We are interested in the relationship between the virtual cohomological dimension (or vcd) of a discrete group Gamma and the smallest possible dimension of a model for the classifying space of Gamma relative to its family of virtually cyclic subgroups. In this paper we construct a model for the virtually cyclic classifying space of the Heisenberg group. This model has dimension 3, which equals the vcd of the Heisenberg group. We also prove that there exists no model of dimension less than 3.
We study the minimal dimension of the classifying space of the family of virtually cyclic subgroups of a discrete group. We give a complete answer for instance if the group is virtually poly-Z, word-hyperbolic or countable locally virtually cyclic. We give examples of groups for which the difference of the minimal dimensions of the classifying spaces of virtually cyclic and of finite subgroups is -1, 0 and 1, and show in many cases that no other values can occur.
In this paper we construct a model for the classifying space, BVCG, of a crystallographic group G of rank n relative to the family VC of virtually-cyclic subgroups of G. The model is used to show that there exists no other model for the virtually-cyclic classifying space of G with dimension less than vcd(G)+1, where vcd(G) denotes the virtual cohomological dimension of G. In addition, the dimension of our construction realizes this limit.
We give a bound for the geometric dimension for the family of virtually cyclic groups in mapping class groups of a compact surface with punctures, possibly with nonempty boundary and negative Euler characteristic.
We give a bound for the virtually cyclic dimension of groups with a normal subgroup of finite index which satisfies that every infinite virtually-cyclic subgroup is contained in a unique maximal such subgroup. As an application we provide a bound for the virtually-cyclic dimension for the braid group of a closed surface with genus greater than 2 and for right-angled Artin groups.
Let G be a discrete group which acts properly and isometrically on a complete CAT(0)-space X. Consider an integer d with d=1 or d greater or equal to 3 such that the topological dimension of X is bounded by d. We show the existence of a G-CW-model E_fin(G) for the classifying space for proper G-actions with dim(E_fin(G)) less or equal to d. Provided that the action is also cocompact, we prove the existence of a G-CW-model E_vcyc(G) for the classifying space of the family of virtually cyclic subgroups such that dim(E_vcyc(G)) is less or equal to d+1.
By examining commensurators of virtually cyclic groups, we show that for each natural number n, any locally finite-by-virtually cyclic group of cardinality aleph_n admits a finite dimensional classifying space with virtually cyclic stabilizers of dimension n+3. As a corollary, we prove that every elementary amenable group of finite Hirsch length and cardinality aleph_n admits a finite dimensional classifying space with virtually cyclic stabilizers.
We show that elementary amenable groups, which have a bound on the orders of their finite subgroups, admit a finite dimensional model for the classifying space with virtually cyclic isotropy.
We prove that, for $n\geq 3$, the minimal dimension of a model of the classifying space of the full braid group $B_n$, and of the pure braid group $P_n$, with respect to the family of virtually cyclic groups is $n$.
We show that every discrete subgroup of $\mathrm{GL}(n,\mathbb{R})$ admits a finite dimensional classifying space with virtually cyclic stabilizers. Applying our methods to $\mathrm{SL}(3,\mathbb{Z})$, we obtain a four dimensional classifying space with virtually cyclic stabilizers and a decomposition of the algebraic $K$-theory of its group ring.
We show that a countable group is locally virtually cyclic if and only if its Bredon cohomological dimension for the family of virtually cyclic subgroups is at most one.
By studying commensurators of virtually cyclic groups, we prove that every elementary amenable group of finite Hirsch length h and cardinality aleph-n admits a finite dimensional classifying space with virtually cyclic stabilizers of dimension n+h+2. We also provide a criterion for groups that fit into an extension with torsion-free quotient to admit a finite dimensional classifying space with virtually cyclic stabilizers. Finally, we exhibit examples of integral linear groups of type F whose geometric dimension for the family of virtually cyclic subgroups is finite but arbitrarily larger than…
We assign real numbers to finite sheeted coverings of compact CW complexes designed as finite counterparts to the Novikov-Shubin numbers. We prove an approximation theorem in the case of virtually cyclic fundamental groups employing methods from Diophantine approximation.
Let G be a virtually cyclic of the form (Z_a x Z_b) x Z or [Z_a x (Z b x Q_{2^i})] x Z. We compute the integral cohomology ring of G, and then obtain the periodicity of the Farell cohomology of these groups.
We calculate the virtually-cyclic dimension of the mapping class group of a sphere with at most six punctures. As an immediate consequence, we obtain the virtually-cyclic dimension of the mapping class group of the twice-holed torus and of the closed genus-two surface.
Let G be an infinite cyclic extension, 1 -> B -> G -> Z -> 1, of a group B where the action of Z on the set of conjugacy classes of non-trivial elements of B is free. This class of groups includes certain ascending HNN-extensions with abelian or free base groups, certain wreath products by Z and the soluble Baumslag-Solitar groups BS(1,m) with |m|> 1. We construct a model for Evc(G), the classifying space of G for the family of virtually cyclic subgroups of G, and give bounds for the minimum dimension of Evc(G). We construct a 2-dimensional model for Evc(G) where G is a soluble Baumslag-Solita…
Classifying spaces for families of virtually abelian subgroups of surface braid groups
Universidad de Sevilla（塞维利亚大学）；Instituto de Matemáticas (IMUS), Universidad de Sevilla（塞维利亚大学数学研究所）；Instituto de Matemáticas, Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）
Let G be a group acting isometrically with discrete orbits on a separable complete CAT(0)-space of bounded topological dimension. Under certain conditions, we give upper bounds for the Bredon cohomological dimension of G for the families of finite and virtually cyclic subgroups. As an application, we prove that the mapping class group of any closed, connected, and orientable surface of genus g greater than 1 admits a (9g-8)-dimensional classifying space with virtually cyclic stabilizers. In addition, our results apply to fundamental groups of graphs of groups and groups acting on Euclidean bui…
We determine the large scale geometry of the minimal displacement set of a hyperbolic isometry of a systolic complex. As a consequence, we describe the centraliser of such an isometry in a systolic group. Using these results, we construct a low-dimensional classifying space for the family of virtually cyclic subgroups of a group acting properly on a systolic complex. Its dimension coincides with the topological dimension of the complex if the latter is at least four. We show that graphical small cancellation complexes are classifying spaces for proper actions and that the groups acting on them…
There is the problem, whether for a given virtually torsionfree discrete group $Γ$ there exists a cocompact proper topological $Γ$-manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. It is related to Nielsen's Realization and to the problem of Brown, whether there is a d-dimensional model for the classifying space for proper actions, if the underlying group has virtually cohomological dimension d. Assuming that the expected manifold model has a zero-dimensional singular set, we solve the problem in the Poincaré category and obtain new results about…
Given a group $G$ and a family of subgroups $\mathcal{F}$, we consider its classifying space $E_{\mathcal F}G$ with respect to $\mathcal{F}$. When $\mathcal F = \mathcal{VC}yc$ is the family of virtually cyclic subgroups, Juan-Pineda and Leary conjectured that a group admits a finite model for this classifying space if and only if it is virtually cyclic. By establishing a connection to conjugacy growth we can show that this conjecture holds for linear groups. We investigate a similar question that was asked by Lück--Reich--Rognes--Varisco for the family of cyclic subgroups. Finally, we constru…
We construct two models for the classifying space for the family of infinite cyclic subgroups of the fundamental group of the Klein bottle. These examples do not fit in general constructions previously done, for example, for hyperbolic groups.
We prove that for every finitely generated subgroup of a virtually connected Lie group which admits a finite dimensional model for the classifying space for proper actions the assembly map in algebraic K-theory is split injective. We also prove a similar statement for algebraic L-theory, which in particular implies the integral Novikov conjecture for such groups.
We prove that every finite connected simplicial complex is homotopy equivalent to the quotient of a contractible manifold by proper actions of a virtually torsion-free group. As a corollary, we obtain that every finite connected simplicial complex is homotopy equivalent to the classifying space for proper bundles of some virtual Poincaré duality group.
This thesis concerns the study of the Bredon cohomological and geometric dimensions of a discrete group $G$ with respect to a family $\mathfrak{F}$ of subgroups of $G$. With that purpose, we focus on building finite-dimensional models for $\operatorname{E}_{\mathfrak{F}} \left( G \right)$. The cases of the family $\mathfrak{Fin}$ of finite subgroups of a group and the family $\mathfrak{VC}$ of virtually cyclic subgroups of a group have been widely studied and many tools have been developed to relate the classifying spaces for $\mathfrak{VC}$ with those for $\mathfrak{Fin}$. Given a discrete gr…
Let $p$ be a prime. We calculate the connective unitary K-theory of the smash product of two copies of the classifying space for the cyclic group of order $p$, using a Künneth formula short exact sequence. As a corollary, using the Bott exact sequence and the mod $2$ Hurewicz homomorphism we calculate the connective orthogonal K-theory of the smash product of two copies of the classifying space for the cyclic group of order two.
By considering the Bredon analogue of complete cohomology of a group, we show that every group in the class $\LHFF$ of type Bredon-$\FP_\infty$ admits a finite dimensional model for $\EFG$. We also show that abelian-by-infinite cyclic groups admit a $3$-dimensional model for the classifying space for the family of virtually nilpotent subgroups. This allows us to prove that for $\mF,$ the class of virtually cyclic groups, the class of $\LHFF$-groups contains all locally virtually soluble groups and all linear groups over $\mathbb C$ of integral characteristic.
The equivariant cohomology for actions of compact connected abelian groups and elementary abelian p-groups have been widely studied in the last decades. We study some of these results on actions of finite cyclic groups over a field of positive characteristic. In particular, we study the module structure of the equivariant cohomology over the cohomology of the classifying space and we provide a criterion for freeness over this ring and the polynomial subring of it. We apply these results to canonical actions of finite cyclic groups on polyhedral products arising from the boundary of a polygo
We define the notion of a hierarchically cocompact classifying space for a family of subgroups of a group. Our main application is to show that the mapping class group $\mbox{Mod}(S)$ of any connected oriented compact surface $S$, possibly with punctures and boundary components and with negative Euler characteristic has a hierarchically cocompact model for the family of virtually cyclic subgroups of dimension at most $\mbox{vcd} \mbox{Mod}(S)+1$. When the surface is closed, we prove that this bound is optimal. In particular, this answers a question of Lück for mapping class groups of surfaces.
We use assembly maps to study $\mathbf{TC}(\mathbb{A}[G];p)$, the topological cyclic homology at a prime $p$ of the group algebra of a discrete group $G$ with coefficients in a connective ring spectrum $\mathbb{A}$. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphism on homotopy groups. For infinite groups, we establish pro-isomorphism, (split) injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, for hyperbolic groups and for virtually finitely generated abelian groups, we…
This is a revised version of the author's PhD thesis, including the corrections by the examiners. It also includes a few additional small corrections. In this thesis the objects of study are classifying spaces of groups with stabilisers in a given family of subgroups. Given a group G and a family of subgroups we study the minimal dimension a classifying space can have. We focus on classifying spaces with virtually cyclic stabilisers.
We compute rationally the topological (complex) K-theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW-model for its classifying space for proper G-actions. For instance word-hyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy this assumption. The answer is given in terms of the group cohomology of G and of the centralizers of finite cyclic subgroups of prime power order. We also analyze the multiplicative structure.
We prove that the Waldhausen nilpotent class group of an injective index 2 amalgamated free product is isomorphic to the Farrell-Bass nilpotent class group of a twisted polynomial extension. As an application, we show that the Farrell-Jones Conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the family of finite-by-cyclic subgroups.
For Artin groups of dihedral type, we compute the Bredon homology groups of the classifying space for the family of virtually cyclic subgroups with coefficients in the K-theory of a group ring.
Very effective algebraic and hermitian K-theory of the cyclic group of order two
University of Kentucky（肯塔基大学）；James Madison University（詹姆斯麦迪逊大学）；Duke University（杜克大学）；University of Virginia（弗吉尼亚大学）；University of Pennsylvania（宾夕法尼亚大学）
Let $G$ be an infinite discrete group. A classifying space for proper actions of $G$ is a proper $G$-CW-complex $X$ such that the fixed point sets $X^H$ are contractible for all finite subgroups $H$ of $G$. In this paper we consider the stable analogue of the classifying space for proper actions in the category of proper $G$-spectra and study its finiteness properties. We investigate when $G$ admits a stable classifying space for proper actions that is finite or of finite type and relate these conditions to the compactness of the sphere spectrum in the homotopy category of proper $G$-spectra a…
The geometric dimension for proper actions $\underline{\mathrm{gd}}(G)$ of a group $G$ is the minimal dimension of a classifying space for proper actions $\underline{E}G$. We construct for every integer $r\geq 1$, an example of a virtually torsion-free Gromov-hyperbolic group $G$ such that for every group $Γ$ which contains $G$ as a finite index normal subgroup, the virtual cohomological dimension $\mathrm{vcd}(Γ)$ of $Γ$ equals $\underline{\mathrm{gd}}(Γ)$ but such that the outer automorphism group $\mathrm{Out}(G)$ is virtually torsion-free, admits a cocompact model for $\underline E\mathrm{…
Using equivariant obstruction theory we construct equivariant maps from certain classifying spaces to representation spheres for cyclic groups, product of elementary Abelian groups and dihedral groups. Restricting them to finite skeleta constructs equivariant maps between spaces which are related to the topological Tverberg conjecture. This answers negatively a question of Özaydin posed in relation to weaker versions of the same conjecture. Further, it also has consequences for Borsuk-Ulam properties of representations of cyclic and dihedral groups.
We consider brave new cochain extensions $F(BG_+,R)\to F(EG_+,R)$, where $R$ is either a Lubin-Tate spectrum $E_n$ or the related 2-periodic Morava K-theory $K_n$, and $G$ is a finite group. When $R$ is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a $G$-Galois extension in the sense of John Rognes, but not always faithful. We prove that for $E_n$ and $K_n$ these extensions are always faithful in the $K_n$ local category. However, for a cyclic $p$-group $C_{p^r}$, the cochain extension $F({BC_{p^r}}_+,E_n) \to F({EC_{p^r}}_+,E_n)$ is not a Galois extensions because it…
The homotopy category of the bordism category $hBord_d$ has as objects closed oriented $(d-1)$-manifolds and as morphisms diffeomorphism classes of $d$-dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of $hBord_d$ for $d = 1$, exhibiting it as a circle bundle over $\mathbb{CP}^\infty_{-1}$. As part of our proof we construct a quotient $Bord_1^{red}$ of the cobordism category where circles are deleted. We show that this category has classifying space $Ω^{\infty-2}\mathbb{CP}^\infty_{-1}$ and moreover that, if one equips these bordisms wit…
