Infinitely Generated virtually free pro- p groups and p -adic representations
aa r X i v : . [ m a t h . G R ] O c t INFINITELY GENERATED VIRTUALLY FREE PRO- p GROUPS AND p -ADIC REPRESENTATIONS P. A. ZALESSKII
Abstract.
We prove the pro- p version of the Karras, Pietrowski,Solitar, Cohen and Scott result stating that a virtually free groupacts on a tree with finite vertex stabilizers. If a virtually free pro- p group G has finite centralizers of all non-trivial torsion elements, astronger statement is proved: G embeds into a free pro- p productof a free pro- p group and a finite p -group. Integral p -adic represen-tation theory is used in the proof; it replaces the Stallings theoryof ends in the pro- p case. MSC classification: 20E18, 20C11Key-words: Profinite groups, pro- p groups, HNN-extensions, profinitemodules. 1. Introduction
Let p be a prime number, and let G be a pro- p group containing anopen free pro- p subgroup F , i.e. G is a virtually free pro- p group. If G is torsion free, then, according to the celebrated theorem of Serre[17], G itself is free pro- p . This motivated him to ask the questionwhether the same statement holds also in the discrete context. Hisquestion was answered positively some years later. In several papers(cf. [19], [20], [21]), J.R. Stallings and R.G. Swan showed that freegroups are precisely the groups of cohomological dimension 1, and atthe same time J-P. Serre himself showed that in a torsion free group G the cohomological dimension of a subgroup of finite index coincideswith the cohomological dimension of G (cf. [18]).One of the major tools for obtaining this type of results in the pres-ence of torsion is the Stallings theory of ends. Using it the descriptionof a virtually free discrete groups as a group acting on a tree with finitevertex stabilizers was obtained by Karras, Pietrowski and Solitar [8] infinitely generated case, D. E. Cohen [2] when the group is countableand by G. P. Scott [16] in the general case.The objective of this paper is to prove a pro- p version of this resultthat generalizes the main result of [3]. partially supported by CNPq Date : October 5, 2018.
Theorem 1.1.
Let G be a virtually free pro- p group. Then G acts ona pro- p tree with finite vertex stabilizers. The theory of ends for pro- p groups has been initiated in [9]. How-ever, it is not known whether an analogue of Stallings’ Splitting Theo-rem holds in this context. Here we use the representation theory over p -adic numbers; it is quite amazing that in the pro- p situation it canreplace the theory of ends.More precisely, first we embed G into a semidirect product of ˜ G =˜ F ⋊ H of a free pro- p group ˜ F and a finite p -group H such that alltorsion elements of ˜ G are conjugate into H . Now we use integral p -adicrepresentations of pro- p groups. Namely, we prove that the abelianiza-tion ˜ F ab is a permutation Z p [ H ]-module. Then we use the results of[10] on infinitely generated p -adic permutation modules and especiallyinfinitely generated pro- p version of the celebrated theorem of A. Weiss[22] to prove the following theorem of independent interest. Theorem 1.2.
Let G = F ⋊ H be a semidirect product of a finite p -group and a free pro- p group F . Suppose the abelianization F ab is a Z p H -permutation module. Then G is an HNN-extension with the basegroup H . Thus ˜ G is such an HNN-extension and as in the classical Bass-Serretheory of groups acting on trees, such an HNN-extension acts on astandard pro- p tree T such that vertex stabilizers are conjugate tosubgroups of H ; hence G acts on T as well. Moreover, this gives thefollowing result. Theorem 1.3.
Let F ⋊ H be a semidirect product of a free pro- p group F and a finite p -group H . Then the action of H extends to the actionon some free pro- p group ˜ F containing F such that H permutes theelements of some basis of ˜ F . The proof of Theorem 1.1 also shortens and simplifies the proof of[3, Theorem 1.3]; it brings the complicated induction in G downstairsto permutation Z p [ H ]-module M = F ab , where it is very transparent.In fact our proof practically does not use the theory of pro- p groupsacting on pro- p trees.If a virtually free pro- p group G have finite centralizers of torsionelements then we show a much more precise result. Theorem 1.4.
Let G be a virtually free pro- p group having finite cen-tralizers of the non-trivial torsion elements. Then G embeds into a freepro- p product G = F ∐ H of a finite p -group H and a free pro- p group F . NFINITELY GENERATED VIRTUALLY FREE PRO- p GROUPS AND p -ADIC REPRESENTATIONS3 This theorem can be considered as a generalization of the main resultof [11] where under the hypothesis of the second countability of G wasshown that G is a free pro- p product of finite p -groups and a free pro- p group. Indeed, this result follows directly from Theorem 1.4 usingthe pro- p version of the Kurosh Subgroup Theorem ([12, Thm. 4.3]or [14, Thm 9.6.2]. In general such a decomposition might not exist.Considering however the natural action of G on G /H and denotingby X the G -subset of G /H whose points have non-trivial stabilizers,we prove the following criterion of the existence of a decomposition of G as a free pro- p product of finite p -groups and a free pro- p group. Theorem 1.5.
Let G be a virtually free pro- p group having finite cen-tralizers of the non-trivial torsion elements. Then G is a free pro- p product of finite p -groups and a free pro- p group if and only if the nat-ural quotient map X −→ X/G by the action of G admits a continuoussection. Such continuous section does not always exist according to [14, Thm10.7.4] or [7, Section 4].
Acknowledgement.
The author thanks the anonymous referee forcareful reading the manusrcript and many suggestions that lead toconsiderable improvement of the paper.
Notation If G is a group and H a subgroup of G then H G denotes the normalclosure of H in G . By G ab we shall denote the abelianization of G . Z p [ G ] shall denote the group ring if G is finite and Z p [[ G ]] the completedgroup ring if G is infinite pro- p . If M is a Z p [ G ]-module, M G denotesinvariants (i.e. fixed points) and M G coinvariants (meaning M G = M/I G M , where I G is the augmentation ideal). By Res GH ( M ) we shalldenote the restriction of M to Z p [ H ].2. Preliminaries
Permutation modules.Definition 2.1.
A boolean or profinite space X is an inverse limit offinite discrete spaces, i.e., a compact, Hausdorff, totally disconnectedtopological space. A pointed profinite space ( X, ∗ ) is a profinite spacewith a distinguished point ∗ . A profinite space X (resp. pointed profi-nite space ( X, ∗ )) with a profinite group G acting continuously on itwill be called a G -space. P. A. ZALESSKII
An example of a pointed profinite space is a one point compactifi-cation X ∪ {∗} of a discrete space X with added point ∗ being thedistinguished point. If R is a commutative pro- p ring with unity and( X, ∗ ) = lim ←− i ∈ I ( X i , ∗ ) is an inverse limit of finite pointed spaces then R [[ X, ∗ ]] = lim ←− i ∈ I R [ X i , ∗ ] is a free pro- p R -module on the pointed space( X, ∗ ). Note that if ( X, ∗ ) is a one point compactification of a discretespace X then R [[ X, ∗ ]] = Q | X | R . Remark also that if ∗ is an isolatedpoint, then R [[ X, ∗ ]] = R [[ X \ {∗} ]], i.e. ∗ becomes superfluous. If G is a pro- p group then R [[ G ]] is a pro- p ring, called a pro- p group ringof G . Definition 2.2.
Let H be a pro- p -group and ( X, ∗ ) a pointed H -space.Then the free abelian group R [[ X, ∗ ]] becomes naturally a pro- p R [[ H ]]-module called permutation module.The next proposition collects facts on permutation pro- p modulesneeded for this paper. Proposition 2.3.
Let R = Z p or F p . Let H be a finite p -group and M be a permutation pro- p R [ H ] -module. (i) ([11, Corollary 2.3]) M = L K ≤ H Q I K R [ H/K ] , where I K issome set of indices. (ii) ([10, Prop 3.4]) Every direct summand A of M is a permutationmodule. Furthermore there exist a subset J K ⊆ I K for each K such that Q K ≤ H Q J K R [ H/K ] complements A in M . (iii) ([10, Cor 6.8]) If U is also permutation Z p [ H ] -module then anextension of M by U splits and so is a permutation Z p [ H ] -module.Remark . Q I K Z p [ H ] is a free Z p [ H ]-module on the pointed profinitebasis ( I K , ∗ ), where ( I K , ∗ ) is a one point compactification of a discretespace I K . Then the composite( I K , ∗ ) −→ Y I K Z p [ H ] −→ Y I K Z p [ H/K ]is injective and so ( I K , ∗ ) can be identified with its homeomorphicimage in Q I K Z p [ H/K ]. So we can regard X = ( S K ≤ H I K , ∗ ) as asubspace of M = L K ≤ H Q I K Z p [ H/K ] that we shall call a pointedbasis of M . Besides, Q I K Z p [ H/K ] = Z p [ H ] ˆ ⊗ Z p [ K ] Z p [[ I K , ∗ ]], where Z p [[ I K , ∗ ]] is a trivial Z p [[ K ]]-module and ˆ ⊗ means the completed tensorproduct.Recall that a Z p [[ G ]]-module M is called a lattice if it is free as Z p -module. One of the most beautiful results of modular representation NFINITELY GENERATED VIRTUALLY FREE PRO- p GROUPS AND p -ADIC REPRESENTATIONS5 theory of the end of 20-th century is the theorem of Weiss [22] whichinfinitely generated pro- p version is the following recently proved Theorem 2.5. [10, Thm 8.8]
Let G be a finite p -group and let M be aleft Z p [ G ] -lattice. Let N be a normal subgroup of G such that (i) res GN ( M ) is a free Z p [ N ] -module, and (ii) M N is a Z p [ G/N ] -permutation module.Then M is a Z p [ G ] -permutation module. The proof of this theorem as well as of the original Weiss’s theoremis quickly reduced (by induction on N ) to the case | N | = p . In this casehowever we can weaken the hypothesis (i) of Theorem 2.5 assuming that M is Z p [ N ]-permutation and that a maximal Z p [ N ]-trivial summandof it is G -invariant. Theorem 2.6.
Let G be a finite p -group and let M be a left Z p [ G ] -lattice. Let N be a normal subgroup of G of order p such that (i) res GN ( M ) is a permutation Z p [ N ] -module M ⊕ M p with M p being Z p [ N ] -free and M being G -invariant and Z p [ N ] -trivial; (ii) M N is a Z p [ G/N ] -permutation module.Then M is a Z p [ G ] -permutation module and M is its direct summand.Proof. We have a Z p -decomposition into direct sum M N = M ⊕ M Np and so modulo p we have an F p -decomposition ¯ M N = ¯ M ⊕ ¯ M Np of F p -modules. Now applying an operator P n ∈ N n to F p [ N ]-module¯ M = ¯ M ⊕ ¯ M p we obtain ¯ M Np and so it is an F p [ G ]-submodule of ¯ M .Hence ¯ M N = ¯ M ⊕ ¯ M Np is an F p [ G ]-decomposition and so by Proposi-tion 2.3(ii) ¯ M and ¯ M Np are F p [ G ]-permutation, since ¯ M N is. By [10,Theorem 8.6] M and M Np ∼ = M N /M are Z p [ G ]-monomial lattices, (i.e.a direct product of lattices induced from a rank 1 lattices). Monomial Z p [ G ]-lattices are permutation Z p [ G ]-lattices for p >
2. For p = 2 weneed to apply [10, Lemma 8.4] to deduce that the Z p [ G ]-module M N decomposes as M N ∼ = M ⊕ M Np and hence by Proposition 3.4 (ii) M and M Np ∼ = M N /M are Z p [ G ]-permutation. Applying Theorem 2.5 wededuce that M p is a Z p [ G ]-permutation module. Therefore M is anextension of permutation Z p [ G ]-modules ( M by M p ) and so by Propo-sition 2.3 (iii) is a permutation Z p [ G ]-module with M being its directsummand. (cid:3) Remark . Theorem 2.6 is the main ingredient of induction in theproof of the main result. We note that hypotheses (i) and (ii) ofTheorem 2.6 are also necessary. Indeed, if M is Z p [ G ]-permutation P. A. ZALESSKII then (i) is clearly satisfied. To see (ii) take a typical direct summand Z p [ G/H ] of M , where H is a subgroup of G . Observe that if N ≤ H then Z p [ G/H ] N = Z p [ G/H ]. Otherwise Z p [ G/H ] is Z p [ N ]-free and so Z p [ G/H ] N = Z p [ G/H ] N = Z p [ G/HN ] (see [11, Lemma 2.4]).The condition of G -invariancy is essential as the following exampleshows. Example . Let G = N × C be a direct product of two groups of order2 with generators c and c respectively. Let M = h x, a, b i be a freeabelian pro-2 group of rank 3. Define an action of G on M as follows: c x = x, c a = b, c b = a , c x = x + a + b , c a = − a, c b = − b . ThenRes GN ( M ) = Z ⊕ Z [ C ] and so is permutation. The Z [ G/N ]-module M N ∼ = Z [ G/N ] and so is permutation. But M is not permutation Z [ G ]-module, since it is not permutation as Z [ C ]-module.2.2. HNN-extensions.
If ( X, ∗ ) is a pointed profinite space, a freepro- p group F = F ( X, ∗ ) is a pro- p group together with a continuousmap ω : ( X, ∗ ) −→ ( F,
1) satisfying the following universal property: F ( X, ∗ ) η ❋❋❋❋❋ ( X, ∗ ) ω O O α / / G for any pro- p group G , any continuous map of pointed profinitespaces α : ( X, ∗ ) −→ ( G,
1) extends uniquely to a continuous homo-morphism η : F ( X, ∗ ) −→ G .If ( X, ∗ ) is a one point compactification of a discrete space X then thefree pro- p group F ( X, ∗ ) on the pointed profinite space ( X, ∗ ) coincideswith the notion of a free pro- p group F ( X ) on the basis X convergentto 1 (see [15, Chapter 3]).We introduce a notion of a pro- p HNN-extension slightly simpler thanthe construction described in [4] to make the paper more accessible tothe reader; this version suffices for our purpose. In the literature usuallythe term HNN-extension stands for the construction with one stableletter only, in our case through the paper we shall have a set of stableletters.
Definition 2.9.
Suppose that G is a pro- p group, and for a finite set { A i | i ∈ I } of subgroups of G and a family of pointed profinite spaces { ( X i , ∗ ) | i ∈ I } there are given continuous maps φ i : A i × X i → G such that ( φ i ) | A i ×{ x } is an injective homomorphism and ( φ i ) | A i ×{∗} = id for each x ∈ X i . The HNN-extension ˜ G := HNN( G, A i , φ i , X i ), i ∈ I NFINITELY GENERATED VIRTUALLY FREE PRO- p GROUPS AND p -ADIC REPRESENTATIONS7 is defined to be the quotient of a free pro- p product ` i ∈ I F ( X i , ∗ ) ∐ G modulo the relations φ i ( a i , x i ) = x i a i x − i for all x i ∈ X i , i ∈ I . Wecall G the base group , A i and B ix := φ i ( A i × { x } ) associated subgroupsand S i ∈ I X i \ {∗} the set of stable letters . Note that if A i and A j are conjugate then the relation φ i ( a i ) = x i a i x − i implies automaticallythe corresponding relation on A j , therefore we can remove j from i enlarging X i ; thus w.l.o.g we shall assume from now on that { A i | i ∈ I } is the family of pairwise non-conjugate subgroups.If each ( φ i ) | A i ×{ x } is an identity map, we call the HNN-extension special . Remark . A pro- p HNN-group ˜ G is a special case of the funda-mental pro- p group Π ( G , Γ) of a profinite graph of pro- p groups ( G , Γ)(see [14, Example (e)]). Namely, a pro- p HNN-group can be thoughtas Π ( G , Γ), where Γ is a bouquet (i.e., a connected profinite graphhaving just one vertex - the distinguished point of the one point com-pactification Γ = ( S i ∈ I X i , ∗ ) of the set of stable letters). In particu-lar, a pro- p HNN-group ˜ G := HNN( G, A i , φ i , X i ) acts on a pro- p tree S = S ( ˜ G ) defined as follows: the vertex set V ( S ) = ˜ G/G and theedge set E ( S ) = S i ∈ I ˜ G/A i × ( X i \ {∗} ) with ˜ gG and ˜ gx i G to be ini-tial and terminal vertices of (˜ gA i , x i ), x i ∈ X i . The fact that S ( ˜ G )is a pro- p tree means that there is an exact sequence of permutation Z p [[ G ]]-modules0 → Z p [[ E ( S ) , ∗ ]] → Z p [[ V ( S )]] → Z p → gA i , x i ) → ˜ gx i G − ˜ gG .The next proposition corrects unessential error in the statementof Lemma 3.8 in [3], namely the description of the centralizers as C ˜ F ( A i ) = ` s ∈ S i F ( Z i , ∗ ) s there was correct only for A i maximal fi-nite (by inclusion), and only for maximal A i used. One can consult [6]for details. We use the left action by conjugation by inverses since itcorresponds better to the left modules obtained after the abelianiza-tion. Proposition 2.11.
Let ˜ G = HNN( K, A i , X i , i ∈ I ) be a special pro- p HNN-extension of a finite p -group K and ˜ F be the normal closure of F ( S i ∈ I X i , ∗ ) in ˜ G . For every i ∈ I choose respectively coset represen-tative sets R i of K/N K ( A i ) and S i of N K ( A i ) /A i . Then ˜ F = a i ∈ I a r ∈ R i a s ∈ S i F ( X i , ∗ ) s − r − . P. A. ZALESSKII
Proof.
By Definition 2.9, one can view ˜ G as the quotient of G := F ( S i ∈ I X i ) ∐ K modulo the relations [ a i , x i ] for all x i ∈ X i and a i ∈ A i ,with i running through the finite set I . By the Kurosh Subgroup The-orem (see [15, Theorem 9.1.9]) applied to the normal closure N of F ( S i ∈ I X i , ∗ ) in G we have a free pro- p decomposition N = a i ∈ I a r ∈ R i a s ∈ S i a a ∈ A i F ( X i , ∗ ) a − s − r − . The relations yield F ( X i , ∗ ) a − = F ( X a − i , ∗ ) = F ( X i , ∗ ). Since for s ∈ S i , a ∈ A i , x ∈ X i one has [ a, x ] = 1 if, and only if, [ a s , x ] = 1 ifand only if [ a, x s − ] = 1, we deduce that˜ F = a i ∈ I a r ∈ R i a s ∈ S i F ( X i , ∗ ) s − r − . (cid:3) Observing that K acts on ˜ F = ` i ∈ I ` r ∈ R i ` s ∈ S i F ( X i , ∗ ) s − r − per-muting the free factors we deduce the following Corollary 2.12. ˜ F ab = M i ∈ I Y z ∈ X i Z p [ K/A i ] , with ( X i , ∗ )[ ˜ F , ˜ F ] / [ ˜ F , ˜ F ] being a pointed basis of Q x ∈ X i Z p [ K/A i ] . More-over, X i ⊂ C ˜ F ( A i )[ ˜ F , ˜ F ] / [ ˜ F , ˜ F ] . The structure of the abelianization
We start this section with the lemma proved for finitely generatedcase in [5] that relates a semidirect product G = F ⋊ C p of a free pro- p group of order p and a group C p of order p with Z p [ C p ] module F ab .The proof however uses only finiteness of the set of conjugacy classes ofgroups of order p in G . We give the proof here as well for convenienceof the reader. Lemma 3.1.
Let G = F ⋊ C be a semidirect product of a finite cyclic p -group C = h c i and free pro- p group F . Suppose G has only oneconjugacy class of groups of order p . Then (i) ([23, Theorem 1.2] , [14, Prop. 10.6.2]) G has a free pro- p prod-uct decomposition G = ( C × H ) ∐ H , such that H and H arefree pro- p groups contained in F . (ii) The Z p [ C ] -module M = F ab decomposes in the form M = M ⊕ M p such that M is trivial and M p is a free Z p [ C ] -module. NFINITELY GENERATED VIRTUALLY FREE PRO- p GROUPS AND p -ADIC REPRESENTATIONS9 Moreover, if X is the basis of H and X is the basis of H then the ba-sis of M is X [ F, F ] / [ F, F ] and the Z p [ C ] -basis of M p is X [ F, F ] / [ F, F ] .Proof. (ii) is a particular case of Corollary 2.12 but the proof in thiscase is much easier so we give it here. Denote by c a generator of C . Bythe pro- p version of the Kurosh subgroup Theorem, [15, Thm. D.3.8]we have F = H ∐ p − a j =0 H c j ! . Factoring out [
F, F ] one arrives at the desired decomposition. (cid:3)
We shall need the following proposition, where tor ( G ) means the setof non-trivial torsion elements. Proposition 3.2. [23, Prop 1.7 (i)]
Let G be a virtually free pro- p group and N a normal subgroup of G generated by torsion. Then tor ( G ) N/N = tor ( G/N ) , where tor ( G ) stands for the torsion of G . Lemma 3.3.
Let G = F ⋊ H be a semidirect product of a finite p -group H and a free pro- p group F . Suppose every torsion element of G is conjugate into H . If C is central subgroup of order p in H , then G/C G = ¯ F ⋊ H/C with ¯ F free pro- p and every finite subgroup of G/C G is conjugate to H/C .Proof.
By Proposition 3.2 every torsion element k of G/C G is the imageof some torsion element g of G . By hypothesis g is conjugate into H so k is conjugate into H/C . By Lemma 3.1(i) ¯ F = F C G /C G is freepro- p . (cid:3) Theorem 3.4.
Let G = F ⋊ H be a semidirect product of a finite p -group H and a free pro- p group F . Suppose every torsion element of G is conjugate into H . Then F ab is a Z p [ H ] -permutation module.Proof. We use induction on the order of H . Let C be a central sub-group of H of order p . By Lemma 3.1 F ab is Z p [ C ]-permutation with F ab = M ⊕ M p , where M is trivial and M p is a free Z p [ C ]-module andmoreover, M is the image of C F ( C ) in F ab . Since C F ( C ) is H -invariantso is M .We want to apply Theorem 2.6 to show that F ab is a Z p [ H ]-permutationmodule and we showed in the previous paragraph that the hypothesis(i) of Theorem 2.6 is satisfied. Consider the commutative diagram F / / (cid:15) (cid:15) F C/C G (cid:15) (cid:15) F ab / / F abC , where the bottom row is just the abelianization of the upper one. Wecan apply Lemma 3.3 and the induction hypothesis to G/C G to deducethat F abC is an Z p [ H/C ]-permutation module. Now F abC /M ∼ = ( M p ) C and by Lemma [11, Lemma 2.4] ( M p ) C ∼ = M Cp , so ( F ab ) C ∼ = F abC is apermutation Z p [ H/C ]-module.Thus the hypothesis (ii) of Theorem 2.6 is also satisfied and so F ab is a permutation Z p [ H ]-module. (cid:3) Permutational abelianization
Proposition 4.1.
Let G = F ⋊ C be a semidirect product of a group C = h c i of order p and a free pro- p group F . Then F ab is Z p [ C ] -permutation if and only if C is the unique subgroup of G of order p upto conjugation.Proof. ‘if’ is a particular case of Theorem 3.4.‘Only if’. By [23, Proposition 1.3] (or by Lemma 7.1) combinedwith Lemma 3.1 G embeds into a pro- p group G = ( C p × H ) ∐ H ,where C p is of order p and H, H are free pro- p . Thus w.l.o.g we mayassume that G ≤ G and C = C p . By the pro- p version of the Kuroshsubgroup theorem (see [15, Theorem 9.1.9]) any open subgroup U of G containing G admits a decomposition U = ` g ∈ S U ( U ∩ ( C × H ) g ) ∐ H U for some free pro- p group H U , where S U is the set of representatives of U \ G / ( C × H ). Let T be a subgroup of G order p which is not conjugateto C . Then we can choose U such that C and T are not conjugate in U and in fact assume w.l.o.g T = C g for some g ∈ S U . Thus we canregard C ∐ T as a subgroup of G and U . Put ¯ F = F ∩ ( C ∐ T ) and notethat it is finitely generated. Then the natural splitting epimorphism U −→ C ∐ T = ¯ F ⋊ C restricted to F gives the following commutativediagram ¯ F / / (cid:15) (cid:15) F / / (cid:15) (cid:15) ¯ F (cid:15) (cid:15) ¯ F ab / / F ab / / ¯ F ab with upper and lower composition maps being identity. So ¯ F ab is afinitely generated direct summand of F ab and by [5, Lemma 6] ¯ F ab is not NFINITELY GENERATED VIRTUALLY FREE PRO- p GROUPS AND p -ADIC REPRESENTATIONS11 a permutation Z p [ C ]-module. By Proposition 2.3(ii) this contradictsto the hypothesis that F ab is Z p [ C ]-permutation. (cid:3) From now on we shall assume in this section that G = F ⋊ H with H = { } and F ab being a Z p [ H ]-permutation module, i.e. F ab = L e ∈ E M e , where M e = Q x ∈ X e Z p [ H/H e ] = Z p [ H/H e ] ˆ ⊗ Z p [[ X e , ∗ ]] and { H e , e ∈ E } is a set of representatives of conjugacy classes of certainsubgroups of H and ( X e , ∗ ) is the one point compactification of X e .Thus we view X e as a subset of M e and in fact ( X e , ∗ ) is a pointedbasis of M e as explained in Remark 2.4.We shall relate centralizers of subgroups of H in F with the Z p [ H ]-module structure of F ab . Lemma 4.2.
If there exists e ∈ E such that H e = H for some e ∈ E , then the decomposition F ab = L e ∈ E M e can be rearranged suchthat C F ( H )[ F, F ] / [ F, F ] = M e = Z p [[ X e , ∗ ]] is a direct Z p [ H ] -trivialsummand of F ab .Proof. We argue by induction on H using Proposition 4.1 as the baseof induction. Let C be a central subgroup of order p in H . ThenRes HC ( F ab ) is a permutation Z p [ C ]-module and so by Proposition 4.1 F C has only one conjugacy class of groups of order p . Therefore byLemma 3.1 (ii) F ab = M ⊕ M p , where M = C F ( C ) ab = C F ( C )[ F, F ] / [ F, F ]is a trivial Z p [ C ]-module and M p is a free Z p [ C ]-module generated by X C [ F, F ] / [ F, F ]. Moreover, M is Z p [ H ]-submodule since C F ( C ) is H -invariant. Note also that M C is Z p [ H/C ]-permutation since M is Z p [ H ]-permutation (see Remark 2.7). Hence by Theorem 2.6 M is a di-rect summand of F ab . Then taking into account that Z p [ H/H e ] is trivial Z p [ C ]-module if and only if C ≤ H e one deduces from Proposition 2.3(ii) that the decomposition F ab = L e ∈ E M e can be rearranged suchthat M = L C ≤ H e M e . Now C F ( H ) ≤ C F ( C ), HC F ( C ) /C = C F ( C ) ⋊ H/C and C F ( C ) ab = M , so by induction hypothesis C F ( H )[ F, F ] / [ F, F ] = M e as required. (cid:3) Proposition 4.3.
Let ( X e , ∗ ) be a pointed profinite basis of M e , e ∈ E .Then the decomposition L e ∈ E M e can be rearranged such that X e ⊂ C F ( H e )[ F, F ] / [ F, F ] . (1) for all e ∈ E . Proof.
We use induction on | H | .If H = H e ∗ for some e ∗ ∈ E then by Lemma 4.2 F ab = L e ∈ E M e canbe rearranged such that C F ( H )[ F, F ] / [ F, F ] = M e ∗ , is a Z p [ H ]-trivialdirect summand of F ab . So we are done if E = { e ∗ } .Let K be a subgroup of H of index p . If H e ≤ K , then Res HK ( M e ) = Q x ∈ X e L r ∈ R Z p [[ K/H re ]], where R is a set of coset representatives of H/K . Otherwise, Res HK ( M e ) = Q x ∈ X e Z p [[ K/H e ∩ K ]]. So putting E K = { e ∈ E | H e ≤ K } one hasRes HK ( F ab ) = M e ∈ E K Y x ∈ X e M r ∈ R Z p [[ K/H re ]] ⊕ M e ∈ E \ E K Y x ∈ X e Z p [[ K/H e ∩ K ]] . By induction hypothesis (applied to
F K ) this decomposition canbe rearranged such that X e ⊂ C F ( H e )[ F, F ] / [ F, F ] for every e ∈ E K (in fact rX e ⊂ C F ( H re )[ F, F ] / [ F, F ] for every r ∈ R and e ∈ E , butwe need it only for r = 1). Since L r ∈ R Q x ∈ X e Z p [[ K/H re ]] = M e thestatement is proved for every e ∈ E K .Let E be a maximal subset of E such that there exists a decompo-sition of F ab with (1) satisfied for all e ∈ E . Since for a given e = e ∗ there exists K of H of index p containing H e , the preciding paragraphshows that if E = { e ∗ } then E is not empty.Pick e ′ ∈ E \ E . Let K be a subgroup of index p in H containing H e ′ . Then by above there is a maximal direct summand M ′ e ′ of F ab isomorphic to M e ′ satisfying the statement of the proposition. Then byProposition 2.3 (ii) F ab = L e = e ′ M e ⊕ M ′ e ′ contradicting the maximalityof E . Thus E = E and the proposition is proved. (cid:3) Special HNN-extensions
We are ready to prove Theorem 1.2, where we shall keep the notation M e = Q x ∈ X e Z p [ H/H e ] viewing X e as a subset of M e (cf. Remark 2.4). Theorem 5.1.
Let G = F ⋊ H be a semidirect product of a finite p -group H and a free pro- p group F . Suppose F ab = L e ∈ E M e is Z p [ H ] -permutation module. Then G = HNN( H, H e , X e ) is a specialHNN-extension with the base group H .Proof. By Proposition 4.3 the decomposition L e ∈ E M e can be rear-ranged such that for all e ∈ E the pointed profinite basis ( X e , ∗ ) of M e is contained in C F ( H e )[ F, F ] / [ F, F ]. Since the natural homomor-phism C F ( H e ) −→ C F ( H e )[ F, F ] / [ F, F ] admits a continuous section(see [15, Lemma 5.6.5]) the injective continuous map η abe : ( X e , ∗ ) −→ M e ; x −→ H e ˆ ⊗ x defined in Remark 2.4 lifts to a continuous map NFINITELY GENERATED VIRTUALLY FREE PRO- p GROUPS AND p -ADIC REPRESENTATIONS13 η e : ( X e , ∗ ) −→ C F ( H e ). Let ˜ G = HNN( H, H e , X e ) be a special HNN-extension and f : ˜ G −→ G be a homomorphism given by the uni-versal property that extends η e and sends H identically to H . Let˜ F = f − ( F ). Then by Corollary 2.12 ˜ F ab = L e ∈ E M e , such that X e ⊂ C ˜ F ( H e )[ ˜ F , ˜ F ] / [ ˜ F , ˜ F ] for every e ∈ E . Thus one has the com-mutative diagram ˜ F ❍❍❍❍❍❍❍❍❍❍ f / / F { { ✈✈✈✈✈✈✈✈✈✈ L e ∈ E M e implying that f | ˜ F induces an isomorphism on the abelianizations andtherefore f has to be an isomorphism. (cid:3) Combining this theorem with Theorem 3.4 we deduce the following
Corollary 5.2.
Let G = F ⋊ H be a semidirect product of finite p -group H and free pro- p group F . Suppose every torsion element of G is F -conjugate into H . Then G is a special HNN-extension with thebase group H . We are now ready to prove Theorem 1.3
Theorem 5.3.
Let F ⋊ H be a semidirect product of a free pro- p group F and a finite p -group H . Then the action of H extends to the actionon some free pro- p group ˜ F containing F such that H permutes theelements of some (pointed) basis of ˜ F .Proof. Use an embedding G = F ⋊ H −→ ˜ G = ˜ F ⋊ H (see Lemma7.1). Every torsion element of ˜ G is ˜ F -conjugate into H so ˜ G =HNN( H, H e , X e ), e ∈ E is a special HNN-extension by Corollary 5.2.By Lemma 2.11 ˜ F = a e ∈ E a r ∈ R e a s ∈ S e F ( X e , ∗ ) s − r − , is a free pro- p product of free pro- p groups F ( X e , ∗ ) on the pointedbasis ( X e , ∗ ) where R e , S e are coset representative sets of H/N H ( H e )and N H ( H e ) /H e respectively. Since H e centralizes X e , the action of H on E by conjugation permutes the free factors F ( X e , ∗ ) s − r − with [ e ∈ E [ r ∈ R e [ s ∈ S e ( X s − r − e , ∗ ) , being the H -invariant pointed basis of ˜ F . (cid:3) Now Theorem 1.1 follows easily.
Theorem 5.4.
Let G be a virtually free pro- p group. Then G acts ona pro- p tree with finite vertex stabilizers.Proof. Let F be an open normal subgroup of G . We can embed G into G = G ∐ G/F = F ⋊ G/F , where F is the kernel of the naturalepimorphism G −→ G/F induced by the natural epimorphism G −→ G/F and the identity map on the second factor. Note that by thepro- p version of the Kurosh subgroup theorem [15, Theorem 9.1.9] F is free pro- p . By Theorem 5.3 G can be embedded in a semidirectproduct ˜ G = ˜ F ⋊ G/F such that
G/F permutes the elements of some(pointed) basis of ˜ F and so ˜ F ab is a Z p [ G/F ]-permutation module.Hence by Theorem 5.1 ˜ G is a special HNN-extension with finite basegroup G/F . Therefore according to Remark 2.10 ˜ G and hence G actson a pro- p tree with finite vertex stabilizers. (cid:3) We finish this section with the result on virtually free pro- p groupsthat resembles the statement of the Weiss theorem for modules. Theorem 5.5.
Let G = F ⋊ H be a semidirect product of finite p -groupand free pro- p group H . Let N be a normal subgroup of H such that(i) all torsion elements of F N are conjugate into N ;(ii) all torsion elements of G/N G are conjugate into HN G /N G .Then G is a special HNN-extension with base group H . In particular,all torsion elements of G are conjugate into H .Proof. We shall use induction on | N | . Suppose | N | = p . Then byLemma 3.1 Z p [ N ]-module M = F ab has a decomposition M ⊕ M p suchthat M is trivial and M p is a free Z p [ N ]-module and moreover M is Z p [ H ]-invariant. Hence M N ∼ = M N . Then observing that N G = N F N we have the following commutative diagram: F / / (cid:15) (cid:15) ¯ F = F N/N G (cid:15) (cid:15) M / / M N , where by Lemma 3.3 ¯ F is free pro- p .Clearly the Z p -module M N = M ⊕ ( M p ) N .By Theorem 3.4 M N ∼ = M N = ¯ F ab is a permutation Z p [ H ]-moduleand so by Theorem 2.6 so is M . Hence by Theorem 5.1 G is a special NFINITELY GENERATED VIRTUALLY FREE PRO- p GROUPS AND p -ADIC REPRESENTATIONS15 HNN-extension of H . In particular, by [14, Cor. 7.1.3] all torsionelements of G are conjugate into H and in fact F -conjugate.Suppose now | N | > p and let C be a central normal subgroup of H contained in N . By Lemma 3.3 G/C G = ¯ F ⋊ H/C , where ¯ F = F C/C G is free pro- p and every torsion element of ¯ F ⋊ N/C is conjugateinto
N/C (since C G = C NF ). Hence by the induction hypothesis alltorsion elements of G/C G are conjugate into H/C . Then by the baseof induction all torsion elements of G are conjugate into H and byCorollary 5.2 G is a special HNN-extension. (cid:3) Finite centralizers of torsion
The main theorem of [11] states that a second countable virtuallyfree pro- p group having finite centralizers of the non-trivial torsionelements is a free pro- p product of finite groups and a free pro- p group.It is not true for virtually free pro- p groups of uncountable rank. Theobjective of this section is to show that a free pro-p groups of arbitraryrank embeds in a free pro- p product of a finite p -group and a free pro- p group. We shall also establish a criterion for its decomposition as afree pro- p product of finite p -groups and a free pro- p group.We shall begin with the following Lemma 6.1.
Let ˜ G = HNN( K, A i , X i , i ∈ I ) be a special pro- p HNN-extension of a finite p -group K and ˜ F be the normal closure of ( S i ∈ I X i , ∗ ) in ˜ G . Let I = { i ∈ I | A i = 1 } and F be a K -invariant freefactor of ˜ F such that C F ( A i ) = 1 for every i ∈ I . Then putting ˜ F K = h X i | i ∈ I i ˜ F we have F ∩ ˜ F K = 1 and ˜ G/ ˜ F K = F ∐ K forsome free pro- p group F .Proof. The second statement follows directly from the presentation ofa special HNN-extension. We shall induct on | I | + | K | to prove thefirst one.If K is of order p (and so | I | ≤
1) then by Lemma 3.1 ˜ G = ( K × F K ) ∐ F , where F K = h X i | i ∈ I i , F are free and ˜ F ab = M ⊕ M p ,where M is the trivial Z p [ K ]-module coinciding with the image of F K in ˜ F ab and M p is a free Z p [ K ]-module. Thus we have the followingcommutative diagram ˜ F / / (cid:15) (cid:15) F (cid:15) (cid:15) ˜ F ab / / M p . Since F is a K -invariant free factor of ˜ F , its abelianization F ab isa pure Z p [ K ]-submodule of ˜ F ab (i.e. a Z p -direct summand). By [13,Theorem B] F ab = M p − ⊕ L , where L is a free Z p [ K ]-module and M p − has no non-zero elements fixed by K . Since L is free it is a directsummand of ˜ F ab (see [1, Proposition 3.6.4]); the proof over Z p worksmutatis mutandis) and so by Proposition 2.3(ii) can be assumed to becontained in M p . Hence M ∩ F ab = 0 and so the bottom map of thediagram is injective on F ab . Therefore the upper map of the diagramis injective on F and the lemma is proved in this case.Suppose | K | > p and let H be a subgroup of index p in K . Put˜ F H = h X i | = A i ≤ H, i ∈ I i ˜ F . By the induction hypothesis appliedto ˜ F H we know that F ∩ ˜ F H = 1 and so the natural epimorphism˜ G −→ ˜ G/ ˜ F H restricted to G is injective. Let I H = { i ∈ I | A i ≤ H } . It follows from the presentation of a special HNN-extension that˜ G/ ˜ F H = HNN( K, A i , X i , i ∈ I \ I H } and so satisfies the premises of thelemma. So if I H = ∅ for such H then we deduce the result from theinduction hypothesis.Thus we left with the case when | I | = 1 and A i = K for i ∈ I . Inthis case ˜ G = ( K × F ( X i , ∗ )) ∐ F for some free pro- p group F . Let C be a normal subgroup of order p of K . Then C ˜ F ( C ) = C ˜ F ( K ) = F ( X i , ∗ ) (see [15, Theorem 9.1.12]). So ˜ F C satisfies the premises ofthe lemma and the result follows from the induction hypothesis appliedto ˜
F C in this case. (cid:3)
Theorem 6.2.
Let G = F ⋊ H be a semidirect product of a free pro- p group F and a finite p -group H . Suppose G have finite centralizersof the non-trivial torsion elements. Then G embeds into a free pro- p product H ∐ F of H and a free pro- p group F .Proof. Consider G as a subgroup of an HNN-extension˜ G = HNN( G, A i , φ i , X i ) , i ∈ I (Lemma 7.1). By Corollary 5.2 ˜ G is a special HNN-extension ˜ G =HNN( H, H e , X e ) , e ∈ E .Since G has finite centralizers of non-trivial torsion elements C F ( A i ) =1 for every i , so applying Lemma 6.1 we obtain the natural epimor-phism ˜ G −→ ˜ G/ h X i ) | A i = 1 i ˜ G = F ∐ H whose restriction to G is aninjection. The result follows. (cid:3) Corollary 6.3.
Let G be a pro- p group possessing an open normal freepro- p group F and having finite centralizers of the non-trivial torsion NFINITELY GENERATED VIRTUALLY FREE PRO- p GROUPS AND p -ADIC REPRESENTATIONS17 elements. Then G embeds into a free pro- p product G/F ∐ F of thefinite quotient group G/F and a free pro- p group F .Proof. Embed G into G = G ∐ G/F = F ⋊ G/F with F free pro- p containing F and apply Theorem 6.2. (cid:3) We are ready to prove Theorem 1.4.
Theorem 6.4.
Let G be a virtually free pro- p group having finite cen-tralizers of the non-trivial torsion elements. Then G embeds into a freepro- p product G = F ∐ H of a finite p -group H and a free pro- p group F .Proof. Let F be a core of an open free pro- p subgroup of G . Then F is open normal and we can apply Corollary 6.3. (cid:3) Let G be a pro- p group and { G x | x ∈ X } be a family of subgroupsindexed by a profinite space X . Following [14, Section 5.2] we say that { G x | x ∈ X } is continuous if for any open subgroup U of G the subset { x ∈ X | G x ⊆ U } is open. Lemma 6.5.
Let G be a virtually free pro- p group having finite cen-tralizers of the non-trivial torsion elements. Then the maximal finitesubgroups of G can be indexed by some profinite G -space X such thatthe family F = { G x | x ∈ X } of them is continuous.Proof. Let G = F ∐ H be a free pro- p product containing G fromTheorem 6.4. Consider the family { G x | x ∈ G /H } of the stabilizersof points of the profinite space G /H on which G acts from the left.By [12, 4.8 (3-d paragraph)] or [14, Lemma 5.2.2]) { G x | x ∈ G /H } is a continuous family of subgroups of G . Since G ∩ F is an openfree pro- p subgroup of G the subset { x ∈ G /H | G x ≤ F ∩ G } isopen and clearly G x F ∩ G iff G x = { } . Therefore the subset X = { x ∈ G /H | G x = { }} is closed and so one deduces fromthe definition of a continuous family that F = { G x | x ∈ X } is acontinuous family of subgroups of G . Clearly X is G -invariant, andsince G x = xHx − is a maximal finite subgroup of G , G x is a maximalfinite subgroup of G . The lemma is proved. (cid:3) We finish the section with the proof of Theorem 1.5.
Theorem 6.6.
Let G be a virtually free pro- p group having finite cen-tralizers of non-trivial torsion elements and X be the space from Lemma6.5. Then G is a free pro- p product of finite p -groups and a free pro- p group if and only if the natural quotient map θ : X −→ X/G withrespect to the action of G admits a continuous section. Proof. ‘If’ follows from the pro- p version of the Kurosh Subgroup The-orem ([12, Thm. 4.3]) applied to the subgroup G of G taking intoaccount that the existence of a continuous section s to θ is only whatis needed to remove the second countability hypothesis from its state-ment. See for example [12, Comment (5.1)] , where this is explicitelywritten. Alternatively one can use [14, Thm 9.6.1 (a)]) by consideringa natural action of G on the standard pro- p tree T for G and as inthe proof of (b) of this theorem one can use [14, Lemma 5.2.2]) andcontinuity of s to show that { G x | x ∈ s ( X/G ) } is a continuous family.‘Only if’. Let G = ` t ∈ T G t ∐ F , where { G t | t ∈ T } is a continuousfamily of finite subgroups of G . Let F be an open free pro- p subgroupof G . By [12, (1.2) Remark] or [14, Lemma 5.2.1] E = S t ∈ T G t is closedin G and so E = E \ F = E \ { } is closed, where F is an open freepro- p subgroup of G . Consider a continuous map η : G × X −→ X × X , η ( g, x ) = ( gx, x ). Let X be the projection of η ( E ) ∩ D on the firstcoordinate, where D is the diagonal of X × X . It suffices to showthat the restriction of θ to X is a homeomorphism. Clearly, θ | X iscontinuous, so we need to check bijectivity. Choose x = x ∈ X .Then η − ( x , x ) = G t × { x } and η − ( x , x ) = G t × { x } for some t = t ∈ T . Since G t and G t are not conjugate in G (see [14, Cor.7.1.5 (a)]) they stabilize points x and x in different G -orbits, so θ | X is injective. The surjectivity of θ | X is clear, since any finite group G x is conjugate to some G t in G see [14, Cor. 7.1.3]) and G t stabilizessome point x ∈ X , i.e. θ ( x ) = θ ( x ). (cid:3) Appendix
We shall give a construction from [4] that uses an HNN-extensionto embed a semidirect product G = F ⋊ H of a free pro- p group F and a finite p -group H into a semidirect product ˜ G = ˜ F ⋊ H of a freepro- p group ˜ F and the same group H such that all finite subgroups of˜ G conjugate into H . In particular it gives another proof of Proposition1.3 in [23].Let π : G −→ H be the natural projection; note that π restricted toany finite subgroup of G is an injection. Let F in ( G ) be the set of allfinite subgroups of G . Since the order of any finite subgroup of G doesnot exceed | H | and G is a projective limit of finite groups, Fin ( G ) isthe projective limit of the respective sets of subgroups of order ≤ | H | – hence it carries a natural topology (the subgroup topology ) – turningit into a profinite space. Equipped with this topology, Fin ( G ) with G acting by conjugation becomes a profinite G -space. NFINITELY GENERATED VIRTUALLY FREE PRO- p GROUPS AND p -ADIC REPRESENTATIONS19 Let A = { A i , i ∈ I } be the set of all subgroups of H . The pro-jection π induces a continuous surjection ρ : F in ( G ) −→ A . Put X i = ρ − ( { A i } ). Define φ i : A i × X i −→ G by setting φ i ( a, x ) to be theunique element a x of the group x ∈ X i such that π ( a x ) = a . Form anHNN-extension ˜ G := HNN( G, A i , φ i , X i ), i ∈ I (since X i are compact,the distinguished point from Definition 2.9 can be omitted). By [14,Theorem 7.1.2] (cf. also [14, Example 6.2.3 (e)]) each finite subgroupof ˜ G is conjugate into G and hence by construction into H . Note alsothat the natural epimorphism π : G −→ H extends by the universalproperty to ˜ π : ˜ G −→ H . By [4, Lemma 10] or [14, Theorem 9.6.1 (a)]˜ F = ker (˜ pi ) is a free pro- p products of conjugates of ˜ F ∩ G = F and afree pro- p group; thus ˜ F is free pro- p . Lemma 7.1.
The natural homomorphism G −→ ˜ G is an injection.Proof. Let ˜ G abs = HNN abs ( G, A i , φ i , X i ), i ∈ I be the abstract HNN-extension. Then it suffices to show that the natural homomorphism˜ G abs −→ ˜ G is injective. By Definition 2.9 ˜ G := HNN( G, A i , φ i , X i ), i ∈ I is defined to be the quotient of G ∐ F ( S i ∈ I X i ), where F ( S i ∈ I X i )is a free pro- p group on ( S i ∈ I X i ), modulo the relations φ i ( a i ) = x i a i x − i for all x i ∈ X i , i ∈ I . Thus we have the following commutative diagram G ⋆ F ( S i ∈ I X i ) / / (cid:15) (cid:15) G ∐ F ( S i ∈ I X i ) (cid:15) (cid:15) ˜ G abs / / ˜ G , where ⋆ means the abstract free product.Note that G ∐ F ( S i ∈ I X i ) is the pro- p completion of G ∗ F ( S i ∈ I X i )with respect to the family of normal subgroups N of finite index suchthat N ∩ G is open in G and ( S i ∈ I X i ) −→ ( S i ∈ I X i ) N/N is continuous.Therefore ˜ G is the pro- p completion of ˜ G abs with respect to the family ofimages of N in ˜ G abs , i.e. with respect to the family of normal subgroups U of finite index in ˜ G abs satisfying the same properties. Moreover,w.l.o.g we may assume that U ∩ G ≤ F .Choose an open normal subgroup V ⊳ o G of G contained in F . Thenthe natural epimorphism G −→ G/V induces the natural continuousmap ρ V : F in ( G ) −→ F in ( G/V ) and therefore continuous maps ρ V,i : X i −→ F in ( G/V ). Then by the universal property of abstract HNN-extensions ρ V,i and G −→ G/V extend to an epimorphism˜ ρ V : ˜ G abs −→ HNN abs ( G/V, A i V /V, ρ V ( X i )) . Note that the natural epimorphism π : G −→ H extends to the epi-morphism ˜ π abs : ˜ G abs −→ H (the samer way as to ˜ π ). Moreover, ˜ π abs factors through ˜ ρ V , i.e. we have the natural epimorphism ϕ V : HNN abs ( G/V, A i V /V, ρ V ( X i )) −→ H whose kernel is an abstract free product of F/V and an abstract freegroup (by the Kurosh subgroup theorem). Moreover, since V ≤ F ,HNN abs ( G/V, A i V /V, ρ V ( X i )) = ker ( ϕ V ) ⋊ HV /V and hence is resid-ually p (as an extension of a residually p group by a finite p -group).Therefore there exists an epimorphism ϕ V K : HNN abs ( G/V, A i V /V, ρ V ( X i )) −→ K to a finite p -group K injective on G/V . Then for the kernel U V K of ϕ V K ˜ ρ V we have U V K ∩ G is open in G and ( S i ∈ I X i ) −→ ( S i ∈ I X i ) U V K /U V K is continuous. 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P. A. Zalesski˘i, Department of Mathematics, University of Brasilia,70.910 Brasilia DF, Brazil
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