Controlled topology and group actions

Our main result provides necessary and sufficient conditions for a finitelygenerated subgroup of GLn(C), n > 0, to have finite virtual cohomological dimension. A group has finite virtual cohomological dimension (VCD) if it has a subgroup of finite index which has finite cohomological dimension; this dimension is, in fact, the same for all torsion-free subgroups of finite index. It is, of course, necessary for a group r with VCD(T) Q), is independent of the choice of solvable series; thus, for a polycyclic group r, h(T) is the number of infinite factors in a normal series with cyclic quotients. We announce our main result.

Our main result provides necessary and sufficient conditions for a finitelygenerated subgroup of GLn(C), n > 0, to have finite virtual cohomological dimension. A group has finite virtual cohomological dimension (VCD) if it has a subgroup of finite index which has finite cohomological dimension; this dimension is, in fact, the same for all torsion-free subgroups of finite index. It is, of course, necessary for a group r with VCD(T) Q), is independent of the choice of solvable series; thus, for a polycyclic group r, h(T) is the number of infinite factors in a normal series with cyclic quotients. We announce our main result.

When is group cohomology finitary?

If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a profinite group, the fibrant objects in the model category of discrete G-spectra are also dicult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in C, under certain finiteness conditions. Similarly, in this paper, we show that if G has finite virtual cohomological dimension and X is a discrete Gspectrum, then there is an explicit fibrant model for X. Also, we give several applications of this concrete model related to closed subgroups of G.

We study the algebraic structure of the outer automorphism group of a general right-angled Artin group. We show that this group is virtually torsion-free and has finite virtual cohomological dimension. This generalizes results proved earlier by the authors and Crisp for 2-dimensional right-angled Artin groups.

The purpose of this paper is to give a necessary and sufficient condition in order that the S limit set of a point consists of exactly one periodic orbit which is both S-Lyapunov stable and S-asymptotically stable. The general reference for most terms used in this paper is [2]. In this paper we assume (X, T, r) is a transformation group. X will always be assumed to be a uniform space where the Hausdorff topology in X is induced by the uniformity. T will be assumed to be generative, that is, T is isomorphic to Rm X I X C where R is the additive group of real numbers, I is the additive group of integers, m and n are nonnegative integers, and C is a compact abelian group [4]. S will denote a closed replete semigroup in T. The S-limit set of yEX, Sy, is defined bySv = n te,s Cl (ytS) [1 ]. DEFINITION 1. The orbit of a point yEX is said to be uniformly S-Lyapunov stable with respect to BCX provided that for each index a of X there is an index ,3 of X such that if weEBnyt'13 for some t' E T then wt Eyt'ta for all t E S.

Certain algebraic invariants of the integral group ring ZG of a group G were introduced and investigated in relation to the problem of extending the Farrell-Tate cohomology, which is defined for the class of groups of finite virtual cohomological dimension. It turns out that the finiteness of these invariants of a group G implies the existence of a generalized Farrell-Tate cohomology for G which is computed via complete resolutions. In this article we present these algebraic invariants and their basic properties and discuss their relationship to the generalized Farrell-Tate cohomology. In addition we present the status of conjecture which claims that the finiteness of these invariants of a group G is equivalent to the existence of a finite dimensional model for EG, the classifying space for proper actions.

Abstract

For a p-toral group G, we answer the question which compact (respectively, open) smooth manifolds M can be diffeomorphic to the fixed point sets of smooth actions of G on compact (respectively, open) smooth manifolds E of the homotopy type of a finite ℤ-acyclic CW complex admitting a cellular map of period p, with exactly one fixed point. In the case where the CW complex is contractible, E can be chosen to be a disk (respectively, Euclidean space).

The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : $\dim (A\cup B)\le \dim A+\dim B+1$. Theorem. Suppose $A,B$ are subsets of a metrizable space and $K$ and $L$ are CW complexes. If $K$ is an absolute extensor for $A$ and $L$ is an absolute extensor for $B$, then the join $K*L$ is an absolute extensor for $A\cup B$. As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension: Theorem. Suppose $A$, $B$ are subsets of a metrizable space. Then \begin{equation*}\dim _{\mathbf {R} }(A\cup B)\le \dim _{\mathbf {R} }A+\dim _{\mathbf {R} }B+1 \end{equation*} for any ring $\mathbf {R}$ with unity and \begin{equation*}\dim _{G}(A\cup B)\le \dim _{G}A+\dim _{G}B+2\end{equation*} for any abelian group $G$. The second part of the paper is devoted to the question of existence of universal spaces: Suppose $\{K_{i}\}_{i\ge 1}$ is a sequence of CW complexes homotopy dominated by finite CW complexes. Then [a.] Given a separable, metrizable space $Y$ such that $K_{i}\in AE(Y)$, $i\ge 1$, there exists a metrizable compactification $c(Y)$ of $Y$ such that $K_{i}\in AE(c(Y))$, $i\ge 1$. [b.] There is a universal space of the class of all compact metrizable spaces $Y$ such that $K_{i}\in AE(Y)$ for all $i\ge 1$. [c.] There is a completely metrizable and separable space $Z$ such that $K_{i}\in AE(Z)$ for all $i\ge 1$ with the property that any completely metrizable and separable space $Z’$ with $K_{i}\in AE(Z’)$ for all $i\ge 1$ embeds in $Z$ as a closed subset.

The articles in these two volumes arose from papers given at the 1991 International Symposium on Geometric Group Theory, and they represent some of the latest thinking in this area. This first volume contains contributions from many of the world's leading figures in this field, and their contributions demonstrate the many interesting facets of geometrical group theory. For anyone whose interest lies in the interplay between groups and geometry, these books will be an essential addition to their library

Let G be a group acting freely, properly discontinuously and cellularly on some finite dimensional CW-complex Σ(2n) which has the homotopy type of the 2n-sphere 𝕊2n. Then, that action induces a homomorphism G → Aut(H2n(Σ(2n))). We classify all pairs (G, φ), where G is a virtually cyclic group and φ: G → Aut(ℤ) is a homomorphism, which are realizable in the way above and the homotopy types of all possible orbit spaces as well. Next, we consider the family of all groups which have virtual cohomological dimension one and which act on some Σ(2n). Those groups consist of free groups and semi-direct products F ⋊ ℤ2 with F a free group. For a group G from the family above and a homomorphism φ: G → Aut(ℤ), we present an algebraic criterion equivalent to the realizability of the pair (G, φ). It turns out that any realizable pair can be realized on some Σ(2n) with dim Σ(2n) ≤ 2n + 1.

We describe degree zero mod-p complete cohomology modulo its radical in purely group-theoretic terms, for members of a class of groups that includes all groups of finite virtual cohomological dimension. We give examples to show that our description does not apply to all discrete groups. We also give examples of discrete groups to which Quillen's description of the ordinary mod-p cohomology ring (up to F-isomorphism) does not apply.

Let $G$ be an infinite discrete group. A classifying space for proper actions\nof $G$ is a proper $G$-CW-complex $X$ such that the fixed point sets $X^H$ are\ncontractible for all finite subgroups $H$ of $G$. In this paper we consider the\nstable analogue of the classifying space for proper actions in the category of\nproper $G$-spectra and study its finiteness properties. We investigate when $G$\nadmits a stable classifying space for proper actions that is finite or of\nfinite type and relate these conditions to the compactness of the sphere\nspectrum in the homotopy category of proper $G$-spectra and to classical\nfiniteness properties of the Weyl groups of finite subgroups of $G$. Finally,\nif the group $G$ is virtually torsion-free we also show that the smallest\npossible dimension of a stable classifying space for proper actions coincides\nwith the virtual cohomological dimension of $G$, thus providing the first\ngeometric interpretation of the virtual cohomological dimension of a group.\n

We study the action of Bianchi groups on the hyperbolic 3-space $$\mathbb {H}^3 $$ . Given the standard fundamental domain for this action and any point in $$ \mathbb {H}^3 $$ , we show that there exists an element in the group which sends the given point into the fundamental domain such that its height is bounded by a quadratic function on the coordinates of the point. This generalizes and establishes a sharp version of a similar result of Habegger and Pila for the action of the Modular group on the hyperbolic plane. Our main theorem can be applied in the reduction theory of binary Hermitian forms with entries in the ring of integers of quadratic imaginary fields. We also show that the asymptotic behavior of the number of elements in a fixed Bianchi group with height at most T is biquadratic in T.