Polynomial braid combing

arXiv:1611.02187v3 [math.AT] 9 Feb 2018 Classifying spaces for the family of virtually cyclic subgroups of braid groups Ram´on Flores and Juan Gonz´alez-Meneses ∗ February 7th, 2018 Abstract We prove that, for n≥3, the minimal dimension of a model of the classifying space of the braid group Bn, and of the pure braid group Pn, with respect to the family of virtually cyclic groups is n. 1 Introduction In the last years, there has been a growing interest in the description of the classifying space of a group with respect to the family of its virtually cyclic subgroups, usually denoted EG. The main reason for this interest is that this space is the target of the Farrell-Jones conjecture [16], which given a group G, intends to compute the (algebraic) K-theory of the group ring ZGthrough an assembly map that uses as a source nonconnective (topological) K-theory of the classifying space. The conjecture has been proved to be true for a number of groups, see [38] for an excellent survey. The knowledge of the algebraic K-theory of ZGis based in part on finding manageable and “small” models for EG, and in particular in understanding which is the minimal possible dimension of such a model. Hence, this problem has been widely studied in the last years, and computations are available for an important number of classes of groups, as for example locally finite groups [13], polycyclic groups [39] and more generally elementary amenable groups [14], [19], CAT(0)-groups [37], [15], linear groups [12], hyperbolic groups [31], [35] or mapping class groups [15], [33]. We are interested in the (full) braid groups Bn, which in fact can be seen as a particular instance of mapping class groups. These objects, aside from being interesting by their own right, have appeared in very different fields of Mathematics, as for example algebra, topology, physics or geometry. In our context, it has been proved by Juan-S´anchez [32] that Farrell-Jones isomorphism holds for them, and moreover their lower algebraic Ktheory groups have been computed [26]. However, not a lot is known about concrete ∗The first author was supported by MEC-FEDER grant MTM2013-42293-P and the second author was supported by MEC-FEDER grant MTM2013-44233-P. Both authors supported by MEC-FEDER grant MTM2016-76453-C2-1-P 1 models for EBn, not even the minimal dimension of these spaces. In his PhD Thesis [17], M. Fluch (Proposition 4.16), estimated this dimension between 3 and 5 in the case of B3. More recently, Juan-Trujillo [33] bounded it for a general Bnwith a factorial bound (more information at the end of Section 3). The main result of our paper (Theorem 5.1) is that the minimal dimension of a model for EBn(and also of the pure version EPn), for every n≥3, is n. In this way, it is proved that the equality in Question 1.2 of [37] holds for these groups, and we expect that our result is helpful for computations of Bredon (co)homology and/or algebraic K-theory. Of course, next challenge in this context will be to find explicit models of EBnthat realize the minimal dimension, and in this sense, a promising line of research concerns the actions of braid groups on CAT(0)-spaces (see [27] and references there). On the other hand, our methods strongly depend on the rich internal structure of the (full) braid groups, so it is likely they can be applied to braid groups over other surfaces for which the virtually cyclic subgroups are understood (see for example [20] for the case of the sphere). The structure of the paper is the following. In Section 2 we provide the background about classifying spaces and geometric dimensions that will be needed later, and in particular, we describe L¨uck-Weiermann model for the case of virtually cyclic subgroups. Section 3 is devoted to braid groups, and in it we recall general facts about these objects, and prove some new results that will be useful for our computations. In Section 4 we describe the structure of the commensurators of the cyclic subgroups. Section 5 contains the proof of our main theorem. The most difficult point of the proof is to bound the dimensions in the reducible non-periodic case, and this goal is achieved by proving that the quotient of certain normalizers in Bnare virtually torsion-free, and then applying an appropriate result of C. Mart´ınez (Theorem 2.5 in [42]) that only works in this case. 2 Background 2.1 Classifying space for families In this section we review the notion of classifying space for a family of subgroups of a group, which will be the main object of study in this note, with special emphasis in the case of the family of virtually cyclic subgroups. We intend to give a brief survey based on the treatment of [14], the reader interested in a more thorough approach is referred to L¨uck excellent monography [36]. Moreover, we assume that the reader is familiar with the notion of G-CW-complex and other fundamental concepts of equivariant G-homotopy; in any case, all the necessary information can be found in [43], Part I, section 2. Definition 2.1 Let Gbe a discrete group, and Fa family of subgroups of Gwhich is closed under conjugation and passing to subgroups. Then a G-CW-complex Xis a classifying space for the family Fif the fixed point set XHis contractible for every H∈ F and empty for every H /∈ F. The classifying space for the family Fis always unique up to G-homotopy equivalence, and is denoted by EFG. Observe that the definition implies that the isotropy groups of 2 the action of Gover any model for EFGare in F. Moreover, as the family is closed under passing to subgroups, the trivial group is always a member of F, and this implies that EFGis always a contractible G-space. Remark also that if G∈ F, the point is a model for EFG. Remark 2.2 Given a family Fof subgroups of Gclosed under conjugation and subgroups and a subgroup H < G, the family F ∩Hwhose elements are the intersections of Hwith the elements of Fis a family of subgroups of Hfor which the same holds. Actually, F∩H is precisely the set of subgroups of Hwhich belong to F. If we consider the action of H over EFGby restriction, EFGbecomes a model also for EF∩HH. Example 2.3 The following are the most important instances of classifying spaces for families. Each one deserves its own notation: •If F{1}is the family whose only element is the trivial subgroup, EF{1}G=EG, the universal space for principal G-bundles. •If FF in is the family of finite subgroups of G, then EFF in Gis usually denoted EG and called the classifying space for proper actions of G. •If Fvc is the family of virtually cyclic subgroups of G, the space EFvc Gis denoted EG. Observe that in a torsion-free group G,F{1}=FF in(G). In particular, in this case, EG =EG. Also, it is well-known that in a torsion-free group G, every virtually cyclic subgroup is cyclic ([40], Lemma 3.2), so Fvc(G) is the set of cyclic subgroups of G. The classifying space for proper actions has been widely studied from the nineties, as it is the target of the Baum-Connes conjecture [6], which remains unsolved in full generality. However, although the classifying space EGis the object of study of the Farrell-Jones conjecture [16] about the algebraic K-theory and L-theory of the group ring of G, also unsolved in general, much less is known about this classifying space, because a construction of a manageable model for it is usually much more difficult than the construction of a good model for EG. In this sense, a big step forward has been the work of L¨uck-Weiermann, which builds a model for EGusing as building pieces models EFH, with H≤Gand Ffamilies of groups which are contained in the family Fvc(G). Because of its importance in our development, we will describe now this model; details can be found in [39]. Let Gbe a group. An equivalence relation in the set Fvc\FF in of infinite virtually cyclic subgroups of Gis defined by establishing that H∼Kif and only if H∩Kis infinite. We will denote by [H] the equivalence class of a subgroup Hby this relation. Notice that conjugation in Gpreserves Fvc,FF in and the relation ∼, hence we can define g−1[H]gas [g−1Hg] for any g∈Gand any Hin Fvc\FF in. Definition 2.4 If [H]is an equivalence class by ∼, the normalizer of [H]is defined as the subgroup NG[H] := {g∈G|g−1[H]g= [H]}. 3 It is important to observe that NG[H] contains the normalizer NG(H) for every representative Hof the class [H], but in general is not equal to any of these normalizers (Example 2.6 in [39]). However, in the groups we will be working with, there always will be a group H0∈[H] such that NG[H] = NG(H0). In general, we will see in Section 4 that for every group G, one has NG[H] = CommG(H), the commensurator of Hin G(see Definition 4.1). Now a family of subgroups of NG[H] can be defined as follows: F[H] := {K < NG[H]|K∼Hor |K|<∞}. Observe that this family is closed under conjugation (in NG[H]) and taking subgroups. Using these subgroups and families, an appropriate model for EGcan be defined: Theorem 2.5 ([39], Theorem 2.3) Let Ibe a complete set of representatives of the Gorbits (under conjugation) of equivalence classes [H]of infinite virtually cyclic subgroups of G. For every [H]∈I, choose models for ENG[H]and EF[H]NG[H]. Choose also a model for EG. Now consider the G-pushout: `[H]∈IG×NG[H]ENG[H]i// `[H]∈IidG×NG[H]f[H]  EG  `[H]∈IG×NG[H]EF[H]NG[H]//X where f[H]is a cellular NG[H]-map for every [H]∈Iand iis the inclusion, or iis a cellular G-map and f[H]is an inclusion for every [H]∈I. Then Xis a model for EG. In practice, this theorem implies that the existence of good models for the proper classifying space of the commensurators and G, and also of the classifying spaces with respect to the families F[H] will lead to the knowledge of good models for EG. Moreover, the push-out implies the existence of a long exact sequence in Bredon homology, and dimensional consequences that we will analyze in next section. Although we have described here only the particular case of finite and virtually cyclic subgroups (as it is the only one that will be needed), it is worth to point out that L¨uckWeiermann model is defined for any two families Fand Gsuch that F ⊆ G and an equivalence relation subject to some conditions is defined in G\F (see [39], Section 2). 2.2 Minimal dimensions for EG In this section we will recall the results about geometric dimensions of classifying spaces that will be necessary in the remaining of this note. Definition 2.6 Let Ga group, Fa family closed under conjugation and taking subgroups. The geometric dimension of Gwith respect to the family Fis the minimal dimension gdFG of a model for EFG. 4 According to the classical notation, we denote gdFGby gd Gwhen F=F{1}; by gd G when F=FF in; and by gd Gwhen F=Fvc. We are mainly interested in the latter. The study of the dimension of a group is a classic topic of research where group theory, homological algebra and geometry overlap via the different geometric and (co)homological versions of the dimension. Usually, these different versions are related, and small values of them allow a sharp knowledge of the structure of the group. Of course, in this paper we are mainly interested in the geometric dimension with respect to the family of virtually cyclic groups, but in our arguments we will also need to compute some proper geometric dimensions, as well as some algebraic counterparts of it. Among them, it will be particularly important the next one: Definition 2.7 Given a group Gand a family Fof subgroups of Gclosed under conjugation and taking subgroups, the (Bredon) cohomological dimension of Gwith respect to the family F, denoted by cdFG, is the maximal dimension of a nonzero Bredon cohomology group with respect to F. The main definitions about Bredon (co)homology can be found in [43] or [17]. In particular, the following fact is described in the second reference, and it will be relevant for our purposes: Proposition 2.8 In the notation of the previous definition, for every Gand Fthe equality cdFG = gdFGholds whenever gdFG6= 3. If gdFG = 3, then 2≤cdFG≤3. The question about the equality when gdFG = 3 is known as Eilenberg-Ganea conjecture, and it is yet unsolved in the case of the trivial family. The paper [10] contains an interesting discussion about this conjecture, including examples of the strict inequality for the family of finite groups; examples for the family of virtually cyclic groups can be found in [18]. The key result that will lead us to identify gd Bnis a bound implied by the previously described model of L¨uck-Weiermann: Corollary 2.9 (Remark 2.5 in [39]) With the above notations, suppose there exists a natural number ksuch that: •gd G≤k. •gd NG[H]≤k−1, for every [H]∈I. •gdF[H]NG[H]≤k, for every [H]∈I. Then gd G≤k. When applying this corollary, it is interesting to realize that according to Remark 2.2, for every H < G and every family of subgroups of Gclosed under conjugation and subgroups, gdF∩HH≤gdFG. Then, gd Gand gd Gwork as upper bounds for the corresponding minimal dimension of any of their subgroups. We should also mention here Proposition 5.1 in [39], which establishes that an upper bound for gd automatically gives a bound for gd: 5 Proposition 2.10 For any discrete group G, gd G≤gd G+ 1. Note that the equality is possible: for example, as Zis virtually cyclic, gd Z= 0, while gd Z= 1. Usually, however, gd G≤gd G, and it is a question of L¨uck [36] to identify families for which the inequality gd G≤gd G+ 1 holds, and also for which this inequality is an equality. For example, Degrijse-Petrosyan (Corollary 4.4 and Example 6.5 in [14]), prove that the inequality holds for elementary amenable groups, and also construct Bestvina-Brady-like counterexamples for which it does not hold. While it is easy to check that the inequality holds for braid groups (see Proposition 3.7 and the previous discussion), to prove that it is in fact an equality has been one of the main motivations of our paper, and is a beautiful and immediate consequence of Theorem 5.1. 3 Braid groups 3.1 A little survey In this section we collect the basic definitions and facts about braid groups. For further information the interested reader is referred to the more detailed treatments of [26] or [23] Given an integer n≥2, the braid group Bnon nstrands can be defined in several different ways. Algebraically, it is given by the following presentation [4, 5]: Bn=σ1,...,σn−1 σiσj=σjσi,if |i−j|>1 σiσjσi=σjσiσj,if |i−j|= 1  Also, if Mnis the configuration space of ndistinct points in the plane C, Mn={(x1,...,xn)∈Cn|xi6=xjfor i6=j}, and Σnis the symmetric group on nelements, which acts on Mnby permuting coordinates, then Bn=π1(Mn/Σn). This can be visualized by choosing nbase points, say {1,...,n} ⊂ C, and considering a braid as a collection of ndisjoint paths in C×[0,1], called strands, where the i-th strand starts at (i, 0), moves monotonically on the second coordinate, and ends at (j, 1) for some j∈ {1,...,n}. This collection of strands (i.e. this braid) is considered up to isotopy of C×[0,1], fixing the boundary pointwise, and the multiplication of braids is given by stacking and rescaling. The generator σjcorresponds to a braid in which the strands j and j+ 1 cross, while the other n−2 strands are constant. More precisely, σjcan be given by the paths (1, t),...,(j−1, t),j+1 2+1 2e(1−t)iπ , t,j+1 2+1 2e−tiπ, t,(j+ 2, t),...,(n, t) for t∈[0,1]. 6 Each braid has a corresponding permutation, induced by the endpoints of its strands, so there is a surjective map Bn→Σn. The kernel of this map, called the pure braid group and denoted Pn, is the subgroup of braids in which the i-th strand starts at (i, 0) and ends at (i, 1) for i= 1, . . . , n. Actually, Pn=π1(Mn). [7] There is a third definition of Bnthat will be important for us in this paper. Let Dn be a n-times punctured disc (for instance, Dn=D\{1,...,n}where Dis the disc in C with diameter [0, n+1] ⊂R). Then a braid can be seen as an orientable homeomorphism of Dnto itself, fixing ∂(Dn) pointwise, up to isotopy relative to ∂(Dn). That is, Bnis the mapping class group of Dn[30, 7]. We remark that an automorphism corresponding to a braid can be obtained from viewing the family of strands as a motion of the npunctures in D, which can be extended to a continuous motion of all points of D, fixing the boundary. The final position of the points yields the homeomorphism, which is unique up to isotopy of Dnrelative to ∂(Dn). Conversely, if an automorphism fof Dnis given which fixes the boundary ∂(Dn) pointwise, it can be uniquely extended to an automorphism fof the whole disk D. As every automorphism of Dfixing the boundary is isotopic to the trivial automorphism (Alexander’s trick), we can continuously deform idDinto f. The trace of the points 1,...,n under this deformation yields a family of npaths in D×[0,1], which can be extended to C×[0,1] by the identity outside the cylinder. These npaths form the braid corresponding to the isotopy class of f. The braid group Bnis torsion-free [11]. Its center is infinite cyclic, generated by ∆2= (σ1(σ2σ1)···(σn−1σn−2···σ1))2 This element corresponds to a Dehn-twist along a curve parallel to the the boundary of Dn(roughly speaking, a rotation of ∂(Dn) by 360 degrees). Hence, to quotient Bnby its center corresponds to collapsing the boundary to a new puncture, so Bn/h∆2iis a subgroup (of index n+ 1) of the mapping class group of the (n+ 1)-times punctured sphere Sn+1. This way of viewing braids as mapping classes allows to use the powerful theory of Nielsen-Thurston [45], and to classify braids into three geometric types. In this way, a braid αis said to be: •Periodic, if some nontrivial power of αbelongs to h∆2i. •Pseudo-Anosov, if there is a pair of transverse measured foliations of Dnpreserved by α, such that the action of a homeomorphism f(representing α) on one of them scales the measure by some real number λ > 1 (called dilatation factor), and the measure of the other one by λ−1. •Reducible non-periodic, if αpreserves a family of isotopy classes of disjoint, essential simple closed curves in Dn. Here essential means enclosing more than one and less than npunctures. Essentially, reducible braids are those which can be reduced into simpler ones. More precisely, suppose that αis a reducible, non-periodic braid. Up to conjugacy in Bn, we can 7 assume that the family of isotopy classes of curves preserved by αcan be represented by a family of circles. Up to replacing αby a power, we can assume that αis represented by an automorphism fwhich sends every circle to itself, so we can restrict the automorphism fto the connected components obtained by removing the family of circles from Dn. As each of these connected components is again homeomorphic to a punctured disc, each restriction corresponds to a braid with fewer strands. Actually, there is a particular family of isotopy classes of curves, called the canonical reduction system of α,CRS(α), such that each of the mentioned restrictions is either periodic or pseudo-Anosov (see [8]). The decomposition of a reducible braid into simpler ones will be used several times in this paper, so we will make it more precise. Given a reducible, non-periodic braid α, let Cαbe the set of outermost isotopy classes of curves in the canonical reduction system CRS(α). Notice that αpreserves Cα(up to isotopy). Up to conjugacy, we can assume that Cαcan be represented by a disjoint union of unnested circles. Now recall that an automorphism representing αcan be extended to an automorphism fof D, and that there is an isotopy from idDto f. The trace of the punctures under this isotopy represent the strands of α, and the trace of the circles representing Cαlooks like a family of tubes, each one enclosing more than one and less than npunctures. See Figure 1. Figure 1: A reducible braid α∈B8, and the external braid αext. In this case, Cαis represented by a family of three circles. The internal braids inside each tube are, respectively, σ1∈B2,σ1σ2∈B3and the trivial braid in B2. The external braid is σ1σ2σ−1 3σ1σ−1 3∈B4. The decomposition of αalong Cαyields an external braid αext, and some internal braids. The external braid αext corresponds to forgetting what happens inside the tubes: one gets a braid made of tubes and (possibly some) strands, which is transformed into a braid by shrinking each tube to form a single strand. The internal braids are those which are contained into each tube. Actually, as a tube does not necessarily end at the same place it started, there is some shifting of the final points of the strands that should be made in order to define the internal braids properly, but it is clear from the picture (see Figure 1). See [24] for details. A further simplification can still be done. Denote C1,...,Cmthe circles representing Cα. The action of αon Cαinduces a permutation on these circles. If Ci1,...,Ciris a cycle under this permutation, then we can conjugate αin such a way that all internal braids in the tubes corresponding to Ci2,...,Cirare trivial, and the only possibly nontrivial one 8 is inside the tube corresponding to Ci1. We can do the same with all cycles, so we can assume (up to conjugating α) that there is just one internal braid αifor each cycle of tubes determined by α. These internal braids α1,...,αkare unique up to conjugacy (see [25]), and can be used to describe the centralizer of α: Theorem 3.1 (Theorem 1.1 in [25]) Let α∈Bn. If n= 2 the centralizer Z(α) = B2≃Z. If n≥3, the centralizer Z(α)is as follows: •If αis periodic, Z(α)either is equal to Bnor is isomorphic to a subgroup of Bm for some m < n. (Actually, in the latter case it is the braid group of an annulus on less than n−1strands, which can be embedded into Bm). •If αis pseudo-Anosov, Z(α)≃Z2, consisting of all elements having a representative which preserves the same two foliations as α. •If αis reducible, there is a split short exact sequence 1−→ Z(α1)×···×Z(αk)−→ Z(α)pα −→ Z0(αext)→1, where α1,...,αkand αext are defined as above, and Z0(αext)is a finite index subgroup of Z(αext). The map pα:Z(α)→Z(αext) mentioned above goes as follows: given β∈Z(α), one has β−1αβ =α. The canonical reduction system of a braid behaves in a natural way with respect to conjugations [8], meaning that CRS(β−1αβ) is the image of CRS(α) under the mapping class β. But CRS(β−1αβ) = CRS(α), hence βsends CRS(α) to itself, and then it sends Cαto itself, too. Therefore, βcan be decomposed along Cα, and pα(β) is just the external braid corresponding to this decomposition. Notice that we do not necessarily have Cβ=Cα(actually, βcould even be periodic and Cβcould be empty), so pα(β) is not necessarily βext (which could be not defined). pα(β) is the tubular braid described by the action of βon Dn, by looking at the motion of the circles representing Cαand of the punctures not enclosed by Cα. 3.2 Centralizers and roots In this section we will state some further properties of centralizers and roots in the pure braid group Pn, that will be useful later. Proposition 3.2 Roots in Pnare unique. That is, if α, β ∈Pnare such that αm=βm for some m6= 0, then α=β. Proof. This can be obtained from the results in [22], and can also be found in [9]. ✷ Corollary 3.3 If α∈Pnis a pure braid, ZBn(α) = ZBn(αm)for every m6= 0. 9 We invoke again Nielsen-Thurston theory to divide the computation in three cases. Notice that if the original generator of ywas periodic (respectively pseudo-Anosov or reducible), then so is the pure element xgiven by Lemma 4.3, as it is a root of a power of y. 1. Periodic case. Suppose that H=hxiwhere xis periodic. According to Lemma 4.4, CommBn(H) = Bn, and as xis pure, periodic and has no proper roots, x= ∆±2. Therefore gd (NBn[H]/H) = gd (Bn/Z(Bn)), being Z(Bn) = h∆2ithe center of Bn. It is known that Bn/h∆2iis a subgroup of Γ0,n+1, the mapping class group of the sphere with n+ 1 punctures, and it is a consequence of work of [29] and [28] that the proper geometric dimension of Γ0,n+1 is n−2. Hence, gd (NBn[H]/H)≤n−2 in this case. 2. Pseudo-Anosov case. Suppose now that xis pseudo-Anosov. By Corollary 4.5, we know that CommBn(H) is either Z2or an index 2 extension of Z2. Without loss of generality, we may assume that xis a power of one of the two generators of Z2. Hence, CommBn(H)/H is isomorphic to a finite extension of Z⊕Z/m (m= 1 is allowed). Applying Theorem 5.26 of [36], we obtain that gd CommBn(H)/H = 1, the Hirsch number, and hence gd (NBn[H]/H) = 1 in this case. 3. Reducible, non-periodic case. Suppose finally that xis reducible and not periodic. We will find an upper bound gd (NBn(H)/H) basing our strategy in the following result of C. Mart´ınez (Theorem 2.5 in [42]): Theorem 5.2 For any discrete group Γ, the following inequality holds: cd Γ≤max F∈FFin(Γ){pdZWΓFB(WΓF) + rk(WΓF)}. Here pd is the projective dimension over the corresponding group ring, WΓF=NΓF/F is the Weyl group, and B(WΓF) is the module of bounded functions over WΓF(see the mentioned paper for details about these concepts). Moreover, rk(WΓF) is the rank of a maximal elementary abelian subgroup of WΓF, as usual. Recall from Proposition 2.8 that for any group G, cd G= gd Gexcept possibly in the case gd G= 3. From now on we assume Γ = NBn(H)/H, and we should bound the previous sum. We first deal with the term pdZWΓFB(WΓF). By Lemma 7.3 in [34], pdZWΓFB(ZWΓF)≤ pdZΓB(ZΓ) for every finite F < Γ. Moreover, if we can prove that Γ is virtually torsionfree, Lemma 3.9 in [41] implies that the latter coincides with vcd(Γ), the virtual cohomological dimension of Γ, and we would have: cdΓ≤max F∈FFin(Γ){vcd(Γ) + rk(WΓF)}. Let us then show that this is true: Lemma 5.3 The group Γis virtually torsion-free. Proof. We will use the short exact sequence 1 →Pn→Bn p →Σn→1, which we can restrict to NBn(H): 1−→ NBn(H)∩Pn−→ NBn(H)p −→ G−→ 1, 16 where G=p(NBn(H)) ⊂Σnis a finite group. Recall that H=hxiwhere xis pure. Then H✂(NBn(H)∩Pn) and H✂NBn(H), so we can quotient the above injection by Hand we obtain: 1−→ (NBn(H)∩Pn)/H −→ Γp −→ G−→ 1. We need to show that (NBn(H)∩Pn)/H is torsion-free. Let αH ∈(NBn(H)∩Pn)/H be a torsion element. That is, α∈NBn(H)∩Pnis such that αm∈Hfor some nonzero integer m. This means that αm=xkfor some nonzero integers mand k. By Corollary 3.4, hα, xiis a cyclic subgroup, which belongs to Pnas it is generated by pure braids. As hxiis a maximal cyclic subgroup in Pn, we get hα, xi=hxi=H, so α∈Hand αH is trivial in (NBn(H)∩Pn)/H. Hence (NBn(H)∩Pn)/H is torsion-free. ✷ Now we can obtain our bound for the projective dimension. Lemma 5.4 The virtual cohomological dimension of Γis bounded above by n−2. Proof. We saw in the previous lemma that (NBn(H)∩Pn)/H is a torsion-free, finite index subgroup of Γ. Hence, we must show that cd ((NBn(H)∩Pn)/H)≤n−2. Notice that NBn(H)∩Pn=NPn(H). Now we point out that the pure braid group Pnis bi-orderable [44]. This means that there is a total order of its elements which is invariant under multiplication on the left and also on the right. In such an order, if an element is positive (greater than the neutral element) its inverse is negative, and viceversa. Recall that H=hxi. If x > 1, then ax > a and hence axa−1>1 for every a∈Pn. Analogously, if x < 1 then axa−1<1 for every a∈Pn. This means that, in Pn,xcannot be conjugated to its inverse. Therefore NPn(H) = ZPn(H). Now recall from Proposition 3.6 that ZPn(H) = Pt1× ··· × Ptr×Zt, where r, t ≥0 and t1+···+tr+t≤n+r−1. Let Cbe the center of ZPn(H). As the center of each Ptiis cyclic (generated by ∆2 ti), it follows that C≃Zr+t. Denote by Githe quotient of Ptiby its center, which is a (finite index) subgroup of the mapping class group Γ0,ti+1 of the (ti+1)-times punctured sphere. Then ZPn(H)/C =G1×···×Gr. We then have the following short exact sequence, which is a central extension: 1−→ C−→ ZPn(H)−→ G1×···×Gr−→ 1. Moreover, as His central in ZPn(H), we can quotient by Hand obtain: 1−→ C/H −→ ZPn(H)/H −→ G1×···×Gr−→ 1. Recall that we want to bound the cohomological dimension of the middle group. It is important to remark that all groups involved in the above exact sequence are torsion-free: First, we have shown in the previous lemma that ZPn(H)/H is torsion-free, hence so is its subgroup C/H. On the other hand, a torsion element in Gi=Pti/h∆2 tiicomes from a 17 periodic element in Pti. But the only pure braids which are periodic in Ptiare the powers of ∆2 ti, hence Giis torsion-free for every i. Now it follows from Theorem 5.15 in [36] that gd G= gd Gis subadditive for extensions of torsion-free groups. Hence gd(ZPn(H)/H)≤gd(C/H) + gd(G1) + ···+ gd(Gr). Now C≃Zr+t, and His a cyclic subgroup which is maximal. Hence C/H ≃Zr+t−1, and gd(C/H) = r+t−1. Also, each Giis a subgroup of the mapping class group Γ0,ti+1, so gd(Gi) = gd (Gi)≤gd (Γ0,ti+1) = ti−2. We then have: gd(ZPn(H)/H)≤(r+t−1) + t1+···+tr−2r≤r−1 + (n+r−1) −2r=n−2. Therefore gd(ZPn(H)/H)≤n−2. As cd G= gd Gexcept in the case cd G= 2 and gd G= 3, it follows that cd(ZPn(H)/H)≤n−2. ✷ Remark 5.5 A different proof of the previous lemma follows from a direct application of the Gysin sequence (see for example [46], 5.12), but we prefer the one we provide because it includes an explicit description of the normalizer. Now, in order to bound the second term of the sum in Theorem 5.2, we need to understand the torsion of Γ. Lemma 5.6 Any finite subgroup of Γis cyclic. Proof. Consider the natural projection p:NBn(H)→NBn(H)/H = Γ. Let Fbe a finite subgroup of Γ, and let G=p−1(F). Consider the restriction of pto G. Its image is finite, and its kernel is a subgroup of H, so is cyclic. Hence Gis a virtually cyclic group. As G⊂Bn, which is torsion-free, Gis cyclic. Hence p(G) = Fis also cyclic. ✷ We are now in a good position to deal with the rank of Weyl groups, and our bound will be a consequence of the following lemma: Lemma 5.7 For every finite F < Γ, every finite group of the Weyl group WΓFis cyclic, and in particular, rk(WΓF) = 1. Proof. Let F < Γ be finite. Consider the short exact sequence: 1−→ F−→ NΓFp −→ NΓF/F −→ 1, where pis the natural projection. Let Xbe a finite subgroup of NΓF/F =WΓF, and let Y=p−1(X). Then Yis an extension of Y∩Fby X, which are both finite groups. Hence Yis finite. As Y⊂NΓF⊂Γ, by Lemma 5.6 Yis cyclic. Hence X=p(Y) is also cyclic. Observe, in particular, that this implies that rk(WΓF) = 1 for every finite Fin Γ. ✷ Now we can join the previous information to obtain: 18 Proposition 5.8 The proper geometric dimension of Γis smaller or equal to n−1, except possibly in the case when gd Γ = 3, in which it can be at most n. Proof. By Theorem 5.2, Lemma 5.4 and Lemma 5.7, we have cd Γ ≤n−1. Hence by Proposition 2.8, cd Γ = gd Γ ≤n−1, except possibly when cd Γ = 2 and gd Γ = 3. ✷ Proof of Theorem 5.1. Let Bnbe the full braid group, for n≥3.First, as Pn< Bn, Corollary 3.8 implies that gd Bn≥n. To check the other equality, we make use of L¨uck-Weiermann model, and in particular the dimensional conditions of Corollary 2.9: •We know (by [2] or [28], see Section 3 above) that the geometric dimension gd Bn is always n−1, so it is smaller than n. •As NBn[H]⊆Bnfor every cyclic H < Bn, gd NBn[H]≤gd Bn≤n−1. •Let n≥3 and H < Bncyclic. If His generated by a periodic or a pseudo-Anosov braid, it was stated above that gdF[H]NBn[H]≤n−2, and in particular smaller than n. If His generated by a reducible non-periodic braid, the previous proposition states that gdF[H]NBn[H]≤n. Now appealing to Corollary 2.9, gd Bn≤n, and thus gd Bn=n. So we are done. ✷ The same is true for pure subgroups of Bn: Corollary 5.9 We have gd Pn=nfor n≥3. Proof. The bound gd Pn≥nwas proved in Corollary 3.8, while gd Pn≤nis a consequence of Theorem 5.1, taking into account Remark 2.2. ✷ References [1] J. Aramayona and C. Mar´ınez-P´erez, The proper geometric dimension of the mapping class group, Algebr. Geom. Topol. 14 (2014), 217–227. [2] V. Arnold, On some topological invariants of algebraic functions, Trans. Moscow Math. Soc. 21 (1970), 33–52. [3] P. Arnoux, J. Yoccoz, Construction de diffeomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris 292 (1981), 75-–78. [4] E. Artin, Theorie der Z¨pfe, Abh. Math. Sem. Univ. Hamburg 4(1925), no. 1, 47—72. 19 [5] E. Artin, Theory of braids, Ann. of Math. (2) 48 (1947), 101-–126. [6] P. Baum, A. Connes and N. Higson, Classifying space for proper actions and Ktheory of group C∗-algebras, in C∗-algebras: 1943-1993, 240–291, Contemp. Math., 167, Amer. Math. Soc., Providence, RI, 1994. [7] J. S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. [8] J. S. Birman, A. Lubotzky and J. McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983), 1107-–1120. [9] C. Bonatti and L. Par´ıs, Roots in the mapping class groups, Proc. Lond. Math. Soc. 98 (2009), 471–503. [10] N. Brady, I. Leary and B. Nucinkis, On algebraic and geometric dimensions for groups with torsion, J. London Math. Soc. 64 (2001), 489-–500. [11] W.-L. Chow. On the algebraical braid group, Ann. of Math. (2), 49 (1948), 654-–658. [12] D. Degrijse, R. K¨ohl and N. Petrosyan, Classifying spaces with virtually cyclic stabilizers for linear groups, Transformation Groups 20 (2015), 381–394. [13] D. Degrijse and N. Petrosyan, Commensurators and classifying spaces with virtually cyclic stabilizers, Groups Geom. Dyn. 7(2013) 543—555. [14] D. Degrijse and N. Petrosyan, Geometric dimension of groups for the family of virtually cyclic subgroups, J. Topol. 7(2014) 697—724. [15] D. Degrijse and N. Petrosyan, Bredon cohomological dimensions for groups acting on CAT(0)-spaces, Groups Geom. Dyn. 9(2015), 1231—1265. [16] T. Farrell and L. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc., 6(1993), 249—297. [17] M. Fluch, On Bredon (co-)homological dimensions of groups PhD Thesis, arXiv:1009.4633. [18] M. Fluch and I. Leary, An Eilenberg-Ganea phenomenon for actions with virtually cyclic stabilisers Groups Geom. Dyn. 8(2014), 135-–142. [19] M. Fluch and B. Nucinkis, On the classifying space for the family of virtually cyclic subgroups for elementary amenable groups, Proc. Amer. Math. Soc. 141 (2013), 3755- –3769. [20] D. Gon¸calves and J. Guaschi, The classification of the virtually cyclic subgroups of the sphere braid groups SpringerBriefs in Mathematics, Springer, 2013. 20 [21] J. Gonz´alez-Meneses, Ordering pure braid groups on compact, connected surfaces, Pacific J. of Math. 203 (2002) 369–378. [22] J. Gonz´alez-Meneses, The n-th root of a braid is unique up to conjugacy, Alg. Geom. Top. 3(2003), 1103–1118. [23] J. Gonz´alez-Meneses, Basic results on braid groups, Annales Math´ematiques Blaise Pascal 18 (2011) 15–59. [24] J. Gonz´alez-Meneses, On reduction curves and Garside properties of braids, Contemp. Math. 538 (2011), 227–244. [25] J. Gonz´alez-Meneses and B. Wiest, On the structure of the centralizer of a braid, Ann. Sci. ´ Ec. Norm. Sup. 37 (2004), 729–757. [26] J. Guaschi and D. Juan-Pineda, A survey of surface braid groups and the lower algebraic K-theory of their group rings, In Handbook of Group Actions 23–76, Advanced Lectures in Mathematics 32, International Press of Boston Inc., 2015. [27] T. Haettel, D. Kielak and P. Schwer, The 6-strand braid group is CAT(0) Geom. Dedicata 182 (2016), 263-–286. [28] J. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math, 84 (1986), 157–176. [29] S. Hensel, D. Osjada and P. Przytycki, Realisation and dismantlability, Preprint arXiv:1205.0513. [30] A. Hurwitz, Uber Riemannsche Fl¨achen mit gegebenen Verzweigungspunkten, Math. Ann. 39, no. 1 (1891), 1–60. [31] D. Juan-Pineda and I. Leary, On classifying spaces for the family of virtually cyclic subgroups Contemp. Math. 407 (2006), 135–145. [32] D. Juan-Pineda and J. S´anchez-Salda˜na, The K and L theoretic Farrell-Jones isomorphism conjecture for braid groups Preprint arXiv:1511.02737. [33] D. Juan-Pineda and A. Trujillo-Negrete, On classifying spaces for the family of virtually cyclic subgroups in mapping class groups, Preprint arXiv:1606.00306. [34] P. Kropholler and G. Mislin, Groups acting on finite-dimensional spaces with finite stabilizers, Comment. Math. Helv. 73 (1998), 122—136. [35] J. Lafont and I. Ortiz, Relative hyperbolicity, classifying spaces, and lower algebraic K-theory Topology 46 (2007), 527–553. [36] W. L¨uck, Survey on classifying spaces for families of subgroups, in Infinite groups: geometric, combinatorial and dynamical aspects, 269—322, Progr. Math., 248, Birkh¨auser, Basel, 2005. 21 [37] W. L¨uck, On the classifying space of the family of finite and of virtually cyclic subgroups for CAT(0)-groups, M¨unster J. of Math. 2(2009), 201–214. [38] W. L¨uck and H. Reich, The Baum-Connes and the Farrell-Jones conjectures in Kand L-theory In Handbook of K-theory, 703–842, Springer. [39] W. L¨uck and M. Weiermann, On the classifying space of the family of virtually cyclic subgroups, Pure Appl. Math. Q. 8, (2012), 497–555. [40] D. Macpherson, Permutation groups whose subgroups have just finitely many orbits, in Ordered groups and infinite permutation groups, 221–230, Kluwer Academic Publishers, 1996. [41] C. Mart´ınez-P´erez, A bound for the Bredon cohomological dimension for groups, J. of Group Theory 10 (2007), 731–747. [42] C. Mart´ınez-P´erez, Euler classes and Bredon cohomology for groups with restricted families of finite subgroups, Math. Z. 275 (2013), 761—780. [43] G. Mislin and A. Valette, Proper group actions and the Baum-Connes conjecture, in Advanced courses in Mathematics, no. 55, CRM Barcelona, 2003. [44] D. Rolfsen, J. Zhu, Braids, Orderings and zero divisors, J. Knot Theory Ramif. 7 (1998), no. 6, 837–˜n841. [45] W. P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc., 19 No. 2 (1988), 417-–431. [46] G.W. Whitehead, Elements of homotopy theory, Graduate Texts in Math., vol 61, Springer-Verlag, 1978. Ram´on Flores: Dpto. Geometr´ıa y Topolog´ıa. Facultad de Matem´aticas Instituto de Matem´aticas (IMUS) Universidad de Sevilla Av. Reina Mercedes s/n 41012 Sevilla (SPAIN) E-mail: ramonjflor[email protected] URL: www.personal.us.es/ramonjflores Juan Gonz´alez-Meneses: Dpto. ´ Algebra. Facultad de Matem´aticas Instituto de Matem´aticas (IMUS) Universidad de Sevilla. Av. Reina Mercedes s/n 41012 Sevilla (SPAIN) E-mail: menes[email protected] URL: www.personal.us.es/meneses 22