induced by φ . Assume a ∈ A with a 6 = 0. W e nee d to find a subgroup of finite index in A that does not cont ain φ i ( a ) for all i ≥ 0. Let ¯ a b e the ¯ A -p ortion of a . If ¯ a = 0, then ¯ A is a subg r oup with the pro per ties we r e quire. Suppose ¯ a 6 = 0. Let p b e a prime such that p do es not divide all of the components of ¯ a and als o do es not divide det ¯ φ . Then, for each in teg e r i ≥ 0, the prime p do es not divide all of the comp onents of ¯ φ i (¯ a ). Hence φ i ( a ) / ∈ pA for all i ≥ 0, so that pA can s erve as the desired subgroup. Now consider the case where φ is not injective. Let C = S ∞ i =1 Ker φ i . Then C ≤ A and φ induces a monomorphism φ ∗ : A/C → A/C . Assume a ∈ A and a 6 = 0. W e need to establish the existence of a subgroup D < f A such that φ i ( a ) ∈ D ⇔ φ i ( a ) = 0 . If a / ∈ C , we can obtain such a subgro up by a pplying the result for monomorphisms to φ ∗ . Now supp ose a ∈ C . Let k b e the smalle st p os itive in teg er such that φ k ( a ) = 0 . Since A is residually finite, there ex is ts D < f A such that φ i ( a ) / ∈ D for 0 ≤ i < k , giving us the des ired subgro up. 15 With the a id of Lemma 4.6, we can ea sily extend the abov e result to p o ly cyclic g roups. Lemma 4.11. If G is a p olycycli c gr oup, then any endomorphism φ : G → G has pr op ert y P . Pr o of. W e pro ceed by induction on the leng th o f the derived series of G , the base case having bee n established in Lemma 4.10. Assume G ′ 6 = 1 and cons ider the extension G ′ ֌ G ։ G ab . This extension satisfies all of the hypo theses of Lemma 4.6 with resp ect to any endo morphism φ : G → G ; thus the c o nclusion follows. As an immediate co nsequence of Lemma 4.11 , we obtain that a scending HNN extensions of p olycy c lic groups ar e highly re s idually finite. Theorem 4.12. If G is a p olycyclic gr oup and φ : G → G a monomorphism, then G φ is highly r esidual ly finite. Theorem 4.12 c an also b e deduced readily from T. Hsu and D. Wise’s [ 7 ] theo rem that ascending HNN extensions of v irtually polycyclic groups are r esidually finite. Nevertheless, we hav e adopted our a ppr oach because, in addition to providing a n a lternative, shorter pro o f for Hsu a nd Wise’s result, it allows us to prov e a theor em that gener alizes b oth Theor em 4.12 and Theor e m 4.8. Before w e can accomplish this, how ever, w e need to prove tha t endomorphisms o f finitely gener ated free gr oups satisfy pr op erty P as w ell. Lemma 4.13. If G is a finitely gener ate d fr e e gr oup and φ : G → G an endomorphism, then φ has pr op erty P . Pr o of. F or each i ≥ 0, φ i ( G ) is a finitely gener a ted free gr oup; moreover, the rank of φ i +1 ( G ) do es no t exceed the rank of φ i ( G ). Hence ther e exists k ≥ 0 such that the rank of φ k ( G ) is equal to the rank o f φ k +1 ( G ). It follows, then, fro m the fact that finitely genera ted free groups are hopfia n that φ induces an injection from φ k ( G ) → φ k +1 ( G ). Hence Ker φ k +1 = Ke r φ k . Now let K = Ker φ k . Then φ ( K ) ≤ K , and φ induces a monomorphism ¯ φ : G/K → G/K . Moreov er, G/K ∼ = φ k ( G ), so that G/K is free of finite rank. Thus, by Theorem 4.2 a nd Prop ositio n 4.5 , ¯ φ has pr o p erty P . Now we a r e re a dy to show that φ , to o, has prope r ty P . Let g ∈ G with g 6 = 1. W e desire a no rmal subgroup of finite index that misses all the nontrivial elements of the form φ i ( g ) for i ≥ 0. If g / ∈ K , then such a subgroup ma y be obtained b y in voking the fact that ¯ φ : G/ K → G/K has prop erty P . Now consider the case where g ∈ K . Let l b e the smalles t po sitive integer such that φ l ( g ) = 1. Since G is r esidually finite, there exists N  f G such that φ i ( g ) / ∈ N for 0 ≤ i < l . The subgro up N , then, enjoys the properties w e s eek. Armed with the a bove lemma, we ar e prepar e d to prov e our most g eneral res ult concer ning ascending HNN ex tensions. Theorem 4 . 14. L et 1 = G 0  G 1  G 2  · · · G r − 1  G r = G b e a normal series in a gr oup G such that e ach fa ctor gr oup G i /G i − 1 is either finite, fr e e of finite r ank or p olycyclic . If φ : G → G is a monomorphism such that φ ( G i ) ≤ G i for 1 ≤ i ≤ r , then G φ is high ly r esidual ly fi nite. Pr o of. This follows immediately from the le mma b elow. 16 Lemma 4. 15. L et 1 = G 0  G 1  G 2  · · · G r − 1  G r = G b e a normal series in a gr oup G such that e ach fa ctor gr oup G i /G i − 1 is either finite, fr e e of finite r ank or p olycyclic. If φ : G → G is a homomorphism such that φ ( G i ) ≤ G i for 1 ≤ i ≤ r , then φ has pr op erty P . Pr o of. W e induct on r , the c ase r = 1 having already been established. Assume r > 1. Consider the extension G r − 1 ֌ G ։ G/G r − 1 . By Lemmas 4.9, 4.11 and 4.13, the endomorphism of G/G r − 1 induced b y φ has prop er ty P ; moreover, the ma p from G r − 1 to G r − 1 induced by φ also ha s pro pe r ty P by the inductive hypothesis. Therefo re, since G/G r − 1 is highly residually finite, the co nclusion holds by Lemma 4 .6 . Two sp ecial ca ses of Theo rem 4.14 ar e worth emphasizing. Corollary 4. 16. If G is a finitely gener ate d virtual ly fr e e gr oup and φ : G → G a monomor- phism, then G φ is high ly r esidual ly fi nite. Corollary 4.17. If G is a virtual ly p olycyclic gr oup and φ : G → G a monomorphism, then G φ is high ly r esidual ly fi nite. In conclusion, we remark that it rema ins unkno wn whether the ascending HNN ex tensions shown to be highly residually finite in this se c tion are a lso in A n for n > 2. Unfortunately , the Ma yer-Vietoris sequence, employ ed in the prev ious s e ction to identify g roups in A n for n > 2, sheds no light on this question, b ecaus e, in an ascending HNN extension, the ba se group may not b e top o lo gically embedded. Ac knowledgemen ts. The author would like to thank the Department of Analysis and Scient ific Computing of the T echnical Universit y o f Vienna for its hos pita lit y while this work was in prog ress. In par ticular, he is dee ply grateful to W olfga ng Her fort for be ing suc h a generous, acco mmo dating host and fo r his perspicacio us sug gestions rega rding this ar ticle. References [ 1] A . Borisov and M. Sapir . P olynomial maps o ver finite fields and residual finiteness of mapping tori of g roup endomor phisms. In vent. Math. 160 (200 5), 3 41-35 6. [ 2] J. Burillo a nd A. Mar tino . Quasi-p otency and cyclic subgroup separability . J. Al - gebr a 298 (2006), 188-2 07. [ 3] J. Corson and T. Ra tko vich . A s tr ong form of residual finiteness for g roups. J. Gr oup The ory 9 (2006), 4 97-50 5 [ 4] P . H all. The F rattini subgroups of finitely genera ted groups. Pr o c. L ondon Math. So c. 11 (1961 ), 32 7-352 . [ 5] P . Hewitt. Extens ions of res idua lly finite gr oups. J. A lgebr a 16 3 (1994 ), 75 7-772 . 17 [ 6] G . Higman. A finitely g enerated infinite simple group. J. L ondon Math. So c. 26 (19 51), 61-64 . [ 7] T. Hsu and D. Wise . Ascending HNN extensio ns of p olyc yclic g roups are r esidually finite. J. Pur e Appl. A lgebr a 182 (20 03), 65-78 . [ 8] A . Rhemtulla and M. Shir v ani . The re s idual finiteness of ascending HNN extensions of certain so luble gr oups. Il linois J. Math. 47 (20 03), 47 7–484 . [ 9] L . Ribes and P. Zalesskii . Pr ofinite Gr oups (Spr ing er, 200 0). [ 10] L. Ribes , P. Zalesski i and D. Segal . Conjugacy separability and free pro ducts of groups with cy clic ama lgamation. J . L ondon Ma th So c. 57 (19 98), 609-6 28. [ 11] L. Ribes and P. Zal esskii . Conjugacy separability of amalga mated pro ducts. J. Al- gebr a 179 (1996), 751-7 74. [ 12] M. S apir and D. Wise . Ascending HNN extensions of res idually finite g roups can b e non-Hopfian and can have very few finite quotients. J . Pur e Appl. A lgebr a 166 (20 02), 191-2 02. [ 13] H. Schneebeli . Group e xtensions whose profinite co mpletio n is exact. Ar ch. Math. 31 (1978), 244 -253. [ 14] S. Schr ¨ oer. T op o lo gical metho ds for complex-analytic Brauer gr oups. T op olo gy 44 (2005), 875 -894. [ 15] J-P. S erre . Galois Cohomo lo gy (Springer , 1997). [ 16] C. T ang . Conjugacy separability o f genera lized free products of certain conjugacy separable gr o ups. Canad. Math Bul l. 38 (1995), 1 20-12 7. [ 17] J. W ilson. Pr ofinite Gr oups. (Cla rendon, 19 98). 18