On the virtually-cyclic dimension of surface braid groups and right-angled Artin groups
aa r X i v : . [ m a t h . A T ] A p r ON THE VIRTUALLY-CYCLIC DIMENSION OF SURFACEBRAID GROUPS AND RIGHT-ANGLED ARTIN GROUPS
ALEJANDRA TRUJILLO-NEGRETE
Abstract.
We give a bound for the virtually cyclic dimension of groupswith a normal subgroup of finite index which satisfies that every infinitevirtually-cyclic subgroup is contained in a unique maximal such sub-group. As an application we provide a bound for the virtually-cyclicdimension for the braid group of a closed surface with genus greaterthan 2 and for right-angled Artin groups. Introduction
Let G be a group. A family F of subgroups of G is a set of subgroups of G which is closed under conjugation and taking subgroups. A model for the classifying space E F G for the family F is a G -CW-complex X such that thefixed point set X H is contractible for H ∈ F and is empty if H
6∈ F .Let
F IN G and VC G be the families of finite and virtually cyclic subgroupsof G , respectively. We abbreviate EG := E FIN G G and EG := E VC G G . Thestudy of models for EG and EG has been motivated by the Baum-Connesand Farrel-Jones Conjectures.A model for E F G always exists and two models for E F G are G -homotopyequivalent (see [9]), however the model may not have finite dimension. Thesmallest possible dimension of a model for E F G is called the geometricdimension of G for the family F and is usually denoted as gd F G . Weabbreviate gd G := gd FIN G G and gd G := gd VC G G , they are called properand virtually-cyclic dimension respectively, and for the trivial family { } ,denote gd G := gd { } G .We remark that in general gd G ≤ gd G +1 (see [10]), and for many familiesof subgroups the following inequality hold:(1) gd G ≤ gd G + 1 . In particular, L¨uck and Weiermann showed in [10] that if a group G has pro-perty Max VC ∞ G (which states that every infinite virtually-cyclic subgroup iscontained in a unique maximal such subgroup), then G satisfies inequality(1). In [6] it was showed that the mapping class group of an orientablecompact surface has normal subgroups of finite index Γ with the propertyMax VC ∞ Γ . The author was supported by research grants from CONACyT-Mexico.
The principal result of this paper is the following:
Theorem 1.
Let → H → G ψ −→ F → be a short exact sequence of groupswhere F is a non-trivial finite group. If G is torsion free, H has property Max VC ∞ H and gd H < ∞ , then gd G ≤ max { , gd H + 1 } . Also we will give a bound for gd G when G has torsion. We remark thatTheorem 1 extends some of the results that we obtained in [1], in whichwe showed that in many cases, inequality (1) holds for the whole mappingclass group, we used that the mapping class groups have the property ofuniqueness of roots and an extra condition.Below are some applications of Theorem 1. Let S be a surface, denote by B n ( S ) the n -braid group of S . Proposition 2. If n ≥ and S is an orientable closed surface of genus g ≥ , then gd B n ( S ) ≤ n + 2 . Proposition 3. If A is a right-angled Artin group, then gd A ≤ gd A + 1 . In addition, for any subgroup H ⊆ A , gd H ≤ gd H + 1 . Classifying spaces for families
A method for constructing a model for EG is to start from a model for EG and then promote the action to a larger space to get a model for EG .In this section we will describe the model of L¨uck and Weiermann given in[10].Define an equivalence relation on the set VC ∞ G = VC G − F IN G of infinitevirtually cyclic subgroups of G . Given V, U ∈ VC ∞ G V ∼ U if and only if | V ∩ U | = ∞ . (2)Let [ VC ∞ G ] denote the set of equivalence classes and by [ V ] the equivalenceclass of V ∈ VC ∞ G . For [ V ] ∈ VC ∞ G defineComm G [ V ] := { g ∈ G | [ gV g − ] = [ V ] } , (3)the commensurator of V in G . Define a family of subgroups of Comm G [ V ]as G G [ V ] = { U ∈ VC ∞ Comm G [ V ] | [ V ] = [ U ] } ∪ F IN Comm G [ V ] . (4) Theorem 2.1. [10, Thm.2.3]
Let VC ∞ G and ∼ be as above. Let I be acomplete system of representatives [ V ] of the G -orbits in [ VC ∞ G ] under the G -action coming from conjugation. Choose arbitrary Comm G [ V ] -CW-modelsfor E (Comm G [ V ]) , E G [ V ] (Comm G [ V ]) and an arbitrary G -CW-model for LASSIFYING SPACES 3 E ( G ) . Define X a G -CW-complex by the cellular G -pushout ` [ V ] ∈ I G × Comm G [ V ] E (Comm G [ V ]) ` [ V ] ∈ I id G × Comm G [ V ] f [ V ] (cid:15) (cid:15) i / / EG (cid:15) (cid:15) ` [ V ] ∈ I G × Comm G [ V ] E G [ V ] (Comm G [ V ]) / / X (5) such that f [ V ] is a cellular Comm G [ V ] -map for every [ V ] ∈ I and i is aninclusion of G -CW-complexes, or such that every map f [ V ] is an inclusionof Comm G [ V ] -CW-complexes for every [ V ] ∈ I and i is a cellular G -map.Then X is a model for EG.
Remark 2.2. [10, Rem .2.5] Suppose that there exists n such that gd G ≤ n ,and for each [ V ] ∈ I ,gd Comm G [ V ] ≤ n − G [ V ] Comm G [ V ] ≤ n, then gd G ≤ n .2.1. Proper dimension.
The following Theorems will be used for the proofof the main Theorem, which is given in Section 3.Let G be a group, and F ∈ F IN G a finite subgroup. The length l ( F )of F is defined as the largest natural number k for which there is a chain1 = F < F < · · · < F k = F . The length of G is l ( G ) = sup { l ( F ) | F ∈ F IN G } . Theorem 2.3. [12, Thm. 3.10, Lem. 3.9]
Suppose that gd G < ∞ . If l ( G ) is finite, then gd G ≤ max { , vcd G + l ( G ) } , where vcd( G ) denotes virtual cohomological dimension of G . Theorem 2.4. [11, Thm. 2.5]
Let G be a group such that any finite subgroupis nilpotent and vcd G < ∞ . Let n = max F ∈FIN G { vcd G + rk ( W G F ) } , where rk ( · ) denotes the biggest rank of a finite elementary abelian subgroup, then gd G ≤ max { , n } . Virtually-cyclic dimension
We say that G has property Max VC ∞ G if every V ∈ VC ∞ G is contained in aunique maximal V max G ∈ VC ∞ G .Suppose G satisfies Max VC ∞ G , let V, U ∈ VC ∞ G , thus V ∼ U if only if V max G = U max G , thereforeComm G [ V ] = N G ( V max G )is the normalizer of V max G in G (see [10]). Lemma 3.1. If G has property Max VC ∞ G and U ∈ VC ∞ G , then( gU g − ) max G = gU max G g − . ALEJANDRA TRUJILLO-NEGRETE
Proof.
We first claim that gU max G g − is maximal in VC ∞ G : let V ∈ VC ∞ G suchthat gU max G g − ⊆ V , hence U max G ⊆ g − V g and by maximality we havethat U max G = g − V g . In this way, gU max G g − = V . So we conclude that( gU g − ) max G = gU max G g − by the uniqueness requirement of the propertyMax VC ∞ G . (cid:3) Lemma 3.2.
Let 1 → H → G ψ −→ F → H satisfies property Max VC ∞ H and F is finite. Let V ∈ VC ∞ G and ( V ∩ H ) max H ∈ VC ∞ H be the maximal containing ( V ∩ H ), thenComm G [ V ] = N G (( V ∩ H ) max H ) . Proof.
Let g ∈ G , we have that[ gV g − ] = [ V ] if and only if [ g ( V ∩ H ) g − ] = [ V ∩ H ]if and only if ( g ( V ∩ H ) g − ) max H = ( V ∩ H ) max H , (6)by Lemma 3.1 ( g ( V ∩ H ) g − ) max H = g ( V ∩ H ) max H g − , then equality (6)holds if and only if g ( V ∩ H ) max H g − = ( V ∩ H ) max H . (cid:3) Theorem 3.3.
Let → H → G ψ −→ F → be a short exact sequence ofgroups where F is a non-trivial finite group. Suppose that H has property Max VC ∞ H and gd H < ∞ , then gd G ≤ max { , gd H + l ( F ) } . If in addition G is torsion free, then gd G ≤ max { , gd H + 1 } .Proof. Let V ∈ VC ∞ G and U = ( V ∩ H ) max H , by Lemma 3.2 we havethat Comm G [ V ] = N G ( U ). Thus a model for EN G ( U ) /U is a model for E G G [ V ] N G ( U ) with the N G ( U )-action induced by the projection p : N G ( U ) → N G ( U ) /U . Hence gd G G [ V ] N G ( U ) ≤ gd N G ( U ) /U .We will use Theorem 2.3 to give a bound for gd N G ( U ) /U . From the exactsequence 1 / / N H ( U ) / / N G ( U ) ψ | / / F ′ / / , where ψ | is the restriction of ψ and F ′ ⊆ F , we have1 / / N H ( U ) /U / / N G ( U ) /U ψ / / F ′ / / , so vcd( N G ( U ) /U ) = vcd( N H ( U ) /U ) ≤ gd( N H ( U ) /U ). Since H satisfiesproperty Max VC ∞ H and gd H < ∞ , by the proof of [10, Thm. 5.8] we havethat gd( N H ( U ) /U ) ≤ gd N H ( U ), hencevcd( N G ( U ) /U ) ≤ gd N H ( U ) ≤ gd H, we remark that gd H ≤ gd H + 1 < ∞ .Further, N H ( U ) /U is torsion free because U is virtually cyclic maximal in LASSIFYING SPACES 5 N H ( U ), therefore the finite subgroups of N G ( U ) /U embeds in F ′ , so that l ( N G ( U ) /U ) ≤ l ( F ′ ) ≤ l ( F ) < ∞ . Applying Theorem 2.3 we have thatgd N G ( U ) /U ≤ max { , gd H + l ( F ) } . By Theorem 2.1 and Remark 2.2, we may conclude thatgd G ≤ max { , gd H + l ( F ) } . If G is torsion free, then the finite subgroups of N G ( U ) /U are cyclic. Ap-plying Theorem 2.4 we havegd N G ( U ) /U ≤ max { , vcd N G ( U ) /U + 1 }≤ max { , gd H + 1 } , again by Theorem 2.1 and Remark 2.2, we obtain the following inequalitygd G ≤ max { , gd H + 1 } . (cid:3) Applications
We will use the following Lemma and the fact that surface braid groupsand right-Artin groups can be embeded in some mapping class group ofa surface, to show that they have normal subgroups of finite index withproperty of maximality in the family of virtually cyclic subgroups, and finallyapply Theorem 3.3.
Lemma 4.1.
Let G be a group that satisfies property Max VC ∞ G , if H ⊆ G ,then H has property Max VC ∞ H .The proof of the Lemma is left to the reader.4.1. Mapping class groups.
Let S be a connected, compact, orientablesurface with a finite set of points removed from the interior (punctures).Denote by Mod( S ) the mapping class group of S .Let m > τ : Mod( S ) → Aut ( H ( S ; Z m ))defined by the action of diffeomorphisms on the homology group. The sub-group Mod( S )[ m ] = ker τ, is called the m - congruence subgroup of Mod( S ). Note that this subgrouphas finite index in Mod( S ). ALEJANDRA TRUJILLO-NEGRETE
Theorem 4.2. [6, Prop. 5.11, Thm. 5.3]
Let S be an orientable com-pact surface with finitely many punctures and χ ( S ) ≤ , then gd M od ( S ) is finite. Furthermore, for m ≥ the group Mod( S )[ m ] satisfies property Max VC ∞ Mod( S )[ m ] . Corollary 4.3.
Let S be an orientable compact surface with finitely manypunctures and χ ( S ) ≤
0. If H is a torsion free subgroup of M od ( S ), thengd H ≤ gd H + 1. Proof.
Let H be a torsion free subgroup of M od ( S ) and H m = H ∩ Mod( S )[ m ],with m ≥
3. By Theorem 4.2 and Lemma 4.1, H m has property Max VC ∞ Hm .Since H m is a normal finite index subgroup of H , by applying Theorem 3.3,we conclude that gd H ≤ gd H + 1. (cid:3) Surface braid groups.
Let S be a compact orientable surface. The n -configuration space of S is defined as follows, F n ( S ) = { ( y , ..., y n ) ∈ S n | y i = y j for all i, j ∈ { , ..., n } , i = j } . We endow F n ( S ) with the topology induced by the product topology fromthe space S n . The configuration space F n ( S ) is a connected 2 n -dimensionalopen manifold. There is a natural free action of the symmetric group Σ n on F n ( S ) by permuting the coordinates. We will denote the quotient by theaction as D n ( S ) = F n ( S ) / Σ n and it may be thought of as the configurationspace of n unordered points. Definition 4.4.
Let n ∈ N . The n -braid group of S is defined as B n ( S ) = π ( D n ( S )) . If S is a closed orientable surface of genus > D n ( S ) is a model for EB n ( S ), [5]. We remark that braid groups B n ( S ) are torsion free if andonly if S is different from S (see [13]), in that case gd B n ( S ) coincides withthe cohomological dimension cd B n ( S ), except possibly for the case wherecd( B n ( S )) = 2 and gd( B n ( S )) = 3. Theorem 4.5. [8, Thm. 1.2] If n ≥ and S is a closed orientable surfaceof genus g ≥ , then cd B n ( S ) = n + 1 . The mapping class groups are closely related to braid groups, see [2]. Let S be a closed orientable surface of genus ≥
2, and Q n = { x , ..., x n } be a setwith n ≥ S , then1 / / B n ( S ) / / Mod( S − Q n ) ρ / / Mod( S ) / / . (7) Proposition 4.6. If n ≥ and S is an orientable closed surface of genus g ≥ , then gd B n ( S ) ≤ n + 2 .Proof. If n = 1, then B ( S ) = π ( S ) is hyperbolic and gd B ( S ) = 2.If n ≥
2, then cd B n ( S ) ≥ B n ( S ) = gd B n ( S ). We consider B n ( S ) LASSIFYING SPACES 7 as a subgroup of
M od ( S − Q n ) via the inclusion of the exact sequence (7).By Corollary 4.3 and Theorem 4.5 we havegd B n ( S ) ≤ gd B n ( S ) + 1 = n + 2 . (cid:3) Right angled Artin groups.
Artin groups arose as a natural gener-alization of braid groups B n = B n ( D ). A right-angled Artin group A is agroup with presentation of the form A = h s , ..., s n | s i s j s i ... | {z } m ij = s j s i s j ... | {z } m ij for all i = j i , where m ij = m ji ∈ { , ∞} for all i, j , when m ij = ∞ we omit the relationbetween s i and s j .We remark that right-angled Artin groups contain many interesting groups:some 3-manifold groups, surface groups [3] and graph braid groups.It is well known that for any right-angled Artin group A , the geometricdimension gd A is finite, and A is torsion free. Proposition 4.7. [7]
Every right-angled Artin group embeds in the mappingclass group of any surface of sufficiently high genus.
Proposition 4.8. If A is a right-angled Artin group, then gd A ≤ gd A + 1 . In addition, for any subgroup H ⊆ A , gd H ≤ gd H + 1 .Proof. The proof follows directly by Corollary 4.3 and Proposition 4.7. (cid:3)
References [1] J. Aramayona, D. Juan-Pineda, A. Trujillo-Negrete. On the virtually-cyclic dimensionof mapping class groups of punctured spheres, arXiv:1708.03709, to appear in Algebraicand Geometric Topology.[2] J. S. Birman. Braids, links and mapping class groups, Ann. Math. Stud. 82, PrincetonUniversity Press, 1974.[3] J. Crisp and B. Wiest. Embeddings of graph braid and surface groups in right-angledArtin groups and braid groups, Algebr. Geom. Topol. 4 (2004), 439472.[4] M. Davis and T. Januszkiewicz. Right-angled Artin groups are commensurable withright-angled Coxeter groups, J. Pure Appl. Algebra 153 (2000), no. 3, 229235.[5] E. Fadell, L. Neuwirth. Configuration spaces, Math. Scand. 10 (1962) 111118.
ALEJANDRA TRUJILLO-NEGRETE [6] D. Juan-Pineda, A. Trujillo-Negrete. On Classifying Spaces for the Family of VirtuallyCyclic Subgroups in Mapping Class Groups, Pure and Applied Mathematics QuarterlyVolume 12, Number 2, 261292, 2016[7] Koberda, T. Right-angled Artin groups and a generalized isomorphism problem forfinitely generated subgroups of mapping class groups, Geom. Funct. Anal. (2012) 22:1541[8] D. Lima Gon¸calves, J. Guaschi, M. Maldonado. Embeddings and the (virtual) cohomo-logical dimension of the braid and mapping class groups of surfaces. 2016. < hal-01377681 > [9] W. L¨uck. Survey on classifying spaces for families of subgroups. In infinite groups:geometric, combinatorial and dynamical aspects, volume 248 of Progr. Math., pages269-322. Birkh¨auser, Basel, 2005.[10] W. L¨uck., and M. Weiermann. On the classifying space of the family of virtually cyclicsubgroups.
Pure and Applied Mathematics Quarterly, Nr. 2 (2012) 497–555.[11] C. Mart´ınez-P´erez. Euler classes and Bredon cohomology for groups with restrictedfamilies of finite subgroups, Math. Z. 275 (2013), 761780.[12] C. Mart´ınez-P´erez,
A bound for the Bredon cohomological dimension.
J. Group The-ory, (2007) 731-747.[13] J. Van Buskirk. Braid groups of compact 2-manifolds with elements of finite order,Trans. Amer. Math. Soc. 122 (1966), 8197. Centro de Investigaci´on en Matem´aticas, A. C. Jalisco S/N, Col. Valen-ciana CP: 36023 Guanajuato, Gto, Mxico
E-mail address ::