aa r X i v : . [ m a t h . G R ] N ov On accessibility for pro- p groups Gareth WilkesNovember 9, 2018
Abstract
We define the notion of accessibility for a pro- p group. We prove thatfinitely generated pro- p groups are accessible given a bound on the size oftheir finite subgroups. We then construct a finitely generated inaccessiblepro- p group, and also a finitely generated inaccessible discrete group whichis residually p -finite. A common thread in mathematics is the search for conditions which tame wildobjects in some way and allow deeper analysis of their structure. In the case ofgroups finiteness properties, finite dimension properties, and accessibility all fallinto this thread. Accessibility is the name given to the following concept. Givena group, it may be possible to split the group over a finite subgroup into pieceswhich could be expected to be somehow ‘simpler’. For a notion of ‘simpler’ tobe of much help, this process should terminate—it should not be possible tocontinue splitting the group ad infinitum . Once we have split as far as possible,the remaining pieces are one-ended and amenable to further study—in this wayaccessibility is the foundation of JSJ theory for groups.We adopt the following definition of accessibility, which is well known to beequivalent to the original classical definition of Wall [Wal71, SW79] in the caseof (finitely generated) discrete groups.
Definition 1.1.
Let G be a discrete group. We say G is accessible if there isa number n = n ( G ) such that any finite, reduced graph of discrete groups G with finite edge groups having fundamental group isomorphic to G has at most | EX | ≤ n edges.That is, ‘one cannot continually split G over finite subgroups’. Examples ofaccessible discrete groups include the finitely generated torsion free groups (byGrushko’s theorem), finitely generated groups with a bound on the size of theirfinite subgroups [Lin83], and all finitely presented groups [Dun85]. However,contrary to a famous conjecture of Wall [Wal71], not all finitely generated groupsare accessible [Dun93].Pro- p groups are a class of compact topological groups with many pleasantfeatures. See [DdSMS03] for an introduction. Being ‘built out of’ finite p -groups,the many special properties of p -groups may be used to analyse pro- p groups.At the same time the category of pro- p groups has surprising similarities to thecategory of discrete groups. There are close relations between the cohomology1heories of the two classes [Ser02], and pro- p groups have a good theory ofactions on trees whose results parallel the Bass-Serre theory for discrete groups[RZ00a, Rib17]. In this way one can often prove results for pro- p groups whichparallel those for discrete groups, even when the discrete groups are studiedgeometrically so that the techniques of the proof do not necessarily translateover. Examples include, for instance, [Lab67, Lub82, Wil17].In this paper we will commence the study of accessibility for pro- p groups.In Section 2 we will give formal definitions of graphs of pro- p groups and theirfundamental groups. We also study the concept of properness, a vital extracomponent in this theory not present in the discrete world. We then derivecertain properties of these notions which are necessary and expedient for therest of the paper.In Section 3 we will prove a pro- p version of Linnell’s theorem [Lin83]. Theorem 3.1.
Let G be a finitely generated pro- p group and let K be an integer.Let ( X, G • ) be a proper reduced finite graph of groups decomposition of G suchthat each edge group G e has size at most K . Then X has at most pKp − G ) −
1) + 1 edges.
In Section 4 we reach the primary purpose of the paper: the explicit con-struction of an inaccessible finitely generated pro- p group. Theorem 4.4.
There exists a finitely generated inaccessible pro- p group J . Finally in Section 5 we note that our construction gives an inaccessible dis-crete group which is residually p -finite. Theorem 5.1.
There exists a finitely generated, inaccessible, residually p -finitediscrete group. p groups In this section we will recall various notions concerning graphs of pro- p groups,and establish some properties of them which do not seem to appear in thestandard literature. In this paper an (oriented) graph X will consist of a set V X of vertices, a set EX of edges, and two functions d k : EX → V X for k ∈ { , } describing the endpoints of e . We will generally ignore the orientationwhen describing a graph, but it is occasionally useful when greater precision isnecessary. Definition 2.1.
A graph G = ( X, G • ) of pro- p groups consists of a finite graph X , a pro- p group G x for each x ∈ X , and monomorphisms ∂ e,k : G e → G d k ( e ) for each edge e ∈ EX and for k = 0 ,
1. A graph of groups is reduced if ∂ e,k isnever an isomorphism when e is not a loop. Remark.
One may define graphs of groups over arbitrary profinite graphs X (see[Rib17, Chapter 6]), but this adds an extra level of complication and subtletywhich is not required in the present paper.2 efinition 2.2. The full definition of a fundamental group of an arbitrarygraph of pro- p groups may be found in [Rib17, Section 6.2]. We give here asimplified version [Rib17, Example 6.2.3(c)] appropriate for the study of finitegraphs of pro- p groups.Let G = ( X, G • ) be a finite graph of pro- p groups and choose a maximalsubtree T of X . A specialisation of G is a triple ( H, ν • , t • ) consisting of a pro- p group H , morphisms ν x : G x → H for each x ∈ X , and elements t e ∈ H for e ∈ EX , such that: • t e = 1 for all e ∈ ET • ν d ( e ) ( ∂ e, ( g )) = ν e ( g ) = t e ν d ( e ) ( ∂ e, ( g )) t − e for all e ∈ EX, g ∈ G e .A pro- p fundamental group of G is a ‘universal’ specialisation—that is, aspecialisation ( H, ν • , t • ) of G such that for any other specialisation ( K, β • , t ′• ) of G there exists a unique morphism φ : H → K such that φν x = β x for all x ∈ X and φ ( t e ) = t ′ e for all e ∈ E . Note that the ν x are not required to be injections.We denote the pro- p fundamental group by Π ( G ) or Π ( X, G • ). We alsouse the notation G x ∐ G e G y for the fundamental group of a graph of groupswith two vertices x and y and one edge e .An equivalent, but slightly less formal, definition would be to define Π ( G )to be the pro- p group given by the pro- p presentation (cid:10) G x ( x ∈ X ) , t e ( e ∈ EX ) (cid:12)(cid:12) t e = 1 (for all e ∈ ET ) ∂ e, ( g ) = g = t e ∂ e, ( g ) t − e (for all e ∈ EX, g ∈ G e ) (cid:11) The fundamental pro- p group of a graph of groups always exists [Rib17, Propo-sition 6.2.1(b)] and is independent of the choice of T [Rib17, Theorem 6.2.4]. Remark.
One can also consider a graph of pro- p groups as a graph of profinitegroups, and consider the above definition of ‘fundamental group’ in the categoryof all profinite groups. The profinite fundamental group is not in general isomor-phic to the pro- p fundamental group, and the two concepts should be carefullydistinguished if both are present. However in this paper we only consider pro- p fundamental groups, so we will not clutter the notation by distinguishing it fromthe profinite fundamental group.Let G = ( X, G • ) be a finite graph of finite p -groups. We may consider G as a graph of pro- p groups as above, or we may consider it as a graph of finitediscrete groups and consider the fundamental group of G in the category ofdiscrete groups [Ser03, Section 5.1]. We denote the discrete fundamental groupby π ( G ) or π ( X, G • ). The pro- p presentation above makes clear the followingrelation between the discrete and pro- p fundamental groups. Proposition 2.3 (Proposition 6.5.1 of [Rib17]) . Let G = ( X, G • ) be a finitegraph of finite p -groups. Then Π ( G ) is naturally isomorphic to the pro- p com-pletion of π ( G ) . In Definition 2.2 we noted that the natural maps G x → Π ( X, G • ) are notrequired to be injections. This is in sharp contrast to the situation for graphs ofdiscrete groups, where injectivity is automatic. However graphs of groups aremost useful when this injectivity does hold, so we give this property a name.3 efinition 2.4. A graph of pro- p groups G = ( X, G • ) is proper if the naturalmaps G x → Π ( G ) are injections for all x ∈ X .A proper, reduced graph of pro- p groups whose fundamental pro- p group isisomorphic to a pro- p group G will be referred to as a splitting of G . Remark.
There is a bifurcation of terminology in the literature. One finds dis-cussions of ‘injective’ graphs of groups [Rib17] and of ‘proper’ amalgamated freeproducts and HNN extensions [RZ00b] (which are of course the basic examplesof graphs of groups). There are also papers with the convention that only properamalgamated free products ‘exist’ [Rib71]. The preference of the present authoris to use the word ‘proper’ uniformly.For a graph of finite p -groups G = ( X, G • ), properness is equivalent both tothe existence of a finite quotient of Π ( G ) into which all G x inject and to theproperty that the discrete fundamental group π ( G ) (that is, the fundamentalgroup when considered as a graph of discrete groups) is residually p . Thereare classical criteria for this property in the case of one-edge graphs of groups[Hig64, Cha94]. Criteria for more general graphs of groups also exist [AF13,Wil18]. Definition 2.5.
Let G be a pro- p group. We say G is accessible if there is anumber n = n ( G ) such that any finite, proper, reduced graph of pro- p groups G with finite edge groups having fundamental group isomorphic to G has at most | EX | ≤ n edges.Generally speaking, proper graphs of groups are the only ones worth con-sidering. An illustration of this principle in the context of accessibility will begiven later (Example 4.1).We define an ‘inverse system of graphs of groups’ in the following way. Fixa graph X and an inverse system ( I, ≺ ). An inverse system of graphs of groups( G i ) i ∈ I is a family of graphs of groups G i = ( X, G i, • ) together with surjectivetransition maps G i,x → G j,x for all x ∈ X and i ≻ j , such that the diagrams G i,e G i,d k ( x ) G j,e G j,d k ( x ) commute for all e ∈ EX , k = 0 , i ≻ j . There is then a natural inverselimit graph of groups H = ( X, H • ) where H x = lim ←− i G i,x for each x ∈ X . Proposition 2.6.
Take an inverse system of graphs of groups G i = ( X, G i, • ) with inverse limit graph of groups H = ( X, H • ) .1. Taking fundamental groups commutes with inverse limits—that is, we have lim ←− Π ( G i ) = Π ( H ) .2. If each G i is proper then H is proper.Proof. By the universal property defining Π ( H ), for each i there is a nat-ural surjection Ψ i : Π ( H ) → Π ( G i ). Hence we have a natural surjectionΨ : Π ( H ) → lim ←− Π ( G i ). 4o verify that Ψ is an isomorphism, take h ∈ Π ( H ) r { } . There is somefinite p -group K and a map φ : Π ( H ) → K such that φ ( h ) = 1. For each x ∈ X ,the natural map lim ←− G i,x → Π ( H ) → K is continuous, and hence factors through some G i x ,x . Since X is a finite graph,we may choose j ∈ I such that j ≻ i x for all x . The natural maps G j,x → K arecompatible with the monomorphisms in the definition of the graph of groups G i , and hence give rise to a natural map Π ( G i ) → K through which φ factors.It follows that Ψ i ( h ) = 1, so that Ψ( h ) = 1 and we have proved part 1 of thetheorem. lim ←− G i,x Π ( H ) Π ( G j ) G j,x K Ψ i φ ∃ The second part of the theorem holds because the map lim ←− G i,x → Π ( H ) is theinverse limit of the maps G j,x → Π ( G j ) and because inverse limits of injectivemaps are injective.When working with graphs of groups one very often conducts proofs in aninductive fashion—for instance by proving a statement for a subgraph of groupswith one fewer edge and then adding in the remaining edge. We will thereforeinclude the following warning: properness is a property of an entire graph ofgroups, and not a property which may be studied edge-by-edge. We give thefollowing example. The graph is simply a line segment with three vertices andtwo edges. The amalgamation over either edge is individually proper, but thefull graph of groups is improper. Example . Take four copies G , . . . , G of the mod- p Heisenberg group (cid:10) x, y (cid:12)(cid:12) x p = y p = [ x, y ] p = 1 , [ x, y ] central (cid:11) whose given generating sets shall be denoted { x , y } , . . . , { x , y } . Furthermoretake two copies H and H of F p , with generating sets { u , v } and { u , v } respectively. Define a graph of groups G G × G G H H (1)with inclusions given by x u [ x , y ] x u [ x , y ][ x , y ] v x [ x , y ] v x This graph of groups is improper: the identifications x = [ x , y ] et cetera inits fundamental group imply a relation x = [[[[ x , y ] , y ] , y ] , y ]5o that x may be expressed as a commutator of arbitrary length, and thereforemust vanish in a pro- p group. So G cannot inject into the fundamental pro- p group of the graph of groups (1). However either amalgamation over just oneedge is proper, as may be readily verified using Higman’s criterion [Hig64].There is thus a warning attached to working with sub-graphs of groups insome cases. However once properness of the whole graph of groups is estab-lished one may indeed manipulate subgraphs in the expected manner. The nextproposition effectively allows us to ‘bracket’ sections of a proper graph of groupsfor individual study.Let X be a connected finite graph and let G = ( X, G • ) be a proper graphof pro- p groups with fundamental group G = Π ( G ). Let Y be a connectedsubgraph of X . Let Z = X/Y be the quotient graph, where we denote theimage of a point x ∈ X r Y by [ x ] and denote the image in Z of Y by [ Y ]. Let G| Y = ( Y, G • ) be the subgraph of groups over Y and denote its fundamentalpro- p group by G Y . Note that G| Y is proper, as the universal property defining G Y immediately gives a commuting diagram G x G Y G for each x ∈ Y . We may define a graph of groups H = ( Z, H • ) by setting H [ x ] = G x for each x ∈ X r Y and setting H [ Y ] = G Y . Note that properness of G| Y implies that the natural maps from an edge group G e of G into G Y , for e having an endpoint in Y , are indeed injections. Proposition 2.8.
With notation and conditions as above, H is proper and hasfundamental group naturally isomorphic to G .Proof. The isomorphism Π ( H ) ∼ = Π ( G ) holds because the two groups may bereadily seen to satisfy the same universal property. It remains to show that H is proper—i.e. that G Y = Π ( Y, G • ) injects into G = Π ( G ).We first show that the proposition is valid for graphs G = ( X, G • ) of finite p -groups. In this case we may also consider G as a graph of discrete groups. Wehave a commuting diagram π ( Y, G • ) π ( G )Π ( Y, G • ) Π ( G )where, by Proposition 2.3, the lower half of the square is the pro- p completionof the upper half.The top horizontal map is an injection by the theory of graphs of discretegroups—for example, this follows from the expression of elements as reducedwords [Ser03, Section I.5.2]. This alone is not enough to show that Π ( Y, G • )injects into Π ( G ): we need further separability properties of π ( Y, G • ) in π ( G ).Specifically [RZ00b, Lemma 3.2.6] we must prove that for each normal subgroup N of π ( Y, G • ) of index a power of p , there is a normal subgroup M of π ( G )whose intersection with π ( Y, G • ) is contained in N . It is enough to prove this6roperty instead for F ∩ π ( Y, G • ) and F for F a normal subgroup of π ( G )with index a power of p .Since G is proper, there is a map φ : π ( G ) → K , for K a finite p -group,whose restrictions to all G x are injections. Then F = ker( φ ) is a free group, ofwhich F ∩ π ( Y, G • ) is a free factor and therefore there is a retraction ρ : F → F ∩ π ( Y, G • ) (see [Ser03, Section I.5.5, Theorem 14] and the remarks followingit). Then for N a normal subgroup of F ∩ π ( Y, G • ) with index a power of p ,the subgroup M = ρ − ( N ) has intersection exactly N with π ( Y, G • ) and weare done.We now deduce the general case. Let U ⊳ o G be any open normal subgroupof G . We can then form a graph of groups G /U = ( X, G • /G • ∩ U ) with finiteedge groups, which is proper since there is a map to the finite p -group G/U restricting to an injection on each G x /G x ∩ U . The universal properties of thevarious graphs of groups then give a natural commuting diagramΠ ( Y, G • ) Π ( X, G • )Π ( Y, G • /G • ∩ U ) Π ( X, G • /G • ∩ U )where the injectivity of the lower arrow follows from the first part of the proof.By naturality and Proposition 2.6, the top arrow is the inverse limit of thebottom arrows over all U ⊳ o G . An inverse limits of injections is an injection,thus proving the theorem. Linnell [Lin83] proved that finitely generated discrete groups satisfy a form ofaccessibility provided one imposes a bound on the size of all edge groups ratherthan simply requiring them to be finite. In this section we prove a similar resultfor pro- p groups. The proof uses residual finiteness to deduce the theoremfrom the case of trivial edge groups, and proceeds using the duality betweenfinite graphs of pro- p groups and actions on pro- p trees. This duality is entirelyanalogous to the classical duality of Bass-Serre theory for discrete groups [Ser03].See [Rib17, Sections 6.4 and 6.6] for an account of the relavant theory. Theorem 3.1.
Let G be a finitely generated pro- p group and let K be an integer.Let ( X, G • ) be a proper reduced finite graph of groups decomposition of G suchthat each edge group G e has size at most K . Then X has at most pKp − G ) −
1) + 1 edges.Proof.
Since the graph of groups is proper, we may treat the G x (for x ∈ X )as (closed) subgroups of G . For each e ∈ EX and each endpoint x of e suchthat G e = G x , there is a finite quotient of G such that the image of G x is notequal to the image of G e . Furthermore the G e are finite. Therefore there existsa finite quotient φ : G → P , with kernel H , with the properties that φ | G e is an7njection for all e ∈ EX and such that φ ( G e ) = φ ( G x ) if x is an endpoint of e for which G e = G x .The action of H on the G -tree T dual to the graph of groups ( X, G • ) givesa graph of groups decomposition ( Y, H • ) of H where Y = T /H . The groups H • are given by H -stabilizers in T , so that for y ∈ Y which maps to x ∈ X under the quotient map T /H → T /G , the group H y is a G -conjugate of H ∩ G x .In particular, since the restriction of φ to G e is injective, the graph of groups( Y, H • ) has trivial edge groups. Furthermore P acts on Y with quotient X , andwith stabilisers which are conjugates of G x /H ∩ G x = φ ( G x ). Thus x ∈ X has | P/φ ( G x ) | preimages in Y , each with group H y isomorphic to H ∩ G x .Factoring out the H y gives a surjection from H to a free group of rank1 − χ ( Y ), giving an inequality1 − χ ( Y ) ≤ rk( H ) . On the other hand, since H is an index | P | subgroup of G , its rank must bebounded above by rk( H ) ≤ (rk( G ) − | P | + 1Calculating χ ( Y ) using the vertex and edge counts of Y from above and can-celling a common factor of | P | gives the inequalityrk( G ) − ≥ X e ∈ EX | G e | − X v ∈ V X | φ ( G v ) | To tame the negative term, choose a maximal rooted subtree Z of X and directit away from the root. Then each vertex x of X other than the root r is theterminal point d ( e ) of exactly one incoming edge e in Z . Since ( X, G • ) isreduced, the only edges for which G e may equal an endpoint are loops, andtherefore are not in Z . Hence p | G e | ≤ | φ ( G t ( e ) ) | for each e ∈ Z . Combining these inequalities, we findrk( G ) − ≥ X e/ ∈ Z G e + X e ∈ Z (cid:18) − p (cid:19) | G e | − φ ( G r )Finally take some edge e adjacent to r . If e ∈ Z then once again p | G e | ≤ | φ ( G r ) | Otherwise, | G e | ≤ | φ ( G r ) | . Either way, one of the | EX | positive terms aboveis at least as great as the final negative term. We gather the other | EX | − | G e | ≤ K for all e , to obtain the final inequalityrk( G ) − ≥ (cid:18) − p (cid:19) | EX | − K as required. 8 emark. From the above calculations one may also find an inequality pp − G ) ≥ X e ∈ EX | G e | . This should be compared with the inequality2d Q G ( Q g ) − ≥ X e ∈ EX | G e | found by Linnell [Lin83, Theorem 2], where Q g = ker( Q G → Q ) is the augmen-tation ideal. p group In this section we will adapt Dunwoody’s construction of an inaccessible discretegroup [Dun93] to the pro- p category to prove the existence of an inaccessiblefinitely generated pro- p group. First let us briefly recall the construction from[Dun93]. Take a diagram of groups as follows, where arrows denote properinclusions. G G G · · · K K K H H H · · · Assume the following hypotheses: • G is finitely generated; • for all i , the group G i +1 is generated by K i and H i +1 ; and • each K i is finite.Then the group P defined as the fundamental group of the infinite graph ofgroups G G G · · · K K K (2)has arbitrarily large splittings over finite groups. It may however not be finitelygenerated. This problem is overcome by noting that the countable group H ∞ = S H n embeds in some finitely generated group H ω . Then by construction J = P ∗ H ∞ H ω is generated by G and H ω , and admits splittings G · · · G n (cid:0) G n +1 G n +2 · · · (cid:1) H ωK K n − K n K n +2 K n +3 H ∞ for all n . In applying this scheme to pro- p groups, there are several additionalconsiderations to worry about. 9. Each graph of groups involved must be confirmed to be proper. If improper‘splittings’ are allowed, the theory quickly becomes absurd. Example p -groups withfundamental group F p ∐ F p ) . Let G n be a copy of the mod- p Heisenberggroup G n = (cid:10) a n , b n (cid:12)(cid:12) a pn = b pn = [ a n , b n ] p = 1 , [ a n , b n ] central (cid:11) and let K n be a copy of F p generated by u n and v n . Take inclusions K n ֒ → G n − , u n b n − , v n [ a n − , b n − ] K n ֒ → G n , u n [ a n , b n ] , v n a n and form the graph of groups G G G · · · G NK K K K N − for N >
1. In the fundamental group G of this graph of groups we haveidentities b n − = [ a n , b n ] , [ a n − , b n − ] = a n for 2 ≤ n ≤ N . Iterating these relations we find that a n and b n − may beexpressed as commutators of arbitrary length; in the pro-nilpotent group G this forces them to be trivial. Thus G is generated only by the order p elements a and b N and is readily seen to be F p ∐ F p .2. Not every increasing union of finite p -groups embeds in a finitely generatedpro- p group. Furthermore, care must be taken to embed not only the unionof the H i in a finitely generated pro- p group, but also their closure in P .Two embeddings of such a union in different pro- p groups could well havenon-isomorphic closures. Example p -groups not embeddable inany pro- p group) . Consider Γ = { z ∈ C | z p n = 1 for some n } , which is aunion of finite cyclic p -groups. We claim that any map from Γ to a pro- p group is trivial. It suffices to check this for finite p -groups. Let φ : Γ → P be a map to a finite p -group. Each g ∈ Γ has a | P | th root h , so that φ ( g ) = φ ( h ) | P | = 1. Example p groups generated by the same union of fi-nite p -groups) . Consider the following two abelian pro- p groups. Let H ′ = F p [[ Z p ]] be the free F p module on the profinite space Z p . Let e Z be Alexandroff compactification of Z —the set Z compactified at a singlepoint ∗ —and let H = F p [[( e Z , ∗ )]] be the free F p -module on the pointedprofinite space ( e Z , ∗ ). See [RZ00b, Chapter 5] for information on freeprofinite modules. By examining the standard inverse limit representa-tions F p [[( e Z , ∗ )]] = lim ←− k F p [( {− k . . . , k, ∗} , ∗ )] , F p [[ Z p ]] = lim ←− n F p [ Z /p n ]of these two groups one verifies that they are generated by the union ofthe finite subgroups F p [ {− k, . . . , k } ].10owever there is no isomorphism from H to H ′ preserving this familyof subgroups. Since 0 is an isolated point of ( e Z , ∗ ) there is a projection H → F p [ { } ] killing the other factors F p [ { k } ] for k = 0. However in theindexing set for H ′ the point 0 is in the closure of Z r { } , hence any mapkilling all the factors F p [ { k } ] for k = 0 must also kill F p [ { } ].3. In dealing with graphs of profinite groups, not only the groups but alsothe graphs involved must be profinite to retain a sensible topology. Henceforming a graph of groups such as (2) is an invalid operation. Our con-struction will use an inverse limit of finite graphs of groups. This canalso be rephrased as a profinite graph of pro- p groups which is a certain‘compactification’ of the graph of groups (2), though we will not make thisexplicit. Now we will describe the promised inaccessible pro- p group. We will constructa diagram of pro- p groups as in Dunwoody’s construction, together with re-traction maps which will allow us to define an inverse limit. Assume for thisdiscussion that all graphs of groups given are proper—we will verify this in thenext subsection.First define the map µ n : { , . . . , p n +1 − } → { , . . . , p n − } by sending an integer to its remainder modulo p n . Define H n = F p [ { , . . . , p n − } ]to be the F p -vector space with basis { h , . . . , h p n − } . There are inclusions H n ⊆ H n +1 given by inclusions of bases, and retractions η n : H n +1 → H n defined by h k h µ n ( k ) .Note also that there is a natural action of Z /p n on H n given by cyclicpermutation of the basis elements, and that these actions are compatible withthe retractions η i . The inverse limit of the H n along these retractions is H ∞ = F p [[ Z p ]]. Since the positive integers are dense in Z p , the group H ∞ is generatedby the images of the H n under inclusion. The continuous action of Z p on thegiven basis of H ∞ allows us to form a sort of ‘pro- p lamplighter group’ H ω = F p [[ Z p ]] ⋊ Z p = lim ←− ( H n ⋊ Z /p n )which is a pro- p group into which H ∞ embeds. As will be seen later, H ω isfinitely generated.Next set K n = F p × H n = h k n i × H n . Set G = K × F p . For n >
1, let G n be the pro- p group with presentation G n = (cid:10) k n − , k n ,h , . . . , h p n − | k pi = h pi = 1 , h i ↔ h j ,k n − ↔ h i for all i = p n − , k n = [ k n − , h p n − ] central (cid:11) where ↔ denotes the relation ‘commutes with’. One may show that in fact G n is isomorphic to the finite p -group whose underlying set is G n = { ( u n − , u n , x , . . . , x p n − ) ∈ F p n p } u n − , u n , x , . . . , x p n − ) ⋆ ( v n − , v n , y , . . . , y p n − ) =( u n − + v n − , u n + v n + u n − y p n − , x + y , . . . , x p n − + y p n − )where the u i -coordinate represents k i and the x i -coordinate represents h i .The choice of generator names describes maps H n → G n , K n − → G n , and K n → G n . Using the explicit description of G n one may easily see that allthree of these maps are injections. Define a retraction map ρ n : G n → K n − by killing k n and by sending h k h µ n − ( k ) . Note that ρ n is compatible with η n : H n → H n − —that is, there is a commuting diagram K n − G n H n − H nρ n η n Let Q n,m be the fundamental pro- p group of the graph of groups Q n,m : G n +1 G n +2 · · · G n + mK n +1 K n +2 K n + m − (3)and let P m = Q ,m . By Proposition 2.8 the group Q n,m includes naturallyin Q n,m +1 . Note that the retractions ρ n : G n → K n − induce retractions π n,m : Q n,m +1 → Q n,m for all n and m : Q n,m +1 : G n +1 · · · G n + m G n + m +1 Q n,m : G n +1 · · · G n + m K n + mπ n,m K n +1 id K n + m − K n + m id ρ n + m +1 K n +1 K n + m − K n + m Define Q n, ∞ to be the inverse limit of the Q n,m along these retractions, and set P = Q , ∞ .By Proposition 2.8 we have an identity P n ∐ K n Q n,m = P n + m . Takingan inverse limit with Proposition 2.6 we find that P has a graph of groupsdecomposition P : G G · · · G m Q n, ∞ K K K m − K m (4)for all n ; so P is inaccessible.To complete the last part of Dunwoody’s construction we are required to takean amalgamated product with the finitely generated group H ω to restore finitegeneration. Because the H n generate H ∞ and there is an injection H ∞ → P defined to be the inverse limit of the injections H n → P n , it follows that theclosure of the union of the H n in P is isomorphic to H ∞ . So consider theamalgamated free product J = P ∐ H ∞ H ω . Both P and H ω are defined asinverse limits whose transition maps both restrict to the maps η n on H n . Hencewe may write J = P ∐ H ∞ H ω = lim ←− n P n ∐ H n ( H n ⋊ Z /p n ) . J is inaccessible are the splittings J : G G · · · G n Q n, ∞ H ωK K K n − K n H ∞ (5)for each n . By Proposition 2.6 these are the inverse limits of the graphs ofgroups G G · · · G n Q n,m H n + m ⋊ Z /p n + m . K K K n − K n H n + m (6) We must now show that the various graphs of groups introduced in the previoussection are proper.First consider the graph of groups (3). It suffices to deal with the case n = 0 since a subgraph of a proper graph of groups is proper. We exhibit anexplicit finite quotient of P n into which all the G i inject for 1 ≤ i ≤ n , therebyestablishing properness. The group presentation F n = (cid:10) k , . . . , k n − , k n , h , . . . , h p n − | k pi = h pi = 1 , k i ↔ k j , h i ↔ h j ,k i ↔ h j for all j = p i , k i +1 = [ k i , h p i ] (cid:11) defines a finite p -group with an explicit form similar to the description of G n given above: specifically the set F n = { (( u , . . . , u n ) , ( x , . . . , x p n − )) ∈ F np × F p n p } with multiplication( u, x ) ⋆ ( v, y ) = (( u i + v i + u i − y p i − ) , x + y ) . The choice of generator names specifies natural maps from the G i and K i into F n , compatible with the inclusions of K i into G i and G i +1 . This thereforedescribes a map P n → F n , and via the explicit description of F n one readilysees that the G i inject into F n under this map. This shows that the graph ofgroups (3) is proper. An application of Proposition 2.6 shows that the inverselimit (4) of these is also proper.To show that the graphs of groups (6) are proper, we require a finite quotientof P n ∐ H n H n ⋊ ( Z /p n ) into which all the G i and H n ⋊ Z /p n inject. Such afinite group E n is given by the presentation below; one could provide an explicitfinite p -group isomorphic to it (similar to the treatments of G n and F n above),but we leave this to the reader with a penchant for such activities. The grouppresentation is E n = (cid:10) k i,r (1 ≤ i ≤ n, ≤ r < p n ) , h , . . . , h p n − , t | k pi,r = h pi = t p n = 1 ,k i,r ↔ k j,s , h i ↔ h j , k i,r ↔ h j if j = p i + r,k i +1 ,r = [ k i,r , h p i + r ] , t − k i,r t = k i,r +1 , t − h j t = h j +1 (cid:11) where the indexing set of the h i should be taken modulo p n , as should the r -coordinate of the indexing set of the k i,r . By Proposition 2.6 we now also knowthat the graph of groups (5) is proper for each n .13 .4 Conclusion By construction, J is expressed as the fundamental group of arbitrarily largereduced graphs of groups (5), which we now know to be proper—that is, J isan inaccessible group. It only remains to argue that J is finitely generated.As per Dunwoody’s construction, J is generated by G and H ω . By definition J is generated by H ω and the G n , and each G n is generated by K n − ⊆ G n − and H n ; so by induction the subgroup of J generated by G and H ω containsall the G n .The finite group G is of course finitely generated. The ‘pro- p lamplightergroup’ H ω is also finitely generated—it is generated by the two elements( h , , (0 , t ) ∈ F p [[ Z p ]] ⋊ Z p where t denotes a generator of the second factor Z p in the semidirect product.That these elements do indeed generate the group follows from the fact thattheir images generate each term in the inverse limit H ω = lim ←− ( H n ⋊ Z /p n )defining H ω .Since both G and H ω are finitely generated, so is J . We have now provedour primary goal, the following theorem. Theorem 4.4.
There exists a finitely generated inaccessible pro- p group J . p -finite inaccessible group The focus of this paper has been pro- p groups. However the constructions madeabove also allow us to give, to the best of the author’s knowledge, the firstexample in the literature of a finitely generated inaccessible discrete group whichalso has the property that it is residually p -finite (or even residually finite). Wenow describe this, with a brief proof.Let the groups G i , K i , H i , and E i be as in the previous section. Define P n to be the (discrete) fundamental group of the graph of groups P n : G G · · · G nK K K n − and let P be the (discrete) fundamental group of the infinite groups P : G G · · · G m · · · K K K m − K n Let H ∞ = F p [ N ] = S n H n , and let H ω = F p [ Z ] ⋊Z . As per Dunwoody’s originalconstruction, the amalgamated free product J = P ∗ H ∞ H ω is an inaccessiblefinitely generated discrete group.To show that J is residually p -finite, let j ∈ J r { } . The element J is givenby a reduced word—that is, an expression j = a b · · · a m b m where 14 a i ∈ P for all i and b i ∈ H ω for all i ; • no a i or b i is trivial, except possibly a or b m ; and • no non-trivial a i or b i is contained in H ∞ except possible a .Now P is the union of the P n , so for all n sufficiently large all a i are containedin P n —and therefore are fixed by the retraction P → P n . For all n sufficientlylarge, the images of the b i under the quotient map H ω → H n ⋊ Z /p n do notlie in H n . Therefore for all n sufficiently large, the image j ′ of j in P n ∗ H n ( H n ⋊ Z /p n ) is given by a reduced word, hence is non-trivial. This latter groupis residually p -finite; this follows from Section 4.3 since the kernel of the map P n ∗ H n ( H n ⋊ Z /p n ) → E n is free, hence is residually p -finite. Therefore thereis a finite p -group quotient φ : P n ∗ H n ( H n ⋊ Z /p n ) → L such that φ ( j ′ ) = 1.Thus the composition J → P n ∗ H n ( H n ⋊ Z /p n ) → L does not kill j . It follows that J is residually p -finite as required. Theorem 5.1.
There exists a finitely generated, inaccessible, residually p -finitediscrete group. Acknowledgements
The author was supported by a Junior Research Fellowship from Clare College,Cambridge.
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