Subgroups and Homology of Extensions of Centralizers of Pro-p Groups
aa r X i v : . [ m a t h . G R ] S e p SUBGROUPS AND HOMOLOGY OF EXTENSIONS OFCENTRALIZERS OF PRO-P GROUPS
D. H. KOCHLOUKOVA AND P.A. ZALESSKII Introduction
Limit groups have been studied extensively over the last ten years and theyplayed a crucial role in the solution of the Tarski problem. The name limit group wasintroduced by Sela. There are different equivalent definitions for these groups. Theclass of limit groups coincides with the class of fully residually free groups; under thisname they were studied by Remeslennikov, Kharlampovich and Myasnikov. Onecan also define limit groups in a constructive manner as finitely generated subgroupsof groups obtained from free groups of finite rank by finitely many extensions ofcentralizers. Starting from this definition, a special class L of pro- p groups (pro- p analogues of limit groups) was introduced by the authors in [5]. The class L consists of all finitely generated subgroups of pro- p groups obtained from free pro- p groups of finite rank by finitely many extensions of centralizers (see Subsection 2.2,for details). In [5] it was shown that many properties that hold for limit groupsare also satisfied by the pro- p groups from the class L : the groups G of the class L have finite cohomological dimension, are of type F P ∞ , have non-positive Eulercharaceristic, if C is maximal procyclic subgroup then its centralizer and normalizercoincide, every finitely generated normal subgroup has finite index, every soluble G is abelian, every two generated G is free or abelian. All these properties areknown to hold for abstract limit groups but the similarity might end there as it isnot known whether the groups from the class L are fully residually free pro- p orwhether the pro- p completion D d of an orientable surface group of odd genus d and p = 2 is in L . Still it was shown by the authors in [6] that for odd d the pro- p group D d is in the class L and D d is fully residually free.The study of the groups of the class L was continued by Snopche, Zalesskii in [10]where it was shown that a group from the class L has Euler characteristic χ ( G ) = 0if and only if G is abelian. A profinite version of the class L was considered byZapata in his PhD thesis [13].In the present paper we study asymptotic behavior of the dimensions of thehomology groups of subgroups of finite index in G ∈ L . In the case of abstractlimit groups this was recently studied by Bridson, Kochloukova in [3] where thecase ∩ i U i = 1 and each U i normal in G was considered. In the case of abstractgroups this is important as permits the calculation of the analytic Betti numbersof abstract limit groups. In the case of pro- p groups little is known about suchlimits, if dimension one homologies are used it is sometimes referred to as the rankgradient of a pro- p group. Partially supported by FAPESP, Brazil.
Theorem A
Let G ∈ L and { U i } i ≥ be a sequence of open subgroups of G suchthat U i +1 ≤ U i for all i and cd ( ∩ i U i ) ≤ . Then (i) lim −→ i dim H j ( U i , F p ) / [ G : U i ] = 0 for j ≥ if [ G : U i ] tends to infinity and def ( U i ) denotes the deficiency we have lim −→ i def ( U i ) / [ G : U i ] = − χ ( G );(iii) if [ G : U i ] tends to infinity and ( ∩ i U i ) = 1 we have lim −→ i dim H ( U i , F p ) / [ G : U i ] = 0 and lim −→ i dim H ( U i , F p ) / [ G : U i ] = − χ ( G ) . This direction of study lead us to prove group theoretic structure properties ofthe pro- p groups from the class L that are of independent interest. For a pro- p group G we denote the minimal number of generators by d ( G ). The followingresult shows that for an open subgroup U of G , the function d ( U ) is a monotonicincreasing function of index [ G : U ]. We do not know whether this holds for abstractlimit groups. Theorem B
Let G ∈ L be non-abelian and U be a normal open subgroup of G of index p . Then d ( U ) > d ( G ) . Theorem B is essentially used in the proof of the following result.
Theorem C
Let N ≤ H ≤ G , with G ∈ L non-abelian or G is non-trivial freepro- p product, H finitely generated and N normal in G . Then [ G : H ] < ∞ . In the case of abstract limit groups Theorem C was proved by Bridson, Howie,Miller, Short in [2]. It is worth noting that in the abstract case the full strengthof Bass-Serre theory was used and in the pro- p case some part of this technique isnot available. In particular geometric methods in the pro- p case failed, whereas theproof in [2] is purely geometric. 2. Preliminaries
Pro- p groups acting on pro- p trees. One of the main tools we use in thispaper is the action for pro- p groups on pro- p trees. By definition a pro- p tree Γ isa profinite graph, where for the pointed space ( E ∗ (Γ) , ∗ ) defined by Γ /V (Γ) withthe image of V (Γ) as a distinguished point ∗ we have a short exact sequence0 → [[ F p (( E ∗ (Γ) , ∗ )]] δ −→ [[ F p V (Γ)]] ǫ −→ F p → δ (¯ e ) = d ( e ) − d ( e ), d , d : E (Γ) → V (Γ) define beginning and end of anedge e ∈ E (Γ), ǫ ( v ) = 1 for every v ∈ V (Γ) and δ ( ∗ ) = 0. We shall assume thatthe action of a pro- p group G on a pro- p tree Γ is always continuous. We denoteby e G the closed subgroup of G generated by all vertex stabilizers G v for v ∈ V (Γ).Then by [9] the pro- p group G/ e G acts freely on the pro- p tree Γ / ˜ G , hence G/ e G isfree pro- p .We note that if a pro- p group G acts on a pro- p tree T cofinitely then we have thedecomposition of G as the fundamental group of a graph of pro- p groups ( G , T /G )i.e. with the underlying graph T /G and vertex and edge groups that are somevertex and edge stabilizers of the action of G on T . By [12, Prop. 4.4] if U isan open subgroup of G then the action of U on T is cofinite and so U is thefundamental group of the graph of pro- p groups ( U , T /U ) i.e. with underlying UBGROUPS AND HOMOLOGY OF EXTENSIONS OF CENTRALIZERS OF PRO-P GROUPS3 graph
T /U . Furthermore as in the Bass-Serre theory of abstract groups the vertexgroups in ( U , T /U ) are G gv ∩ U where v runs through the vertices of T /G and g ∈ G v \ G/U and the edge groups are G ge ∩ U where e runs through the edges of T /G and g ∈ G e \ G/U .Another important fact is that a pro- p group G acting on a pro- p tree with trivialedge stabilizers and H is a pro- p subgroup, possibly of infinite index, then resultsfrom the classical Bass-Serre theory work in their pro- p version. The followingresult follows from the proof of [11, Thm. 3.6]; see also the last section of [7]. Theorem 2.1.
Let G be a pro- p group acting on a pro- p tree T with trivial edgestabilizers such that there exists a continuous section σ : V ( T ) /G → V ( T ) . Then G is isomorphic to a free pro-p product ( a v ∈ V ( T ) /G G σ ( v ) ) a ( G/ h G w | w ∈ V ( T ) i The class of pro- p groups L : extensions of centralizers. The class ofpro- p groups L was defined in [5]. The groups in the class L are finitely generatedpro- p subgroups of G n , n ≥
0, where G can be any finite rank free pro- p groupand for n ≥ G n = G n − ` C ( C × Z sp ) for some s ≥
1, where C can beany self-centralized pro- p subgroup in G n − .Let G ∈ L . Then G ≤ G n , where n is the height of n i.e. is the smallest n possible. Then G n acts on the pro- p tree T associated to the decomposition G n = G n − ` C ( C × Z sp ). By restriction G acts on T but in general T /G is notfinite and hence applications of the pro- p version of Bass-Serre theory for pro- p groups is difficult for this action. Still in [10] Snopche, Zalesskii proved that T canbe modified to a pro- p tree Γ with cofinite G -action. We shall use this several timesin the paper and therefore state in the form we need. Theorem 2.2.
There is a pro- p tree Γ on which G acts cofinitely (i.e. Γ /G isfinite) such that the vertex stabilizers and the non-trivial edge stabilizers of theaction of G on Γ are vertex and edge stabilizers of the action of G on T . Thus G = π ( G , Γ /G ) is the fundamental group of pro- p groups, where vertex and edgegroups are certain stabilizers of vertices and edges of Γ in G . Proof of Theorem B
We start with a simple lemma about finite p -groups. Lemma 3.1.
Let K be a finite p -group with a normal subgroup A ≃ F dp and [ K : A ] = p . Let y, z ∈ K \ A such that Φ( K ) z = Φ( K ) y . Then z p = y p and theelements of the set { ( zy − ) x j } ≤ j ≤ p − are not linearly independent (over F p ) forany x ∈ K \ A .Proof. Note that since Φ( K ) z = Φ( K ) y we have that z = yc , c ∈ Φ( K ). We view A as a right F p [ K/A ]-module via conjugation, denote by upper right index the actionof F p [ K/A ] on A . Then c ∈ Φ( K ) = [ K, K ] K p = h A I , z p i ≤ A where I is theaugmentation ideal of of F p [ K/A ].Note that [
K, K ] = h a z − | a ∈ A i and ( az j ) p = a z + ... + z p − z jp for 1 ≤ j ≤ p − y p = ( zc ) p = z p c z + ... + z p − . We use ¯ z for the image of z in K/A . Since theaugmentation ideal I annihilates the socle of F p [ K/A ], the element c z + ... + z p − ∈ Φ( K ) (1+¯ z + ... +¯ z p − ) is the trivial element, hence y p = z p . D. H. KOCHLOUKOVA AND P.A. ZALESSKII
Since ( zy − ) z j = ( zcz − ) z j = c z j − and c z + ... + z p − is the trivial element,the elements of the set { ( zy − ) z j } ≤ j ≤ p − are not linearly independent (over F p ).Finally note that { ( zy − ) z j } ≤ j ≤ p − = { ( zy − ) x j } ≤ j ≤ p − . (cid:3) Lemma 3.2. a) Let A = G ` C G be an amalgamated free pro- p product where C is procyclic. Then d ( G ) + d ( G ) − ≤ d ( A ) ≤ d ( G ) + d ( G ) and d ( A ) = d ( G ) + d ( G ) if and only if C ⊆ Φ( G ) ∩ Φ( G ) .b) Let A = HN N ( G , C, t ) = h G , t | c t = c i be a proper pro- p HNN extensionwith procyclic associated subgroups h c i = C ≃ C = h c i and stable letter t . Then d ( G ) ≤ d ( A ) ≤ d ( G ) + 1 and d ( A ) = d ( G ) + 1 if and only if c Φ( G ) = c Φ( G ) .c) Let A be a pro- p group built from the pro- p group B after applying finitely manypro- p HNN extensions with stable letters t , . . . , t k and associated pro- p subgroups C i = h c i i and e C i = h ˜ c i i for ≤ i ≤ k , i.e. c t i i = ˜ c i . Assume further that C i ∪ e C i ⊆ B for all ≤ i ≤ k . Then d ( B ) ≤ d ( A ) ≤ d ( B ) + k. Furthermore1. d ( A ) = d ( B ) + k if and only if c i Φ( B ) = ˜ c i Φ( B ) for all ≤ i ≤ k ;2. d ( A ) = d ( B ) if and only if the images of the elements { c − i ˜ c i } ≤ i ≤ k in B/ Φ( B ) are linearly independent (over F p ).Proof. Parts a) and b) follow directly from the fact that for a pro- p group G we have d ( G ) = dim F p G/ Φ( G ), where Φ( G ) = G ′ G p . Note that by [8] pro- p amalgamatedproducts over procyclic subgroup are proper. Part c) follows from part b) byinduction on k . (cid:3) Theorem 3.3.
Let G ∈ L be non-abelian and U be a normal open subgroup of G of index p . Then d ( U ) > d ( G ) .Proof. Let G ≤ G n , where n is the height of G . Then by Theorem 2.2 G = Π ( G , Γ)is the fundamental group of a finite connected graph of groups with procyclic edgegroups and with vertex groups from the class L all of smaller height. Inductingon the height n of G we can assume that the theorem holds for all vertex groups.Furthermore we can induct on the size of Γ.Assume that e is an edge of Γ. Note that the edge stabilizer C of e is eitherinfinite procyclic or trivial. Then G is either a free pro- p product with amalgamation G ` C G or a pro- p HNN extension h G , t | C t = C i . By induction the result holdsfor both G and G .1. Assume that G = G ` C G . Note that if C ≃ Z p then at least one of G and G is not abelian. Indeed if not χ ( G ) = χ ( G ) + χ ( G ) − χ ( C ) = 0 + 0 − G is abelian, a contradiction.Let T be the standard pro- p tree on which G acts i.e. T /G has one edge andtwo vertices, the edge stabilizers are conjugates of C and the vertex stabilizers areconjugates of G or of G . By restriction U acts on T and since [ G : U ] < ∞ we can decompose U as the fundamental group of a finite graph of pro- p groups( U , T /U ) with underlying graph T /U (see Proposition 4.4 in [12] or subsection 2.1in the preliminaries). Then we have three cases.
UBGROUPS AND HOMOLOGY OF EXTENSIONS OF CENTRALIZERS OF PRO-P GROUPS5 CU = G . Then C = 1, U G = G = U G and T /U hasone edge with two vertices. The vertex groups of the graph of groups ( U , T /U )are U ∩ G and U ∩ G and the edge group is U ∩ C . Thus U is the proper pro- p amalgamated product ( U ∩ G ) ` C ∩ U ( U ∩ G ). By symmetry we can assume that G is not abelian but G might be abelian. Then by induction d ( U ∩ G ) ≥ d ( G )and d ( U ∩ G ) ≥ d ( G ), so d ( U ) ≥ d ( U ∩ G ) + d ( U ∩ G ) − ≥ d ( G ) + 1 + d ( G ) − d ( G ) + d ( G ) ≥ d ( G ) . But if d ( U ) = d ( G ) then d ( G ) = d ( G ) + d ( G ), so by Lemma 3.2 C ⊆ Φ( G ) ∩ Φ( G ) ≤ Φ( G ) ⊆ U , in particular C ⊆ U a contradiction. Thus d ( U ) > d ( G ).1.2. Suppose that G ⊆ U and G U = G . Then U is the fundamental group ofthe graph of groups ( U , T /U ) with vertex groups G ∩ U and G , G x , . . . , G x p − ,where x ∈ G \ U , so we can assume that x ∈ G \ U . The edge group linking thevertex groups G ∩ U and G x i is C x i . Then d ( U ) ≥ d ( G ∩ U ) + p.d ( G ) − p. Note that C is a direct factor in its centralizer in G and this centraliser is a finitelygenerated abelian group (it might be procyclic). Thus if d ( G ) = 1 we get that C = G and G = G and by induction the lemma holds for G , so that we aredone. Then we can assume d ( G ) ≥ G is not abelian, so by induction d ( G ∩ U ) ≥ d ( G ) + 1. Then d ( U ) ≥ d ( G ∩ U )+ p.d ( G ) − p ≥ d ( G )+1+ p.d ( G ) − p ≥ d ( G )+ d ( G )+1 ≥ d ( G )+1 . Assume now that G is abelian. Since d ( G ) ≥ d ( U ) ≥ d ( G ∩ U ) + p.d ( G ) − p = d ( G ) + p. ( d ( G ) − ≥ d ( G ) + 2 d ( G ) − ≥ d ( G ) + d ( G ) ≥ d ( G ) . If d ( G ) = d ( U ) then d ( G )+ d ( G ) = d ( G ) and by Lemma 3.2 C ⊆ Φ( G ) ∩ Φ( G ) ⊆ Φ( G ) contradicting C being a direct summand in its centraliser C G ( C ) = G .Hence d ( U ) > d ( G ).1.3 Suppose that C ⊆ U , G U = G U = G . Then U is the fundamental groupof the graph of groups ( U , T /U ) with two vertex groups U ∩ G and U ∩ G and p edges linking this two vertices with edge groups C, C x , . . . , C x p − , where x ∈ G \ U .If at least one of G and G is non-abelian we can assume (by symmetry) that G is non-abelian. Then since d ( U ∩ G ) ≥ d ( G ) + 1 and d ( U ∩ G ) ≥ d ( G ) d ( U ) ≥ d ( U ∩ G ) + d ( U ∩ G ) + ( p − − p ≥ d ( G ) + 1 + d ( G ) + ( p − − p ≥ d ( G ) + d ( G ) ≥ d ( G )If d ( U ) = d ( G ) then d ( G ) = d ( G ) + d ( G ) and d ( U ∩ G ) = d ( G ), hence byLemma 3.2 C ⊆ Φ( G ) ∩ Φ( G ) and by induction hypothesis G is abelian. Butthis contradicts the fact that C is a direct summand of G . Thus d ( U ) > d ( G ).If both G and G are abelian then C is trivial (since G ∈ L ), hence d ( U ) ≥ d ( U ∩ G ) + d ( U ∩ G ) + ( p − ≥ d ( G ) + d ( G ) + 1 > d ( G )and this completes the case of G splitting as an amalgamated free pro- p product.2. Now assume that G = HN N ( G , C, t ) = h G , t | c t = c i i.e. G is an HNNextension with a base group G , stable letter t and associated procyclic subgroups C = h c i and C = h c i . If C is trivial then G = G ` h t i , hence we can apply the D. H. KOCHLOUKOVA AND P.A. ZALESSKII case of amalgamated pro- p products considered above. Thus we can assume that C is infinite.If G is abelian then χ ( G ) = χ ( G ) − χ ( C ) = 0, so by [10, Thm. 3.4] G is abelian,a contradiction. Thus we can assume that G is not abelian.Let T be the standard pro- p tree on which G acts i.e. T /G has one edge (aloop), the edge stabilizers of the action of G on T are conjugates of C and thevertex stabilizers of the action of G on T are conjugates of G . By restriction U acts on T and since [ G : U ] < ∞ we can decompose U as the fundamental group ofa finite graph of pro- p groups ( U , T /U ) with underlying graph T /U (see Proposition4.4 in [12] or subsection 2.1 from the preliminaries). Then we have three cases.2.1. Suppose that
U C = G , hence G U = G . Then U is the fundamental groupof the graph of groups ( U , T /U ) with one vertex group U ∩ G and one edge group U ∩ C = C p . By Lemma 3.2 d ( G ) ≤ d ( G ) ≤ d ( G ) + 1, where d ( G ) = d ( G ) + 1 ifand only if c Φ( G ) = c Φ( G ). Similarly d ( G ∩ U ) ≤ d ( U ) ≤ d ( G ∩ U ) + 1. Since G is non-abelian by induction d ( G ∩ U ) ≥ d ( G ) + 1. Then d ( U ) ≥ d ( G ∩ U ) ≥ d ( G ) + 1 ≥ d ( G ) . If d ( U ) = d ( G ) then d ( G ) + 1 = d ( G ) and so by Lemma 3.2 c Φ( G ) = c Φ( G ).Also d ( U ) = d ( G ∩ U ) and U ∩ C = h c p i , U ∩ C = h c p i , hence by Lemma 3.2we have c p Φ( G ∩ U ) = c p Φ( G ∩ U ). This contradicts Lemma 3.1 for the group K = G / Φ( U ∩ G ). Thus d ( U ) > d ( G ).2.2. Suppose C ⊆ U and G U = G . Then U is the fundamental group ofa graph of groups ( U , T /U ) with one vertex group U ∩ G and p edge groups C, C x , . . . , C x p − , where x / ∈ U . Since G is not abelian we have d ( U ∩ G ) ≥ d ( G ) + 1, so by Lemma 3.2 d ( U ) ≥ d ( U ∩ G ) ≥ d ( G ) + 1 ≥ d ( G ) . If d ( U ) = d ( G ) then d ( G ) + 1 = d ( G ) hence by Lemma 3.2 c Φ( G ) = c Φ( G ). Aswell d ( U ) = d ( U ∩ G ) then since U ∩ C = h c i , U ∩ C = h c i we deduce by Lemma 3.2that { ( c − c ) x j Φ( U ∩ G ) } ≤ j ≤ p − are linearly independent in U ∩ G / Φ( U ∩ G ),contradicting Lemma 3.1.2.3. Suppose that G ⊆ U . Then U is the fundamental group of the graph ofgroups ( U , T /U ) with p vertex groups G x j ∩ U = G x j and p edge groups C x j ∩ U = C x j such that the underlying graph of groups is a circuit of p edges. Then since G is not abelian we have d ( G ) ≥ d ( U ) ≥ − p + p X j =1 d ( G x j ∩ U ) ≥ − p + pd ( G ) ≥ d ( G ) + 1 ≥ d ( G ) . If d ( U ) = d ( G ) then pd ( G )+1 − p = d ( G )+1, so d ( G ) = p/ ( p − ∈ Z , p = 2 and d ( G ) = 2. As well d ( G )+ 1 = d ( G ), so by Lemma 3.2 we have c Φ( G ) = c Φ( G ).Let ¯ G be the quotient group of G modulo the normal closure of Φ( G ) in G .Then ¯ G = A ` ¯ C ( ¯ C × h ¯ t i ), where A = G / Φ( G ) and overlining stands for theimage of a subgroup or an element of G in ¯ G .Assume first that c ∈ Φ( G ). Then ¯ G = A ` h ¯ t i . Since G ⊆ U we deduce that A ¯ G ⊆ ¯ U , hence ¯ U = A ` A ¯ t ` h ¯ t i . Thus d ( U ) ≥ d ( ¯ U ) = 2 d ( A ) + 1 = 2 d ( G ) + 1 =5 > d ( G ) + 1 = d ( G ), a contradiction.Now suppose that c / ∈ Φ( G ). Define K as the quotient group of ¯ G by the centralsubgroup ¯ C . Then K = B ` h f i , where f is the image of ¯ t in K and B = A/ ¯ C . UBGROUPS AND HOMOLOGY OF EXTENSIONS OF CENTRALIZERS OF PRO-P GROUPS7
Since B ⊆ W , where W is the image of ¯ U in K , we get W = B ` B f ` h f i , so d ( W ) = 2 d ( B ) + 1 = 3. On other hand since ¯ C is a direct factor in ¯ G we get that¯ U ≃ W × ¯ C , hence d ( U ) ≥ d ( ¯ U ) = d ( W ) + 1 = 4 > d ( G ) + 1 = d ( G ) = d ( U ),a contradiction. (cid:3) Remark.
We do not know whether Theorem 3.3 holds in the abstract case.
Corollary 3.4.
Let G ∈ L be non-abelian. Let H be a finitely generated of infiniteindex and C a procyclic subgroups of G . Then HC is not open in G .Proof. Suppose HC is open in G . Then by going down to a subgroup of finite indexin G we can assume that HC = G , hence for every open subgroup U of G thatcontains H we have d ( U ) ≤ d ( H ) + 1 contradicting Therem 3.3. (cid:3) By Theorem 2.2 a pro- p group G from class L acts cofinitely on a pro- p tree Γ.If U i is a chain of open subgroups of G then every U i is the fundamental group ofa graph of pro- p groups over Γ i = Γ /U i . Proposition 3.5.
Let G ∈ L be non-abelian and let Γ be a pro- p tree on which G acts cofinitely with procyclic edge stabilizers. Let H be a finitely generated subgroupof G such that H/ e H is not abelian. Then for every sequence { U i } i ≥ of opensubgroups of G such that U i +1 is a normal subgroup of index p in U i , U = G and H = T i U i such that d ( U i / f U i ) = rank ( π (Γ /U i )) tends to infinity when i tends toinfinity.Proof. Let { U i } i ≥ be a sequence of open subgroups of G such that U i +1 is anormal subgroup of index p in U i , U = G and H = T i U i . Let Γ i = Γ /U i . Thus U i = Π ( U i , Γ i ) is the fundamental group of a graph of groups with vertex and edgegroups being certain vertex and edge stabilizers of the action of U i on Γ, so edgegroups are procyclic.Observe that U i / e U i are free pro- p groups of rank d ( π (Γ i )) and Γ i +1 / ( U i /U i +1 ) =Γ i hence there is a natural projection map ϕ i +1 : Γ i +1 → Γ i that induces a homomorphism ϕ ∗ i +1 : π (Γ i +1 ) → π (Γ i ). Note that by Schreier’sformula d ( U i +1 / ] U i +1 ) ≥ d ( U i +1 ) ≥ d ( U i / f U i ), where U i +1 is the image of the map U i +1 / ] U i +1 → U i / f U i induced by the incluion of U i +1 in U i . Since H/ e H is theinverse limit of U i / f U i we may assume that U i / f U i ≃ π (Γ i ) is non-procyclic freepro- p group, in particular, Γ i is not a tree.Assume that d ( U i / f U i ) = rank ( π (Γ /U i )) does not tend to infinity when i tendsto infinity. Then for some large i , for all i ≥ i , the ranks of U i / f U i are the same,in particular since U i / e U i is not procyclic we have U i +1 = U i / e U i , so the map ϕ ∗ i +1 is an epimorphism ( hence an isomorphism). From now on consider only i ≥ i .Note that if for a sufficiently large i , say i ≥ i , we have that Γ i = Γ i +1 then T /H = Γ i is finite. Then the number of edges of T /H is finite, hence we havefinitely many double coset classes H \ G/C where C is some edge stabilizer of theaction of H on T . This implies that HC is open in G contradicting Corollary 3.4.Thus we can assume that Γ i +1 = Γ i for infinitely many i .The main ingredient of the proof is a description of how Γ i +1 and Γ i relate toeach other when Γ i +1 = Γ i . We recall that U i +1 is the fundamental group of a D. H. KOCHLOUKOVA AND P.A. ZALESSKII graph of group over the graph Γ i +1 and U i is the fundamental group of a graph ofgroups over the graph Γ i and the decomposition of U i +1 as a graph of groups isinduced by the decomposition of U i as a graph of groups as explained in the firstparagraph of subsection 2.1.We split V (Γ i ) as a disjoint union V ∪ V , where U i /U i +1 fixes every elementof ϕ − i +1 ( V ) and U i /U i +1 acts freely on ϕ − i +1 ( V ). Then the preimage of V in Γ i +1 has the same cardinality as V ; the preimage of V in Γ i +1 has cardinality p | V | . Itfollows that V = ∅ since otherwise ϕ i +1 is a covering and so ϕ ∗ i +1 is not surjectivecontradicting to the above.Now there are 3 types of edges in Γ i : 1) those that have their vertices in V , 2)those that have their vertices in V and 3) those that have one of the vertices in V and the other in V . We denote them by E , E and E respectively. Note that U i /U i +1 acts freely on the preimages of E and E so that they have cardinalities p | E | and p | E | . Observe that since ϕ i +1 is injective the preimage under ϕ i +1 of anedge e of E contains exactly one edge , otherwise we can have a non-contractableclosed path (of two edges) in E i +1 whose image in Γ i is e ¯ e , contradicting injectivityof ϕ ∗ i +1 .Let M j be the graph obtained from Γ j contracting the connected componentsof the subgraph ∆ = V ∪ E to a point, where j = i or j = i + 1. Assumethat ∆ has k points. Since the restriction of ϕ i +1 on ∆ is a bijection the map π ( M i +1 ) → π ( M i ) induced by ϕ i +1 is an isomorphism. Then | E | + | E | − k − | V | + 1 = | E ( M i ) | − ( V ( M i ) | + 1 = rank ( π ( M i )) = rank ( π ( M i +1 )) = | E ( M i +1 ) | − ( V ( M i +1 ) | + 1 = p | E | + p | E | − k − p | V | + 1Hence ( p − | E | + ( p − | E | = ( p − | V | and 0 ≤ rank ( π ( M i )) = | E | + | E | − k − | V | + 1 = 1 − k ≤
0, so M i and M i +1 are trees, ∆ is connected. Thus Γ i is obtained from ∆ by attaching several trees and Γ i +1 is obtained from ϕ − i +1 (∆)by attaching several trees, the restriction of ϕ i +1 to ϕ − i +1 (∆) is a bijection. Sincethe inverse limit Γ of Γ i is a pro- p tree we get that the fundamental group of ∆does not survive in the inverse limit, thus ∆ and hence Γ i are simply connected, acontradiction. (cid:3) Corollary 3.6.
Let G ∈ L be non-abelian and let Γ be a pro- p tree on which G acts cofinitely with procyclic edge stabilizers. Let H be a finitely generated subgroupof G such that H/ e H is non abelian. Let { V i } i ≥ be open subgroups of G such that V i +1 ≤ V i , V = G such that H = T i V i . Then d ( V i / e V i ) tends to infinity when i tends to infinity.Proof. We can refine the sequence . . . V i +1 ≤ V i ≤ . . . ≤ V ≤ G to get a sequence . . . U i +1 ≤ U i ≤ . . . ≤ U ≤ G as in the previous theorem and apply this theorem. (cid:3) The core property
Theorem 4.1.
Let N ≤ H ≤ G , where G ∈ L is not abelian, H finitely generatedand N non-trivial normal in G . Let G act on a pro- p tree Γ with Γ /G finite andprocyclic edge stabilizers. Assume that N acts freely on Γ . Then [ G : H ] < ∞ . UBGROUPS AND HOMOLOGY OF EXTENSIONS OF CENTRALIZERS OF PRO-P GROUPS9
Proof.
Suppose that [ G : H ] = ∞ . Then by Theorem 6.5 [5] N is not finitelygenerated, we will need only that it is not procyclic.By Proposition 3.5 H/ e H is the inverse limit of U/ e U where U runs througha set { U i } i ≥ of open subgroups of G such that U i +1 is a normal subgroup ofindex p in U i , U = G and H = T i U i such that d ( U i / f U i ) = rank ( π (Γ /U i ))tends to infinity when i tends to infinity. Then H ( H/ e H, F p ) is the inverse limitof H ( U i / f U i , F p ) and H ( H/ e H, F p ) is finite. So for some U i we have that theinclusion map H → U i induces an injective map H ( H/ e H, F p ) → H ( U i / f U i , F p ).Hence for every H ≤ U i ≤ U i the map H ( H/ e H, F p ) → H ( U i / f U i , F p ) is injective.Since H/ e H and U i / f U i are free pro- p we get that the map H/ e H → U i / f U i is injectiveand so f U i ∩ H = e H . Since N is infinitely generated (hence is not procyclic) andacts freely on T by replacing H with some of its open subgroup containing N wemay assume that N e H/ e H is not abelian. Indeed if for every open subgroup H i of H that contains N we have that H i / f H i is procyclic then N = N/ e N is inverse limit ofthe procyclic groups H i / f H i , so is procyclic, a contradiction. Thus we may assumethat N e H/ e H is not abelian and in particular H/ e H is not abelian.Consider N ≤ U i ≤ U i such that rank ( H/ e H ) < rank ( U i / f U i ) and N f U i .This is possible since 1 = e N = ∩ i ≥ i f U i and rank ( U i / f U i ) tends to infinity. Considerthe groups N f U i / f U i ≤ H/ e H ≤ U i / f U i . Since N f U i / f U i is non-trivial normal subgroup of a free pro- p group U i / f U i , H/ e H isa subgroup of finite index in U i / f U i and by Euler characteristic formula (Schreierformula) d ( H/ e H ) − U i / f U i : H/ e H ]( d ( U i / f U i ) −
1) a contradiction with rank ( H/ e H ) < rank ( U i / f U i ). (cid:3) Theorem 4.2.
Let N ≤ H ≤ G , with G ∈ L non-abelian, H finitely generated and N non-trivial normal in G . Then [ G : H ] < ∞ .Proof. Let n be the height of G i.e. n is the smallest number such that G is aclosed pro- p subgroup of G n = G n − ` C ( C × B ), where C ≃ Z p is self-centralizedin G n − and B is Z kp for some k ≥
1. We induct on n .Let T be the standard pro- p tree on which G n acts i.e. T /G n has one edge andtwo vertices, the edge stabilizers are conjugates of C and the vertex stabilizers areconjugates of G n − and C × B . Claim.
There is a non-trivial normal subgroup N of G such that N ≤ N and N acts freely on T .Proof of claim. If h B i G n ∩ G = 1 then G embeds into G n − ∼ = G n = G n / h B i G n contadictingminimality of n . So h B i G n ∩ G = 1. By Theorem 2.7 in [5] h B i G n = a g ∈ G n − /C B g so the commutator subgroup K of h B i G n acts freely on T . Since G ∩ h B i G n is notabelian (by Theorem 6.5 in [5]), K ∩ G = 1. If N ∩ ( K ∩ G ) = 1 then in G theygenerate N × ( K ∩ G ) and so by commutative transitivity it is abelian of rank at least2. Then by Corollary 5.5 in [5] N × ( K ∩ G ) is conjugate in G n into C × B , therefore is a finitely generated abelian normal subgroup of G . But this is impossible becausefor any g ∈ G \ ( C × B ) we have N × ( K ∩ G ) ≤ ( C × B ) ∩ ( C × B ) g is containedin a conjugate of C . Hence N ∩ K = 1. Then it suffices to set N = N ∩ K . Thiscompletes the proof of the claim.Let Γ be the pro- p tree of Theorem 2.2. Then a subgroup of G acts freely on Γif and only if it acts freely on T . Thus the claim and Theorem 4.1 complete theproof of the theorem. (cid:3) Aproximating homologies
Theorem 5.1.
Let G be a profinite (pro- p ) group acting on a profinite (pro- p ) tree T such that T /G is finite and all vertex and edge stabilizers are of type
F P ∞ . Let M be a finite pro- p F p [[ G ]] -module. Let { U i } i ≥ be a sequence of open subgroups of G such that for all i we have U i +1 ≤ U i and lim −→ i dim H j ( U i ∩ G gv , M ) / [ G gv : ( G gv ∩ U i )] = ρ ( v, g ) , lim −→ i dim H j − ( U i ∩ G ge , M ) / [ G ge : ( G e ∩ U i )] = ρ ( e, g ) , where ρ ( v, g ) , ρ ( e, g ) are continuous functions with domains V ( T ) × G and E ( T ) × G respectively. Then sup −→ i dim H j ( U i , M ) / [ G : U i ] ≤ X v ∈ V ( T ) /G sup g ∈ G ( ρ ( v, g )) + X e ∈ E ( T ) /G sup g ∈ G ( ρ ( e, g )) . In particular, if ρ ( v, g ) and ρ ( e, g ) are the zero maps, then lim −→ i dim H j ( U i , M ) / [ G : U i ] = 0 . Remark. If U i are normal then the functions ρ ( v, g ) and ρ ( e, g ) are constant on g . Proof.
Since G v and G e are all of type F P ∞ all the groups H j ( U i ∩ G gv , M ) and H j ( U i ∩ G ge , M ) are finite dimensional over F p . Furthermore since T /G is finite wededuce that G is of type F P ∞ .Consider the Mayer-Vietoris long exact sequence in homology for the action of G on T. . . → M e ∈ E ( T ) /G M g ∈ G e \ G/U i H j ( U i ∩ G ge , M ) → M v ∈ V ( T ) /G M g ∈ G v \ G/U i H j ( U i ∩ G gv , M ) → H j ( U i , M ) → M e ∈ E ( T ) /G M g ∈ G e \ G/U i H j − ( U i ∩ G ge , M ) → . . . It follows thatdim H j ( U i , M ) ≤ X v ∈ V ( T ) /G X g ∈ G v \ G/U i dim H j ( U i ∩ G gv , M )+(5.1) X e ∈ E ( T ) /G X g ∈ G e \ G/U i dim H j − ( U i ∩ G ge , M ) < ∞ . By hypothesis ρ ( i, v, g ) = dim H j ( U i ∩ G gv , M ) / [ G gv : U i ∩ G gv ]and ρ ( i, e, g ) = dim H j − ( U i ∩ G ge , M ) / [ G ge : U i ∩ G ge ] UBGROUPS AND HOMOLOGY OF EXTENSIONS OF CENTRALIZERS OF PRO-P GROUPS11 tend to ρ ( v, g ) and ρ ( e, g ) respectively as i goes to infinity. Since G is compact for afixed v and e the sequences { ρ ( i, v, g ) } i and { ρ ( i, e, g ) } i tend respectively to ρ ( v, g )and ρ ( e, g ) uniformly . Hence for a fixed ǫ > i such that for i ≥ i , g ∈ G and v ∈ V ( T ) /G, e ∈ E ( T ) /G we have(5.2) H j ( U i ∩ G gv , M ) ≤ ( ǫ + ρ ( v, g ))[ G gv : U i ∩ G gv ]and(5.3) H j − ( U i ∩ G ge , M ) ≤ ( ǫ + ρ ( e, g ))[ G ge : U i ∩ G ge ] . Counting G v and G e -orbits and the sizes of these orbits in G/U i we have(5.4) X g ∈ G v \ G/U i [ G gv : U i ∩ G gv ] = [ G : U i ] and X g ∈ G e \ G/U i [ G ge : U i ∩ G ge ] = [ G : U i ]Then by (5), (5.2), (5.3) and (5.4) we havedim H j ( U i , M ) ≤ X v ∈ V ( T ) /G X g ∈ G v \ G/U i dim H j ( U i ∩ G gv , M )+ X e ∈ E ( T ) /G X g ∈ G e \ G/U i dim H j − ( U i ∩ G ge , M ) ≤ X v ∈ V ( T ) /G X g ∈ G v \ G/U i ( ǫ + ρ ( v, g ))[ G gv : U i ∩ G gv ]+ X e ∈ E ( T ) /G X g ∈ G e \ G/U i ( ǫ + ρ ( e, g ))[ G ge : U i ∩ G ge ] ≤ [ G : U i ] (cid:0) X v ∈ V ( T ) /G ( ǫ + sup g ( ρ ( v, g ))) + X e ∈ E ( T ) /G ( ǫ + sup g ( ρ ( e, g ))) (cid:1) . Therefore for i ≥ i dim H j ( U i , M ) / [ G : U i ] ≤ ǫ ( | V ( T /G ) | + | E ( T /G ) | )+ X v ∈ V ( T ) /G sup g ∈ G ( ρ ( v, g )) + X e ∈ E ( T ) /G sup g ∈ G ( ρ ( e, g )) . It follows thatsup −→ i dim H j ( U i , M ) / [ G : U i ] ≤ X v ∈ V ( T ) /G sup g ∈ G ( ρ ( v, g )) + X e ∈ E ( T ) /G sup g ∈ G ( ρ ( e, g )) . (cid:3) Corollary 5.2.
Under the assumptions of Theorem 5.1 with j = 1 suppose furtherthat [ G ge : ( G ge ∩ U i )] tends to infinity for every g ∈ G and every fixed e ∈ E ( T ) such that G e = 1 . Then if lim −→ i dim H ( U i , M ) / [ G : U i ] exists we have lim −→ i dim H ( U i , M ) / [ G : U i ] ≤ X v ∈ V ( T ) /G sup g ∈ G ( ρ ( v, g )) . Proof.
Since H ( U, M ) =
M/J ( U ) M , where J ( U ) is the augmentation ideal of Z p [[ U ]] and U is a pro- p group we have ρ ( e, g ) = lim −→ i dim H ( U i ∩ G ge , M ) / [ G ge : ( G ge ∩ U i )] =lim −→ i dim( M/J ( U i ∩ G ge ) M ) / [ G ge : ( G ge ∩ U i )] ≤ lim −→ i dim M/ [ G ge : ( G ge ∩ U i )] = 0 . Thus we can apply Theorem 5.1 to deduce the result. (cid:3)
Recall that for a finitely presented pro- p group S the deficiency def ( S ) =dim H ( S, F p ) − dim H ( S, F p ). Theorem 5.3.
Let G ∈ L and { U i } i ≥ be a sequence of open subgroups of G suchthat U i +1 ≤ U i for all i and cd ( ∩ i U i ) ≤ . Then (i) lim −→ i dim H j ( U i , F p ) / [ G : U i ] = 0 for j ≥ if [ G : U i ] tends to infinity we have lim −→ i def ( U i ) / [ G : U i ] = − χ ( G );(iii) if [ G : U i ] tends to infinity and ( ∩ i U i ) = 1 we have lim −→ i dim H ( U i , F p ) / [ G : U i ] = 0 and lim −→ i dim H ( U i , F p ) / [ G : U i ] = − χ ( G ) . Proof. (i) We induct on the height of G . First if height of G is 0, then G iseither free or abelian. If G is free H j ( U i , F p ) = 0 for j ≥ G is abelian then U i ≃ G for every i , so dim H j ( U i , F p ) = dim H j ( U i +1 , F p ),hence if { [ G : U i ] } i tends to infinity we are done. If { [ G : U i ] } i does not tend toinfinity then U i = U i +1 = U i +2 = . . . , so ∩ t U t = U i has finite index in G and so cd ( G ) = cd ( ∩ j U j ) ≤
2. Then H j ( U t , F p ) = 0 for j ≥ L of smaller height. Let n be the height of G and G ⊆ G n = G n − ` C n − A n − , where A n − = Z mp = C n − × B . Then G n acts cofinitely on a pro- p tree T with vertex stabilizers conjugates of G n − and A n − and vertex stabilizers conjugates of C n − . By Theorem 2.2 G acts cofinitely on a pro- p tree Γ with vertex stabilizers that are intersections of thevertex stabilizers of G n (in T ) with G and edge stabilizers that are either trivial orinfinite cyclic.Note that the height of G v is smaller than the height of G since G v is inside ofa conjugate of G n − or a conjugate of A n − . By induction applied for the group G gv for a fixed j > ρ ( i, v, g ) = dim H j ( U i ∩ G gv , F p ) / [ G gv : U i ∩ G gv ]tends to 0 as i goes to infinity. Observe that since edge stabilizers are procyclic wehave for any j > H j − ( U i ∩ G ge , F p ) = 0, hence ρ ( i, e, g ) = dim H j − ( U i ∩ G ge , F p ) / [ G ge : U i ∩ G gv ] = 0 . Then by Theorem 5.1 the result follows for j > χ ( G ) = χ ( U i ) / [ G : U i ] = X j ≥ ( − j dim H j ( U i , F p ) / [ G : U i ] UBGROUPS AND HOMOLOGY OF EXTENSIONS OF CENTRALIZERS OF PRO-P GROUPS13 we have χ ( G ) = X j ≥ lim −→ i dim H j ( U i , F p ) / [ G : U i ]) − lim −→ i def ( U i ) / [ G : U i ] + lim −→ i / [ G : U i ]= − lim −→ i def ( U i ) / [ G : U i ](iii) By induction applied for the group G gv ρ ( v, g ) = lim −→ i dim H ( U i ∩ G gv , F p ) / [ G gv : U i ∩ G gv ] = 0 . Furthermore if G e = 10 ≤ ρ ( e, g ) = lim −→ i dim H ( U i ∩ G ge , F p ) / [ G ge : U i ∩ G ge ] ≤ lim −→ i / [ G ge : U i ∩ G ge ] = 0and if G e = 1 by the definition of ρ ( e, g ) we have ρ ( e, g ) = 0. Then by Theorem 5.1lim −→ i dim H ( U i , F p ) / [ G : U i ] = 0and hence by (ii)lim −→ i dim H ( U i , F p ) / [ G : U i ] = lim −→ i dim def ( U i ) / [ G : U i ]+lim −→ i dim H ( U i , F p ) / [ G : U i ]= − χ ( G ) + 0 = − χ ( G ) . (cid:3) Remark.
The proof shows that (iii) also holds if we just assume [ G ge : ( U i ∩ G ge )]tends to infinity for every e ∈ E ( T ) /G such that G e = 1 and for every g ∈ G .The Theorem 5.3 allows to obtain another proof of Theorem 6.5 in [5] that westate as the following Corollary 5.4.
Let H be a pro-p group from the class L with a non-trivial finitelygenerated normal pro- p subgroup N of infinite index. Then H is abelian.Proof. We give two new proofs one of which is an application of the results fromthis section.1. By Proposition 13 in [1] if N is a non-trivial finitely generated normal sub-group of a finitely generated residually finite group G of infinite index, then rankgradient of G is zero. The pro- p version of it is also valid with changing all thegroups in the proof to pro- p groups. Applying this for G = H and by Theorem 5.3the rank gradient for pro- p groups islim −→ i dim H ( U i , F p ) / [ H : U i ] = − χ ( H ) . Then χ ( H ) = 0 and by [10] H is abelian.2. Note that the corollary is a particular case of Theorem 4.2. The only placeits proof relies on Theorem 6.5 in [5] is the fact that in Theorem 4.1 N cannot beprocyclic. But if N was procyclic then by [4] by substituting H with a subgroupof finite index we can assume that H/N has finite cohomological dimension, thus χ ( H/N ) is well defined. Then χ ( H ) = χ ( N ) χ ( H/N ) = 0 and we are done as in theprevious proof. (cid:3)
References [1] M. Abert, A. Jaikin-Zapirain, N. Nikolov , The rank gradient from a combinatorial viewpoint,Groups Geometry and Dynamics, 2011, Vol. 5, p. 213-230.[2] M. R. Bridson, J. Howie, C. F. Miller III, H. Short, Subgroups of direct products of limitgroups. Ann. of Math. (2) 170 (2009), no. 3, 1447–1467[3] M. Bridson, D. Kochloukova, Volume gradients and homology in towers of residually-freegroups, arXiv:1309.1877.[4] A. Engler, D. Haran, D. Kochloukova, P. Zalesskii, Normal subgroups of profinite groups offinite cohomological dimension. J. of London Math. Soc., v. 69, (2004) p. 317-33[5] D. Kochloukova and P. Zalesskii,
On pro- p analogues of limit groups via extensions of cen-tralizers , Math. Z. 267 (2011), 109–128.[6] D. Kochloukova and P. Zalesskii, Fully residually free pro- p groups, J. Algebra 324 (2010),no. 4, 782 - 792[7] O.V. Mel’nikov, Subgroups and the homology of free products of profinite groups Math.USSR-Izv., 34, (1990), no. 1, 97-119.[8] L. Ribes, On amalgamated products of profinite groups. Math. Z. 123 (1971), 357 - 364[9] L. Ribes and P.A. Zalesskii, Pro- p Trees,
New Horizons in pro- p Groups (eds M du Sautoy,D. Segal and A. Shalev), Progress in Mathematics 184 (Birkh¨auser, Boston, 2000).[10] I. Snopche, P.A. Zalesskii
Subgroup properties of pro- p extensions of centralizers. SelectaMathematica (to appear).[11] P.A. Zalesskii, Normal subgroups of free constructions of profinite groups and the congruencekernel in the case of positive characteristic, Izv. Russ. Acad. Sciences, Ser Math., (1996).[12] P.A. Zalesskii, O.V. Melnikov, Fundamental Groups of Graphs of Profinite Groups,
Algebrai Analiz