Elementary amenable groups of cohomological dimension 3
aa r X i v : . [ m a t h . G R ] F e b ELEMENTARY AMENABLE GROUPS OFCOHOMOLOGICAL DIMENSION 3
JONATHAN A. HILLMAN
Abstract.
We show that torsion-free elementary amenable groupsof Hirsch length 3 are solvable, of derived length
3. This classincludes all solvable groups of cohomological dimension
3. Weshow also that groups in the latter subclass are either polycyclic,semidirect products with base a solvable Baumslag-Solitar group,or properly ascending HNN extensions with base Z or π ( Kb ). We show that finitely generated, torsion-free elementary amenablegroups of Hirsch length 3 are in fact solvable minimax groups, of derivedlength
3. We show also that such a group is finitely presentable if andonly if it is constructible, and such groups are either polycyclic, semidi-rect products with base a solvable Baumslag-Solitar group, or properlyascending HNN extensions with base Z or π ( Kb ). Our interest in thisclass of groups arose from recent work on aspherical 4-manifolds withnon-empty boundary and elementary amenable fundamental group [4].Such groups have cohomological dimension F P ,and thus are in the class considered here. (One of the results of [4]is that the groups arising there are all either polycyclic or solvableBaumslag-Solitar groups, and so may be considered well understood.)1. background
Let G be a torsion-free elementary amenable group of finite Hirschlength h = h ( G ). Then G is virtually solvable [6], and so has a subgroupof finite index which is an extension of a finitely generated free abeliangroup Z v by a nilpotent group [3]. Since v h < ∞ we may assumethat v is the virtual first Betti number of G , i.e., the maximum of theranks of abelian quotients of subgroups of finite index in G . If G = 1then 0 < v h = h ( G ) c.d.G h + 1.We recall that the Hirsch-Plotkin radical √ G of a group G is the(unique) maximal locally nilpotent normal subgroup of the group. (Forthe groups G considered below, either √ G is abelian or G is virtually Key words and phrases. cohomological dimension, elementary amenable, finitelypresentable, Hirsch length, solvable, torsion-free. nilpotent.) A solvable group is minimax if it has a composition serieswhose sections are either finite or isomorphic to Z [ m ], for some m > constructible if it is in the smallest class containing the trivialgroup which is closed under finite extensions and HNN extensions [1]. If G is a torsion-free virtually solvable group group then c.d.G = h ⇔ G is of type F P ⇔ G is constructible [7].Let BS ( m, n ) be the Baumslag-Solitar group with presentation h a, t | ta m t − = a n i , and let BS ( m, n ) be the metabelian quotient BS ( m, n ) / hh a ii ′ , where hh a ii ′ is the commutator subgroup of the normal closure of the imageof a in BS ( m, n ). (When m = 1 and n = ± Z and π ( Kb ).)2. hirsch length 2 In this section we shall consider groups of Hirsch length 2, whicharise naturally in the analysis of groups of Hirsch length 3. (Note alsothat some groups of Hirsch length 2 have cohomological dimension 3.)
Theorem 1.
Let G be a torsion-free elementary amenable group ofHirsch length . Then √ G is abelian, and either √ G has rank and G ∼ = √ G ⋊ Z or √ G has rank and [ G : √ G ] .Proof. Since G is virtually solvable [6] and the lowest non-trivial termof the derived series of a solvable group is a non-trivial abelian normalsubgroup, √ G = 1. Since any two members of √ G generate a torsion-free nilpotent group of Hirsch length √ G isabelian, of rank r = 1 or 2, say, and h ( G/ √ G ) = 2 − r .Let C = C G ( √ G ) be the centralizer of √ G in G . If N C isa normal subgroup of G with locally finite image in G/ √ G then N ′ is locally finite, by an easy extension of Schur’s Theorem [8, 10.1.4].Hence N ′ = 1, so N is abelian, and then N √ G , by the maximalityof √ G . Therefore any locally finite normal subgroup of G/ √ G mustact effectively on √ G .If √ G has rank 1 then G/ √ G can have no non-trivial torsion nor-mal subgroup. If C = √ G is infinite then it has an infinite abeliannormal subgroup (since it is non-trivial, virtually solvable, and hasno non-trivial torsion normal subgroup). But the preimage of anysuch subgroup in G is nilpotent (since it is a central extension ofan abelian group). This contradicts the maximality of √ G . Hence= √ G and so G/ √ G acts effectively on √ G . Since h ( G/ √ G ) = 1 and Aut ( √ G ) Q × , and G/ √ G has no normal torsion subgroup, we seethat G/ √ G ∼ = Z . LEMENTARY AMENABLE GROUPS 3 If √ G has rank 2 then G/ √ G is a torsion group, and Aut ( √ G ) isisomorphic to a subgroup of GL (2 , Q ). If G/ √ G is infinite then it musthave an infinite locally finite normal subgroup (since it is a virtuallysolvable torsion group). But finite subgroups of GL (2 , Q ) have orderdividing 24, and so G/ √ G is finite. If g in G has image of finite order p > G/ √ G then conjugation by g fixes g p ∈ √ G . It follows that g must have order 2 and its image in GL (2 , Q ) must have determinant −
1. Hence [ G : √ G ] (cid:3) If G is finitely generated then √ G is finitely generated as a moduleover Z [ G/ √ G ], with respect to the action induced by conjugation in G .If h ( √ G ) = 1 then √ G is not finitely generated as an abelian group,while G/ √ G ∼ = Z . Hence Z [ G/ √ G ] ∼ = Z [ t, t − ], and the action of t is multiplication by some nm ∈ Q \ { , ± } , since √ G is torsion-freeand of rank 1. After replacing t by t − , if necessary, we may assumethat √ G ∼ = Z [ t, t − ] / ( mt − n ), for some m, n with ( m, n ) = 1 and | n | > m >
0. Hence G ∼ = BS ( m, n ). Then c.d.G = 2 ⇔ G is finitelypresentable ⇔ m = 1 [5].If G is finitely generated and h ( √ G ) = 2 then G ∼ = Z or π ( Kb ),and so c.d.G = 2.Let Z (2) be the localization of Z at 2, in which all odd integers areinvertible, and let Z (2) act on Q through the surjection to Z (2) / Z (2) ∼ = Z × = {± } . Let Q ⊗ Kb be the extension of Z (2) by Q with this action.Then if h = 2 and G is not finitely generated it is either a subgroupof Q ⋊ mn Z , for some nonzero m, n with ( m, n ) = 1 (if h ( √ G ) = 1),or is a subgroup of Q ⊗ Kb (if h ( √ G ) = 2). Every such group hascohomological dimension 3.3. hirsch length 3 Suppose now that h ( G ) = 3. Then h ( √ G ) = 1, 2 or 3. Theorem 2.
Let G be a torsion-free elementary amenable group ofHirsch length . If h ( √ G ) = 1 then √ G is abelian and G/ √ G ∼ = Z .If h ( √ G ) = 2 then √ G is abelian and G/ √ G ∼ = Z , D ∞ or Z ⊕ Z / Z .If h ( √ G ) = 3 then G is virtually nilpotent. In all cases, G has derivedlength at most .Proof. If h ( √ G ) = 1 then √ G is isomorphic to a subgroup of Q and(as in Theorem 1) G/ √ G has no locally finite normal subgroup. Since C G ( √ G ) is virtually solvable, it follows that C G ( √ G ) = √ G and so G/ √ G embeds in Aut ( √ G ), which is isomorphic to a subgroup of Q × .Hence G/ √ G ∼ = Z , and so G has derived length 2. JONATHAN A. HILLMAN If h ( √ G ) = 2 then √ G is abelian and (as in Theorem 1 again) themaximal locally finite normal subgroup of G/ √ G has order at most2. Since G/ √ G is virtually solvable and h ( G/ √ G ) = 1, it has anabelian normal subgroup A of rank 1, which we may assume torsion-freeand of finite index in G/ √ G . Moreover, G/ √ G embeds in Aut ( √ G ),which is now isomorphic to a subgroup of GL (2 , Q ). No nontrivialelement of A can have both eigenvalues roots of unity, for otherwise C G ( √ G ) > √ G . Since the eigenvalues of A have degree Q , itfollows that no nontrivial element of A can be infinitely divisible in A .Hence G/ √ G is virtually Z , and so it is either Z or the infinite dihedralgroup D ∞ = Z / Z ∗ Z / Z , or an extension of one of these by Z / Z .If G has a normal subgroup H such that H/ √ G ∼ = Z / Z then conju-gation in G must preserve the filtration 0 < H ′ < √ G of √ G . There-fore elements of G ′ act nilpotently on √ G , and so G/H cannot be D ∞ .Thus if h ( √ G ) = 2 then G/ √ G ∼ = Z , D ∞ or Z ⊕ Z / Z , and G hasderived length 2, 3 or 2, respectively.If h ( √ G ) = 3 then h ( G/ √ G ) = 0, and so G is virtually nilpotent.Since iterated commutators live in finitely generated subgroups, thederived length of G is the maximum of the derived lengths of its finitelygenerated subgroups. Finitely generated torsion-free virtually nilpotentgroups of Hirsch length 3 are polycyclic, and are fundamental groupsof N il -manifolds. These are Seifert fibred over flat 2-orbifolds withoutreflector curves, and so these groups have derived length
3. Hence G has derived length (cid:3) Corollary 3. If G is finitely generated then it is a minimax group.Proof. If h ( √ G ) = 1 and G is finitely generated then √ G is finitelygenerated as a Z [ Z ]-module. Since it is also torsion-free and of rank 1as an abelian group, it is in fact a cyclic Z [ Z ]-module. Hence √ G ∼ = Z [ D ] for some D > h ( √ G ) = 2 then G has a subgroup K of index K/ √ G ∼ = Z . If G is finitely generated then K is also finitely generated.Then √ G is again finitely generated as a Λ-module, and is torsion-freeand of rank 2 as an abelian group. Hence it is isomorphic as a groupto a subgroup of Z [ m ] , for some m > G is finitely generated and h ( √ G ) = 3 then G is polycyclic. In allcases it is clear that G is a minimax group. (cid:3) If h ( √ G ) = 1 and G is finitely generated then G has a presentation h a, t, u | ta m t − = a n , ua p u − = a q , utu − = ta r , hh a ii ′ i . LEMENTARY AMENABLE GROUPS 5 for some nonzero m, n, p, q with ( m, n ) = ( p, q ) = 1 and some r . Hence √ G ∼ = Z [ D ], where D is the product of the prime factors of mnpq . Aftera change of basis for G/ √ G , if necessary, we may assume that mn hasa prime factor which does not divide pq . We may further arrange that p divides m and q divides n , after replacing t by tu N or tu − N for N large enough, if necessary. Hence D is the product of the prime factorsof mn . We then see that G is a semidirect product BS ( m, n ) ⋊ Z .We shall assume for the remainder of this section that h ( √ G ) = 2.If G/ √ G ∼ = Z then the action of G/ √ G on √ G by conjugation in G determines a conjugacy class of matrices M in GL (2 , Q ). Hence G ∼ = √ G ⋊ M Z . Lemma 4.
A matrix M ∈ GL (2 , Q ) is conjugate to an integral matrixif and only if det M and tr M ∈ Z .Proof. These conditions are clearly necessary. If they hold then thecharacteristic polynomial is a monic polynomial with Z coefficents. If x ∈ Q is not an eigenvector for M then the subgroup generated by x and M x is a lattice. Since M preserves this lattice, by the Cayley-Hamilton Theorem, it is conjugate to an integral matrix. (cid:3) If G is finitely generated then √ G is finitely generated as a Z [ G/ √ G ]-module. Hence it is finitely generated as an abelian group (and so G ispolycyclic) if and only if det M = ± tr M ∈ Z . If G is F P and G/ √ G ∼ = Z then G is an HNN extension with base a finitely generatedsubgroup of √ G [2], and the HNN extension is ascending, since G issolvable. Hence M (or M − ) must be conjugate to an integral matrix.On the other hand, if G ∼ = √ G ⋊ M Z and neither M nor M − isconjugate to an integral matrix then G cannot be F P .Torsion-free polycyclic groups G with h ( √ G ) = 2 are S ol -manifoldgroups. There are such groups with G/ √ G ∼ = Z , D ∞ or Z ⊕ Z / Z . (Theexamples with G/ √ G ∼ = D ∞ are fundamental groups of the unions oftwo twisted I -bundles over a torus along their boundaries.)For instance, the group G with presentation h u, v, y | uyu − = y − , vyv − = v − y − , v = u y i is a generalized free product with amalgamation A ∗ C B where A = h u, y i ∼ = B = h v, u y i ∼ = π ( Kb ) and C = h u , y i ∼ = Z . It is clear that G/C ∼ = D ∞ , and it is easy to check that C = √ G .If G is the group with presentation h t, x, y | tx = xt, tyt − = y n , xyx − = y − i JONATHAN A. HILLMAN then √ G is normally generated by x and y , so h ( √ G ) = 2 and G/ √ G ∼ = Z ⊕ Z / Z .If G/ √ G ∼ = D ∞ then G is generated by √ G and two elements u , v with squares in √ G . The matrices in GL (2 , Q ) corresponding to theactions of u and v have determinant −
1. Hence t = uv corresponds toa matrix with determinant 1. There are finitely generated examples ofthis type which are not polycyclic. For instance, let F be the groupwith presentation h u, v, x, y | u = x, uyu − = y − , v = xy, vy v − = x y − i , and let K be the normal closure of the image of { x, y } in F . Then F/K ∼ = D ∞ and K/K ′ ∼ = Z [ ] , and F/K ′ is torsion-free, solvable and h ( F/K ′ ) = 3.However, if such a group G is F P then so is the subgroup generatedby √ G and t . Hence this subgroup is an ascending HNN extensionwith finitely generated base H √ G [2]. Since t maps H ∼ = Z intoitself and has determinant 1 it must be an automorphism of H , and so G is polycyclic.4. finitely presentable implies constructible In this section we shall show that if a torsion-free solvable group G of Hirsch length 3 is finitely presentable then it is in fact constructible,and we shall describe all such groups.If G is F P and G/G ′ is infinite then G is an HNN extension H ∗ ϕ with finitely generated base H [2], and the extension is ascending since G is solvable. Clearly h ( H ) = h ( G ) −
2, and c.d.G c.d.H + 1.In fact h ( H ) must be 2, for otherwise H ∼ = Z and c.d.G = 2. In ournext theorem we shall need the stronger hypothesis that G be finitelypresentable. (Homological methods do not seem to be useful here; aspectral sequence argument shows that H i ( G ; Z ) is finite for all i > Theorem 5.
Let G be a torsion-free solvable group of Hirsch length .Then G is finitely presentable if and only if it is constructible.Proof. If G is constructible then it is finitely presentable. Assume that G is finitely presentable. If √ G has rank 1 then G has a presentation h a, t, u | ta m t − = a n , ua p u − = a q , utu − t − = C ( a, t, u ) , R i , for some nonzero m, n, p, q with ( m, n ) = ( p, q ) = 1 and word C ( a, t, u )of weight 0 in each of t and u , and some finite set of relators R . Let D be the product of the prime factors of mnpq . Then √ G ∼ = Z [ D ], andcontains the image of c in G . As observed after Corollary 3, we may LEMENTARY AMENABLE GROUPS 7 assume that p and q divide m and n , respectively and that mn has aprime factor which does not divide pq .We may assume that each of the relations in R has weight 0 in eachof t and u . Then we may write C ( a, t, u ) and each relator in R asa product of conjugates b i,j = t i u j au − j t − i of a . Since R is finite theexponents i, j involved lie in a finite range [ − L, L ], for some L > a in G is √ G ∼ = Z [ D ]. Hence the images of the b i,j s in G commute, and arepowers of an element α represented by a word w = W ( a, t, u ) which isa product of powers of (some of) the b i,j s. In particular, a = α N and b i,j = α e ( i,j ) , for some exponents N and e ( i, j ). Clearly N = e (0 , tα m t − = α n and uα p u − = α q . Hence adjoininga new generator α and new relations(1) a = α N ;(2) tα m t − = α n ;(3) uα p u − = α q ;(4) α = W ( a, t, u ); and(5) t i u j au − j t − i = α e ( i,j ) , for all i, j ∈ [ − L, L ].gives an equivalent presentation.We may use the first relation to eliminate the generator a . Since theimage of α in G generates an infinite cyclic subgroup, the relations R must be consequences of these, and so we may delete the relations in R . Moreover the relation α = W ( a, t, u ) collapses to a tautology, andso may also be deleted, and we may use the final set of relations towrite C ( a, t, u ) as a power of α . Since tb i,j t − = b i +1 ,j and ub i,j u − = b i,j +1 , we see that e ( i, j ) = ( nm ) i ( qp ) j e (0 , i, j ∈ [ − L, L ]. Since α generates the subgroup spanned by the b i,j s it follows that N =( mnpq ) L and e ( i, j ) = N ( nm ) i ( qp ) j for i, j ∈ [ − L, L ]. Hence the final setof relations are consequences of the second and third relations.Thus G has the finite presentation h t, u, α | tα m t − = α n , uα p u − = α q , utu − t − = α c i , for some c ∈ Z . Since the subgroup generated by the images of t and α is isomorphic to BS ( m, n ) and is solvable, either m or n = 1 [2].If h ( √ G ) = 2 then G has a subgroup J of index H √ G . Since h ( H ) = 2, we have H ∼ = Z . Hence J is constructible, and G is alsoconstructible.If h ( √ G ) = 3 then G is virtually nilpotent, and so is again con-structible. (cid:3) JONATHAN A. HILLMAN
Theorem 6.
Let G be a torsion-free elementary amenable group ofHirsch length . Then G is constructible if and only if either(1) G ∼ = BS (1 , n ) ⋊ θ Z for some n = 0 or ± and some θ ∈ Aut ( BS (1 , n )) ;(2) G ∼ = H ∗ ϕ is a properly ascending HNN extension with base H ∼ = Z or π ( Kb ) ; or(3) G is polycyclic.Proof. It shall suffice to show that if G is constructible then it is oneof the groups listed here, as they are all clearly constructible. We mayalso assume that G is not polycyclic, and so h ( √ G ) = 1 or 2.Since G is constructible it has a subgroup J of finite index which isan ascending HNN extension with base a constructible solvable groupof Hirsch length 2. Since G is not polycyclic, we may assume that J = G , by Theorem 2 (when h ( √ G ) = 1) and by Theorem 2 with theobservations towards the end of § h ( √ G ) = 2). Constructiblesolvable groups of Hirsch length 2 are in turn Baumslag-Solitar groups BS (1 , m ) with m = 0.If h ( √ G ) = 1 then | m | > G ∼ = BS (1 , m ) ∗ ϕ , for some injectiveendomorphism of BS (1 , m ). We shall use the presentation for BS (1 , m )given in §
2. After replacing a by t − k at k , if necessary, we may assumethat ϕ ( a ) = a q and ϕ ( t ) = ta r , for some q = 0 and r in Z . Then G hasa presentation h a, t, u | tat − = a m , uau − = a q , utu − = ta r i . Let s = tu and n = mq . Then sas − = a n , and the subgroup H ∼ = BS (1 , n ) generated by a and s is normal in G . Conjugation by u generates an automorphism θ of H , since q is invertible in Z [ n ]. Hence G ∼ = BS (1 , n ) ⋊ θ Z , and so G is of type (1)If h ( √ G ) = 2 then m = ± G is not polycyclic, and so G is of type (2). (cid:3) We have allowed an overlap between classes (1) and (2) in Theorem6, for simplicity of formulation. Polycyclic groups of Hirsch length3 are virtually semidirect products Z ⋊ Z , and hence are ascendingHNN extensions, but the extensions are not properly ascending, andso classes (2) and (3) are disjoint. Corollary 7.
A finitely generated torsion-free elementary amenablegroup G of Hirsch length is coherent if and only if h ( √ G ) > .Proof. If G is coherent then it is finitely presentable, and hence isconstructible. Suppose that G is in class (1), and let A be the normalclosure of a in G . If the subgroup of Aut ( A ) = Z [ n ] × generated by LEMENTARY AMENABLE GROUPS 9 conjugation by t and u has rank 2 then it contains a proper faction pq with p, q = ±
1. Hence G has a subgroup isomorphic to BS ( p, q ).Since this subgroup is not finitely presentable [2], G is not coherent.Otherwise h ( √ G ) = 2 and G is also in class (2). It is easy to see that if G is an ascending HNN extension with finitely generated abelian basethen any finitely generated subgroup of G is either a subgroup of thebase or is itself an ascending HNN extension with finitely generatedbase. Hence all groups in classes (2) and (3) are coherent. (cid:3) We may summarize the main results of this section as follows.
Corollary 8. If G is a torsion-free elementary amenable group ofHirsch length then c.d.G = 3 ⇔ G is constructible ⇔ G is finitelypresentable ⇔ G is one of the groups listed in the theorem above. (cid:3) The first equivalence is a very special case of one of the main resultsof [7], but we have not needed that general result here.It remains an open question whether an
F P torsion-free solvablegroup G with h ( G ) = 3 and h ( √ G ) = 1 must be finitely presentable. References [1] Baumslag, G. and Bieri, R. Constructable solvable groups,Math. Z. 151 (1976), 249–257.[2] Bieri, R. and Strebel, R. Almost finitely presentable soluble groups,Comment. Math. Helv. 53 (1978), 258–278.[3] ˇCarin, On soluble groups of type A ,Mat. Sbornik 94 (1960), 895–914.[4] Davis, J. F. and Hillman, J. A. Aspherical 4-manifolds with elementaryamenable fundamental group, in preparation.[5] Gildenhuys, D. Classification of soluble groups of cohomological dimensiontwo, Math. Z. 166 (1979), 21–25.[6] Hillman, J. A. and Linnell, P. A. Elementary amenable groups of finite Hirschlength are locally-finite by virtually solvable,J. Aust. Math. Soc. 52 (1992), 237–241.[7] Kropholler, P. H. Cohomological dimension of soluble groups,J. Pure Appl. Alg. 43 (1986), 281–287.[8] Robinson, D. J. S. A Course in the Theory of Groups ,GTM 80, Springer-Verlag, Berlin – Heidelberg – New York (1982).
School of Mathematics and Statistics, University of Sydney, NSW2006, Australia
Email address ::