Constructing groups of type FP_2 over fields but not over the integers
aa r X i v : . [ m a t h . G R ] F e b Constructing groups of type
F P over fields but not over theintegers Robert KrophollerMarch 1, 2021
Abstract
We construct examples of groups that are
F P ( Q ) and F P ( Z /p Z ) for all primes p but notof type F P ( Z ). We begin with a definition:
Definition 1.1.
A group G is of type F P n ( R ) if there is an exact sequence: P n → · · · → P → P → P → R → RG -modules such that P i is finitely generated and R is the trivial RG -module.Using the chain complex of the universal cover of a presentation 2-complex we see that finitelypresented groups are of type F P ( Z ). Moreover, if a group is of type F P ( Z ), then it is of type F P ( R ) for any ring R .In [1], the first examples of groups that are of type F P ( Z ) but not finitely presented are given.More recently, there have been many new constructions of groups of type F P with interestingproperties, see [1, 10, 9, 8, 7, 4]. In particular, there are various constructions of uncountablefamilies of groups of type F P ( Z ) [10, 7, 4].It is also possible to use the examples of [1, 10] to give examples of groups that are of type F P ( R ) but not F P ( Z ) for certain rings R . The construction of [1], takes in a connected flagcomplex L and constructs a group BB L that is of type F P ( R ) if and only if H ( L ; R ) = 0. Sincethese flag complexes are finite, it follows that if H ( L ; Z /p Z ) = 0 for all primes p , then H ( L ; Z ) = 0.Thus, if BB L is F P ( Z /p Z ) for all primes p , then it is type F P ( Z ). Similar results can be obtainedfor the groups constructed in [10].In this paper we build on the work of [10]. Leary built uncountably many groups of type F P by taking branched covers of a cube complex X . Leary’s construction takes as input a flag complex L and a set S ⊂ Z . It outputs a cube complex X ( S ) L with a height function f ( S ) . These have theproperty that if a vertex has height in S , then the ascending and descending links at v are L . If theheight of a vertex is not in S , then the ascending and descending links are ˜ L , the universal cover of L . We build on this construction by varying the covers that can be taken at each height. Ourconstruction is the following: 1 onstruction 1.2. Let L be a flag complex. Let σ : Z → C be a function, where C is the collectionof normal covers of L . Then there is a cube complex X σL and a height function f σ such that if f σ ( v ) = n , then the ascending and descending links of v are exactly σ ( n ).This cube complex arises as a branched cover and there is a group of deck tranformations G L ( σ ).Thus we can use the cube complex X σL to investigate the finiteness properties of G L ( σ ). We obtainthe following theorem: Theorem 3.4.
Let σ, L, C be as above. Suppose that π ( L ) /π ( σ ( n )) is of type F P k ( R ) for all n .Then G L ( σ ) is type F P k ( R ) if and only if ˜ H i ( σ ( n ); R ) vanishes for all but finitely pairs ( i, n ) with n ∈ Z and i < k .Similarly, suppose π ( L ) /π ( σ ( n )) is of type F P ( R ) for all n . Then G L ( σ ) is type F P ( R ) ifand only if ˜ H i ( σ ( n ) , R ) vanishes for all but finitely pairs ( i, n ) with n ∈ Z and i ∈ N .We can use this to construct new examples of groups of type F P ( R ) over various rings R . Herewe detail two such constructions.As pointed out previously, if G is of type F P ( Z ), then G is of type F P ( R ) for all rings R .One may hope that there is a collection of rings R such that if G is of type F P ( R ) for all R ∈ R ,then G is of type F P ( Z ). One possible candidate is the collection of fields. We show that this isnot the case, proving the following. Theorem 1.3.
There are groups that are of type
F P ( F ) for all fields F , but not of type F P ( Z ) . In fact, it is enough to study Q and Z /p Z for all primes p . Since if P is the prime subfield of F , then F P ( P ) implies F P ( F ). Thus, we prove the following: Theorem 6.1.
There exist groups that are of type
F P ( Q ) and F P ( Z /p Z ) for all primes p thatare not F P ( Z ).Moreover, we are able to prove the above theorem for arbitrary sets of primes. Theorem 6.2.
Let S be a set of primes. Then there exists a group which is type F P ( Z /p Z ) ifand only if p / ∈ S .We highlight one particularly novel corollary to this theorem: Corollary 1.4.
There exists a group G that is type F P ( Q ) but not of type F P ( Z /p Z ) for anyprime p . The second theorem of this paper concerns the constructions from [10, 7]. In both papers,uncountably many groups of type
F P ( Z ) are constructed by considering subpresentations of aninitial group that is known to be of type F P ( Z ). One may believe removing relators from an almostfinitely presented group should result in an almost finitely presented group. It is clear that one hasto be careful when removing relations from a group. Indeed, every n generated group appears as asubpresentation of the trivial group presented as h x , . . . , x n | F ( x , . . . , x n ) i .However, in the examples from [10, 7] care is taken when considering subpresentations. In bothcases a generating set for the relation module of the initial group is retained and this is enough toensure the resulting group is of type F P ( Z ). For finitely presented groups, it is true that there isa finite set of relations, that if retained ensure that the subpresentation gives a finitely presentedgroup.It is then of interest to know whether if one retains an appropriate finite set of relations, saygenerators for the relation module, does one retain the property of being of type F P ( Z )? We showthat this is not the case: 2 heorem 5.3. There exists a presentation G = h X | R i of a group of type F P ( Z ) such that forany finite subset T ⊂ R , we can find S with T ⊂ S ⊂ R and h X | S i is not of type F P ( R ) for anyring R . Acknowledgements:
The author is thankful to Kevin Schreve for posing questions leadingto this article. The author is also grateful to Peter Kropholler and Ian Leary for helpful conversa-tions. The author was funded by the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) under Germany’s Excellence Strategy EXC 2044–390685587, Mathematics M¨unster:Dynamics–Geometry–Structure.
Definition 2.1.
Let L be flag complex. The spherical double of L , denoted S ( L ), is defined byreplacing every simplex of L by an appropriately triangulated sphere of the same dimension, in thefollowing way.Let { v , . . . , v n } be the vertices of L . The vertex set of S ( L ) is a set { v +1 , v − , . . . , v + n , v − n } . Thus,each 0–simplex { v i } of L corresponds to a 0–sphere { v + i , v − i } in S ( L ). If τ ⊂ L is an m –dimensionalsimplex of L , it can be represented as the join τ = { v i } ∗ · · · ∗ { v i m } of a collection of m + 1 verticesof L . Let S ( τ ) be the join S ( τ ) = { v + i , v − i } ∗ · · · ∗ { v + i m , v − i m } . We have that, S ( τ ) is homeomorphicto an m –dimensional sphere. We define S ( L ) = S τ ⊆ L S ( τ ), where τ ranges through all simplices of L . It can be shown that S ( L ) is a simplicial complex, which is flag if and only if L is flag [1,Lemma 5.8]. Also the map v ti v i , gives a retraction S ( L ) → L .The following two results are analogous to Proposition 7.1 and Corollary 7.2 of [10], we includethe proofs for completeness. Proposition 2.2.
Let L be a flag complex. Let ¯ L be a cover of L . Then S ( ¯ L ) is a cover of S ( L ) .Proof. This follows since S ( ¯ L ) can be seen as the pullback in the square: S ( ¯ L ) S ( L )¯ L L r The lower map is a covering this, we have a pull back of a covering map which is also a coveringmap.
Corollary 2.3.
Let r : S ( L ) → L be the retraction above. Then π ( S ( ¯ L )) = r − ∗ ( π ( ¯ L )) .Proof. Since the diagram in the proof of Proposition 2.2 commutes we see that π ( S ( ¯ L )) ⊂ r − ∗ ( π ( ¯ L )).Let γ be a loop in S ( L ) such that r ◦ γ is an element of π ( ¯ L ). Then we can lift r ◦ γ to a loop γ ′ in ¯ L . Then ( γ, γ ′ ) defines a loop in S ( ¯ L ) which maps to γ . Thus π ( S ( ¯ L )) ⊃ r − ∗ ( π ( ¯ L )).3 .2 Morse theory For full details, we refer the reader to [1].A map f : X → R defined on a cube complex X is a Morse function if • for every cell e of X , with characteristic map χ e : [0 , m → e , the composition f ◦ χ e : [0 , m → R extends to an affine map R m → R and f ◦ χ e is constant only when dim e = 0; • the image of the 0–skeleton of X is discrete in R .Suppose X is a cube complex, f : X → R a Morse function. The ascending link of a vertex v ,denoted Lk ↑ ( v, X ) is the subcomplex of Lk( v, X ) corresponding to cubes C such that f | C attainsits minimum at v . The descending link , Lk ↓ ( v, X ), is defined similarly replacing minimum withmaximum. Definition 2.4.
Let L be a flag complex. The right-angled Artin group , or RAAG, associated to L is given by the presentation: A L = h v ∈ Vertices( L ) | [ a i , a j ] = 1 if { a i , a j } is an edge of L i . Given a right-angled Artin group A L , the Salvetti complex , S L associated to A L is a cubecomplex S L defined as follows. For each v ∈ Vertices( L ) let S v be a circle endowed with a structureof a CW complex having a single 0–cell and a single 1–cell. Let T = Q v S v be an n –dimensionaltorus with the product CW structure. For every simplex K ⊂ L , define a k –dimensional torus T K as a Cartesian product of CW complexes: T K = Q v ∈ K S v and observe that T K can be identifiedas a combinatorial subcomplex of T . Then the Salvetti complex is S L = [ (cid:8) T K ⊂ T | K is a simplex of L (cid:9) . The link of the single vertex of S L is S ( L ). This is a flag simplicial complex, and hence S L is anon-positively curved cube complex. It follows that the universal cover X L = ˜ S L is a CAT(0) cubecomplex.We can define a homomorphism φ : A L → Z by sending each generator to 1. We can realisethis topologically as a map f : S L → S by restricting the map ˆ f : T → S given by ( x , . . . , x n ) x + · · · + x n . Let BB L := ker( φ ). We can lift f to a map which we also call f : X L /BB L → R .This is a Morse function. The ascending and decending links are copies of L spanned by v + i and v − i respectively. In [10], branched covers of X L /BB L were taken to obtain uncountably many groups G L ( S ) de-pending on S ⊂ Z , whose finiteness properties are controlled by the topology of L . In this section,we generalise this machinery to construct various new groups.Throughout, let L be a flag complex. Let C be the set of normal covers of L . Let Z = C Z bethe set of functions σ : Z → C . Define a partial ordering on Z by σ (cid:22) σ ′ if σ ( n ) is a cover of σ ′ ( n ).We will associate to each element σ ∈ Z a group as follows.4 onstruction of G L ( σ ) : For each integer n there is a single vertex v of X L /BB L such that f ( v ) = n . Let γ i,n be a sequence of loops in S ( L ) that normally generate G n = π ( S ( σ ( n ))) ≤ π ( S ( L )). Let V be the vertex set of X L /BB L and let V ( σ ) := { v ∈ X L /BB L | σ ( f ( v )) = L } ⊂ V .By identifying Lk( v, X ) with the boundary of the neighbourhood of v we can consider γ i,n asloops in X L /BB L . Let Y L ( σ ) be the complex obtained from ( X L /BB L ) r V ( σ ) by attaching disksto all the loops γ i,n . Let G L ( σ ) = π ( Y L ( σ )).We can also view the group G L ( σ ) as a group of deck transformations of a branched cover ofcube complexes. Theorem 3.1.
There is a CAT(0) cube complex X σL with a branched covering map b : X σL → X L /BB L such that G L ( σ ) is the group of deck transformations of this branched cover.Proof. We construct X σL as follows. Let ^ Y L ( σ ) be the universal cover of Y L ( σ ). Let D be thecollection of open disks added to ( X L /BB L ) r V ( σ ) to obtain Y L ( σ ) and let ˜ D be the collection oflifts of D to ^ Y L ( σ ). Let Z L ( σ ) be ^ Y L ( σ ) r ˜ D . Then the covering map p : ^ Y L ( σ ) → Y L ( σ ) restrictsto a covering map Z L → ( X L /BB L ) r V ( σ ). We can now lift the metric and complete to obtaina branched cover b : X σL → X L /BB L . The deck group is exactly the deck group of the covering ^ Y L ( σ ) → Y L ( σ ), this is G L ( σ ).Given a vertex v ∈ X σL we obtain a covering map b v : Lk( v, X σL ) → Lk( b ( v ) , X L ). Since thecover of a flag complex is a flag complex we see that X σL is non-positively curved.Taking the completion adds in the missing vertices of Z L . The vertices added cone off theirlinks. As such the boundary of each disk in ˜ D is trivial in X σL and thus π ( ^ Y L ( σ )) surjects π ( X σL ).We conclude, X σL is simply connected. Thus, X σL is non-positively curved and simply connectedand hence CAT(0).There is a Morse function f σ : X σL → R given by composition f ◦ b . Since G L ( σ ) is the coveringgroup of Z L → X L r V ( X L ), we see that it acts on X σL cellularly and freely away from the vertexset. Moreover, G L ( σ ) acts properly, freely and cocompactly on f − σ ( ). Thus by understandingthis level set we can understand finiteness properties of G L ( σ ). We will proceed by understandingthe ascending and descending links of the Morse function f σ . Lemma 3.2.
Let v be a vertex of X σL such that f σ ( v ) = n . Then Lk( v, X σL ) = S ( σ ( n )) and Lk ↑ ( v, X σL ) = Lk ↓ ( v, X σL ) = σ ( n ) .Proof. Let D v be the collection of disks glued at v in ^ Y L ( σ ). There is a retraction X σL r { v } → N ( v, X σL ) r { v } where N ( v, X σL ) is a neighbourhood of v . Let γ be the attaching map of a disk in˜ D r D v , then this bounds a disk in X σL r { v } . Thus we can extend the retraction over elements of˜ D r D v . This gives a retraction ^ Y L ( σ ) → Y L ( σ, v ), where Y L ( σ, v ) = ( N ( v, X σL ) r { v } ) ∪ D v Sincethe former is simply connected, so is the latter.We see that Y L ( σ, v ) is homotopy equivalent to a cover S ( L ) of S ( L ) together with disks glued toeach lift of γ i,n . Since this is simply connected we see that S ( L ) is the cover corresponding hh γ i,n ii .Since { γ i,n } normally generates π ( S ( σ ( n ))) we see that S ( L ) = S ( σ ( n )) and the ascending link isthe preimage of L which is exactly σ ( n ). Similarly the descending link is σ ( n ).This allows us to understand the finiteness properties of G L ( σ ).Firstly, we recall a simplified version of Brown’s criterion [3] (from [10]) for a group to be oftype F P k ( R ). 5 heorem 3.3. Suppose that X is a finite-dimensional R -acyclic G-CW-complex, and that G actsfreely except possibly that some vertices have isotropy subgroups that are of type F P ( R ) (resp. F P k ( R ) ). Suppose also that X = ∪ m ∈ N X ( m ) where X ( m ) ⊂ X ( m + 1) ⊂ · · · ⊂ X is an ascendingsequence of G -subcomplexes, each of which contains only finitely many orbits of cells. In this case G is F P ( R ) (resp. F P k ( R ) ) if and only if for all i (resp. for all i < k ) the sequence ˜ H i ( X ( m ); R ) of reduced homology groups is essentially trivial. Theorem 3.4.
Let σ, L, C be as above. Suppose that π ( L ) /π ( σ ( n )) is of type F P k ( R ) for all n .Then G L ( σ ) is type F P k ( R ) if and only if ˜ H i ( σ ( n ) , R ) vanishes for all but finitely pairs ( i, n ) with n ∈ Z and i < k .Similarly, suppose π ( L ) /π ( σ ( n )) is of type F P ( R ) for all n . Then G L ( σ ) is type F P ( R ) ifand only if ˜ H i ( σ ( n ) , R ) vanishes for all but finitely pairs ( i, n ) with n ∈ Z and i ∈ N .Proof. We focus on the proof for
F P k ( R ), the proof for F P ( R ) is similar.The group G L ( σ ) acts on X σL freely away from vertices. For vertices at height n , the stabiliser isthe deck group of the covering σ ( n ) → L . This is exactly π ( L ) /π ( σ ( n )) which is of type F P k ( R ).Let X ( m ) = f − σ ([ − m − , m + ]). We are now in the situation of Theorem 3.3, thus G L ( σ ) isof type F P k ( R ) if and only if for all i < k the sequence ˜ H i ( X ( m ); R ) of reduced homology groupsis essentially trivial.If ˜ H i ( σ ( n ) , R ) vanishes for all but finitely pairs ( i, n ) with n ∈ Z and i < k , then by [1, Corollary2.6] we can find an l such that for all m > l the inclusion X ( l ) → X ( m ) induces an isomorphism onall H i for i < k . Since homology commutes with direct limits we see that H i ( X ( l )) = H i ( X σL ) = 0for all i < k . Thus the system is essentially trivial.Now conversely suppose that there are infinitely many n such that H i ( σ ( n ) , R ) = 0 for some i < k . Thus for each l > l ′ > l + 1 and v such that H i ( σ ( l ′ ); R ) or H i ( σ ( − l ′ ); R )is non-trivial for some i < k . We will assume that σ ( l ′ ) has the non-trivial homology group. Let v be a vertex at height l ′ . There is a map from X ( l ′ −
1) to Lk( v, X σL ). We can further composewith the retraction Lk( v, X σL ) → σ ( l ′ ). By extending geodesics from v downwards, we can view σ ( l ′ ) as a subspace of X ( l ′ −
1) and this composition will be a retraction. However, the inclusion σ ( l ′ ) → X ( l ′ ) induces the trivial map on homology, thus we have an element in the kernel of themap H i ( X ( l ′ − R ) → H i ( X ( l ′ ); R ). Thus the system of homology groups is not essentially trivialand G L ( σ ) is not of type F P k ( R ).We can also prove similar results about finite presentability of G L ( σ ). Theorem 3.5.
Suppose that π ( L ) /π ( σ ( n )) is finitely presented for all n . Then G L ( σ ) is finitelypresented if and only if π ( σ ( n )) vanishes for all but finitely many n .Proof. For one direction, suppose that G L ( σ ) is finitely presented. Since it acts freely and cocom-pactly on f − σ ( ), there are finitely many orbits of loops which normally generate π ( f − σ ( )). Sincethe limit of X ( m ) is simply connected, we see that there is an l such that each of these loops istrivial in X ( l ). Thus the inclusion f − σ ( ) → X ( l ) is trivial on fundamental groups. By [1, Corol-lary 2.6], we have that f − σ ( ) → X ( l ) is also a surjection. Thus, X ( l ) is simply connected. Nowusing the retraction from the proof of Theorem 3.4 we obtain a π -surjective map X ( l ) → σ ( m )for all m such that | m | > l . Thus, we see that for all m such that | m | > l , we must have σ ( m ) issimply connected.For the other direction, suppose that there is an l such that if | m | > l we have that σ ( m ) issimply connected. Then, by [1, Corollary 2.6], X ( l ) is simply connected and has a cocompact action6y G L ( σ ). We now have a cocompact action of G L ( σ ) on a simply connected CW complex wherestabilisers of cells are finitely presented. Thus G L ( σ ) is finitely presented. G L ( σ ) This section closely follows Section 14 of [10].We begin by describing presentations of the groups G L ( σ ) obtained in the previous section. Definition 4.1.
Let L be a simplicial complex. Let E be the set of edges of L . For a loop c = ( e , . . . , e l ) in L . Let c [ k ] denote the word e k e k . . . e kl in F ( E ). Theorem 4.2.
Let γ i,n be a collection of loops that normally generate π ( σ ( n )) . Let E be the edgesof L . Suppose that σ (0) = L . Then G L ( σ ) has the following presentation: h E | ef g, gf e for each triangle e, f, g in L , γ [ n ] i,n for n ∈ Z and all i i . Proof.
Recall that we have a Morse function f : X L /BB L → R . Let Y = f − (0). Since σ (0) = L we have that Y ⊂ Y L ( σ ) By [10, Corollary 10.4], we have that the inclusion Y → Y L ( σ ) is asurjection on π . From [10, Theorem 14.1], we have a presentation for π ( Y ) which is exactly, h E | ef g, gf e for each triangle e, f, g in L i . Let Z = X L r { v ∈ X (0) L | f ( v ) = 0 } . Then π ( Z ) = π ( Y ).Thus to obtain a presentation of G L ( σ ) we add in the relations coming from the disks in Y L ( σ ).By [10, Lemma 14.3], we see that if D is the disk glued to the word γ i,n , then this correspondsexactly to the relation γ [ n ] i,n . Thus we arrive at the desired presentation.Later we will show that there are uncountably many such groups. For this purpose it will beuseful to know the following: Lemma 4.3.
Let γ be an edge loop in L . Then γ [ n ] is trivial in G L ( σ ) if and only if γ lifts to aloop in σ ( n ) .Proof. The loop γ [ n ] in Y L ( σ ) is homotopic to the loop γ in the link of the vertex v of X L at height n . We will label this γ n . Note that γ n belongs to the ascending or descending link of v (dependingon the sign of n ) and this subspace is exactly L .If γ lifts to a loop in σ ( n ), then γ defines an element of π ( σ ( n )) and is homotopic to a productof the loops γ i,n . Thus together with the triangle relations we see that γ [ n ] is trivial in π ( Y L ( σ )).Now suppose that γ [ n ] is trivial in π ( Y L ( σ )), then the loop γ [ n ] lifts to a loop in ^ Y L ( σ ). Wecan now lift the homotopy and see that γ n also lifts to a loop in ^ Y L ( σ ) which we will call λ n . Thisloop must be in the ascending or descending link of some vertex w of ^ Y L ( σ ). However, this link isexactly σ ( n ), thus λ n is a loop in σ ( n ) which is a lift of γ .7 Maps between G L ( σ ) Recall that we partially order the set Z by σ (cid:22) σ ′ if σ ( n ) is a cover of σ ′ ( n ). Let γ i,n be the set ofloops that generate π ( σ ( n )) and γ ′ i,n be the set of loops that normally generate π ( σ ′ ( n )). In thecase that σ ( n ) is a cover of σ ′ ( n ) we can assume that { γ i,n } ⊂ { γ ′ i,n } . Thus, one can see from thepresentations in Theorem 4.2 that there are surjective maps G L ( σ ) → G L ( σ ′ ) whenever, σ (cid:22) σ ′ .This map can also be realised at the level of cube complexes. Theorem 5.1.
Let σ, σ ′ ∈ Z . Suppose that σ ( n ) (cid:22) σ ′ ( n ) . Then there is a branched cover of cubecomplexes X σL → X σ ′ L which preserves level sets.Proof. To see this recall Y L ( σ ) is obtained from X L by removing all vertices and gluing disks inthe link at level n to generators for π ( S ( σ ( n ))). Thus take as our generating set for π ( S ( σ ( n )))a generating set for π ( S ( σ ′ ( n ))) along with extra generators. This way we obtain an inclusion Y L ( σ ) → Y L ( σ ′ ).Let D be the set of disks added to obtain Y L ( σ ). Let D ′ be the set of disks added to Y L ( σ ) toobtain Y L ( σ ′ ).Let ^ Y L ( σ ′ ) be the universal cover of Y L ( σ ′ ). Let Y be the space obtained from ^ Y L ( σ ′ ) byremoving the lifts of disks in D ′ . Then Y is the cover of Y L ( σ ) corresponding to the kernel of thesurjection G L ( σ ) → G L ( σ ′ ). Thus, the universal cover of Y is the universal cover of Y L ( σ ). Whenwe remove the disks and complete we get a branched cover of cube complexes X σL → X σ ′ L . Corollary 5.2.
Let σ, σ ′ ∈ Z with σ ≺ σ ′ . Suppose that σ (0) = σ ′ (0) = L . Then the Cayley graphof G L ( σ ) is a cover of the Cayley graph for G L ( σ ′ ) , where both groups have as generating sets theedges of L .Proof. In the case that σ (0) = L we have that G L ( σ ) acts freely on f − σ (0). Moreover, it actstransitively on vertices. Thus, the 1-skeleton of f − σ (0) is the Cayley graph for G L ( σ ). Similarstatements hold for G L ( σ ′ ).Now Theorem 5.1, we have a covering map X σL → X L ( σ ′ ) which preserves level sets. Thus weget a covering of Cayley graphs. Theorem 5.3.
There exists a presentation G = h X | R i of a group of type F P ( Z ) such that forany finite subset T ⊂ R , we can find S such that T ⊂ S ⊂ R and h X | S i is not of type F P ( R ) forany ring R .Proof. Throughout the proof let G = Z ⋊ SL ( Z ) and N = Z ⊳ G . There are two properties ofthis pair we will use namely, that G is perfect (see for instance [5]) and so H ( G ; R ) = 0 for allrings R and H ( N ; R ) = R for all rings R .Let L be a flag complex with no local cut points and fundamental group G . Let K be the covercorresponding to N . Let E be the set of edges of L . Let α i = ( e ,i , . . . , e n i ,i ) be a sequence of loopsin L that generate N . Let β i = ( f ,i , . . . , f m i ,i ) be a sequence of loops in L that generate G . Wecan obtain the following presentation for BB L from [6]: h E | ef g, gf e for each triangle e, f, g in L , α [ n ] i , β [ n ] i for n ∈ Z and all i i . Since G is perfect, we obtain from [1] that BB L is of type F P ( Z ).Let T be a finite subset of the relations of BB L . Let F = { n | ∃ i such that β [ n ] i ∈ T } . Let S bethe the union of the following sets of relations: 8 T , • all triangle relations, • α [ n ] i for all n ∈ Z , • β [ n ] i for n ∈ F Consider the subpresentation h E | S i This is a presentation of G L ( σ ), where σ ( n ) = ( K, if n / ∈ F,L, if n ∈ F. Since F is a finite set there are infinitely many vertices such that the ascending and descendinglink have non-trivial first homology with coefficients in R . Thus G L ( σ ) is not of type F P ( R ) byTheorem 3.4. F P over fields We are now ready to prove the following theorem.
Theorem 6.1.
There exists groups that are of type
F P ( Q ) and F P ( Z /p Z ) for all p but not oftype F P ( Z ) .Proof. To do this we find a finitely presented group G with a sequence of subgroups G n such that H ( G n ; Z ) = ( Z mn , if n is prime,0 , if n is not prime.Let G = SL ( Z ). Let G p be the level p congruence subgroup. By [11], we have that H ( G p ; Z ) = Z /p Z .Now let L be a flag complex with no local cut points with fundamental group G . Let G n be thelevel n congruence subgroup if n > L n be thecover corresponding to G n . Let σ be the function assigning n to L n .In this case L n is a trivial or finite cover of L . In either case, the quotient is finitely presentedand hence of type F P over any ring.Since all the homology groups considered are finite we see that H ( σ ( n ); Q ) vanishes for all n .Also H ( σ ( n ); Z /p Z ) is non-trivial if and only if n = p . Thus we can apply Theorem 3.4 to see that G L ( σ ) is of type F P ( Q ) and F P ( Z /p Z ) for all p .However, there are infinitely many n with H ( σ ( n ); Z ) non-trivial. Thus, by Theorem 3.4 wesee that G L ( σ ) is not of type F P ( Z ).One would imagine that it is possible to prove the corresponding result for F P k or even F P .The above theorem gives a template for how to do this. To prove the above theorem for type
F P k one would need a flag complex L and a sequence of normal covers L n satisfying the followingconditions: 9 For infinitely many n , there is an i < k such that H i ( L n ; Z ) does not vanish. • For all but finitely many pairs ( i, n ) with i < k , we have that H i ( L n ; Q ) vanishes. • For each prime p , we have that for all but finitely many pairs ( i, n ), with i < k , we have that H i ( L n ; Z /p Z ) vanishes. • For all n and all p the quotient π ( L ) /π ( L n ) is of type F P k ( Q ) and F P k ( Z /p Z ).For the F P result replace
F P k by F P and remove the i < k assumption throughout.In a similar way to Theorem 6.1, we can prove the following theorem.
Theorem 6.2.
Let P be the set of primes. For each subset S of P there is a group which is type F P ( Z /p Z ) if and only if p / ∈ S .Moreover, we can construct such a group that has a proper action on a 3-dimensional CAT(0)cube complex.Proof. Let L be a flag complex with fundamental group GL ( Z ). For p >
2, let G p denote thelevel p congruence subgroup in SL ( Z ). Note that G p is still finite index and normal in GL ( Z ).Let G = GL ( Z ). Let ¯ L be the cover of L corresponding to SL ( Z ). Let L p be the cover of L corresponding to G p . Thus H ( L p ; Z ) is a finite p -group.Let S = { p , p , p , . . . } Let ( a n ) n ∈ N be any sequence which contains p i infinitely many timesfor each p i ∈ S . For instance, we can take the sequence p , p , p , p , p , p , p , p , p , p , p , . . . . Define σ : Z → C as follows σ ( n ) = ( ¯ L, if n < ,L a n , if n ≥ . Since each p i appears infinitely many times in ( a n ) we have by Theorem 3.4, that G L ( σ ) is F P ( Z /p Z ) if and only if p / ∈ S .Since all the covers taken were finite index we see that all the vertex stabilisers are finite. Thusthe action of G L ( σ ) on X σL is proper. We can show that there are uncountably many quasi-isometry classes of groups as in Theorem 6.2.The proof given here closely follows that of [8]. The idea of the proof is to interleave the sequenceof covers from Theorem 6.2, with the sequences of universal covers as in [8]. Thus, we can use thesequence from covers from Theorem 6.2 to obtain the desired finiteness properties and we use therelations (or lack thereof) from the universal covers to obtain uncountably many quasi-isometryclasses.Let σ, σ ′ ∈ Z with σ ≺ σ ′ . Let M ( σ, σ ′ ) = min {| n | | σ ( n ) = σ ′ ( n ) } . Lemma 7.1.
Suppose that L is d -dimensional and σ, σ ′ ∈ Z with σ ≺ σ ′ and σ (0) = σ ′ (0) = L ,and take the standard generating set for G L ( σ ) and G L ( σ ′ ) . The word length of any non-identityelement in the kernel of the map G L ( σ ) → G L ( σ ′ ) is at least M ( σ, σ ′ ) p /d + 1 . roof. This follows from Lemmas 3.1 and 3.2 of [8].We will use the taut loop length spectrum of Bowditch [2].
Definition 7.2.
Let Γ be a graph and l ∈ N . Let Γ l denote the 2-complex with 1-skeleton Γ and a2-cell attached to each loop of length < l . An edge loop of length l is taut if it is not null-homotopicin Γ l . Bowditch’s taut loop length spectrum , H (Γ) is the set of lengths of taut loops. Definition 7.3.
Let
H, H ′ be two sets of natural numbers. We say that H and H ′ are k -related iffor all l ≥ k + 2 k + 2, whenever l ∈ H then there is some l ′ ∈ H ′ such that lk ≤ l ′ ≤ lk and viceversa.The key element from [2] is the following relating the taut loop length spectrum to quasi-isometries. Lemma 7.4.
If (connected) graphs Γ and Λ are k -quasi-isometric, then H (Γ) and H (Λ) are k -related. We are now ready to prove the main theorem of this section. The proof is similar to that ofTheorem 5.2 in [8]. Here is a brief outline. For each F ⊂ N we will construct a function σ F . We willthen give a rough computation of the taut loop length spectrum for the Cayley graph for G L ( σ F ).This will show that if G L ( σ F ) is quasi-isometric to G L ( σ F ′ ), then F △ F ′ is finite and conclude thedesired result. The key change from [8] is that due to the sequence of covers we cannot take a singleconstant C and must take a sequence of constants C i satisfying certain conditions relating to thefunction σ F . Comparing to the proof of Theorem 5.2 in [8] we have replaced the constant β by thesequence of constants r p and chosen C i to ensure the arguments contained there still work. Theorem 7.5.
Let S be a set of primes. Then there are uncountably many quasi-isometry classesof groups that are of type F P ( Z /p Z ) if and only if p ∈ S .Proof. Let L be a flag complex with fundamental group GL ( Z ). For each prime p , let L p bethe cover of L corresponding to the level- p congruence subgroup in SL ( Z ). Let L be the covercorresponding to SL ( Z ). Let L be the universal cover ˜ L of L .Given a finite set Ω of loops that normally generate π ( L p ), define r p (Ω) as the maximum lengthof a loop in Ω. Define r p to be the minimum of r p (Ω) as Ω runs over all possible normal generatingsets for π ( L p ).Let ( b n ) be the following sequence: b n = ( a m , if n = 2 m + 1 , , if n = 2 m, Let α = p / ( d + 1), where d is the dimension of L . Let C n be a sequence of integers satisfyingthe following conditions: • C α > • C n α > r b n − , • C n α > r b n • C n > C n − . 11rom this we can deduce that αC n n > r b n − C n − n − .Let F ⊂ N . Let F = { n | n ∈ F } ∪ { n + 1 | n ∈ N } . Define σ F as follows: σ F ( n ) = L b i , if n = C i i and i ∈ F , L, if n = 0, L , otherwise.Let Γ F be the Cayley graph of G L ( σ F ) with generating set the edges of L . We will prove thefollowing two propositions about H (Γ F ): • If k ∈ F , then H (Γ F ) ∩ [ C k k α, C k k r b k ] = ∅ . • If k > k / ∈ ∪ l ∈ F [ C l l α, C l l r b l ], then k / ∈ H (Γ F ).Suppose k ∈ F , let F ′ = F r { k } . There is a surjection G L ( σ F ) → G L ( σ F ′ ). Let K be thekernel of this surjection. By Lemma 7.1, any non-identity element of K has length at least αC k k .We also know that there is an element of length r k C k k in K . The length of the shortest element of K defines a member of H (Γ F ). Thus we obtain the first statement.For the second statement, let k / ∈ F . Choose n ∈ N maximal such that r b n C n n < k or − F ′ = F ∩ [0 , n ]. Let K be the kernel of the map G L ( σ F ′ ) → G L ( σ F ). Consider thecovering map Γ F ′ → Γ F coming from Theorem 5.1. Every relator in G L ( σ F ′ ) has length ≤ r b n C n n .Now suppose that γ is a loop of length k in Γ F . We can lift γ to Γ F ′ . If γ lifts to a loop in Γ F ′ ,then it must be a consequence of loops of length ≤ r b n C n n . Thus, γ cannot be taut.Now suppose that γ lifts to a non-closed path. In this case γ defines an element of K . Theshortest such element has length ≥ αC m m where m = M ( σ F , σ F ′ ). By choice of n we have that k ≤ r m C m m . Thus, we obtain that k ∈ [ αC m m , r b m C m m ]. However, this contradicts the choice of k and thus γ is not taut.To complete the proof, suppose that l ∈ [ αC m m , r b m C m m ] and l ′ ∈ [ αC m + n m + n , r b m + n C m + n m + n ] forsome n >
0. Then l ′ l ≥ αC m + n m + n r b m C m m > αC m + n m r b m C m m ≥ αC m − m r b m > C m − m . Suppose that H (Γ F ) and H (Γ F ′ ) are k -related. If n ∈ F △ F ′ , then C n − n ≤ k . Thus if F △ F ′ is infinite, then H (Γ F ) and H (Γ F ′ ) are not k -related. Hence, G L ( σ F ) is not quasi-isometric to G L ( σ F ′ ).While we have used the framework of Bowditch’s taut loop length spectrum it is also possible touse the work of [12]. Let G be the space of marked groups. Combining Lemma 4.3 and Lemma 7.1we see that Z → G given by σ G L ( σ ) is a continuous injection with perfect image. Thus, by [12,Theorem 1.1] we obtain uncountably many quasi-isometry classes of groups of the form G L ( σ ). Bycarefully choosing subsets of Z to ensure that the image is still perfect one can proves analogues ofTheorem 7.5 for other properties satisfied by the G L ( σ ). References [1] Mladen Bestvina and Noel Brady. Morse theory and finiteness properties of groups.
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