Some remarks concerning degree zero complete cohomology

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?

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

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.

This is a report on our work during the last few years on extending the Bieri-Neumann-Strebel-Renz theory of “geometric invariants” of groups to a theory of group actions on non-positively curved (= CAT(0)) spaces. With the exception of Theorem 8, which is proved here, and the related material in §5.3, proofs of all our theorems can be found in our papers [BG I ] (controlled connectivity and openness results), [BG II ] (the geometric invariants) and [BG III ] ( SL 2 actions on the hyperbolic plane). An earlier expository paper [BG 98] is also relevant. The geometric invariants we recall the “geometric” or “Σ-” invariants of groups developed during the 1980's by Bieri, Neumann, Strebel and Renz (abbrev. BNSR); see [BNS 87], [BR 88], [Re 88]. We set things out in a way which leads directly to generalizations which were not anticipated in the original literature. Let G be a group of type F n , n ≥ 1. Let X be a contractible G -CW complex which is either (a) free with cocompact n -skeleton, or (b) properly discontinuous and cocompact. Case (a) exists by the definition of F n ; Case (b) is often useful but can only exist when G has finite virtual cohomological dimension. Controlled connectivity Let χ : G → ℝ be a non-zero character, i.e., a homomorphism to the additive group of real numbers. Reinterpret ℝ as the group of translations, Transl, of the Euclidean line, and thus reinterpret χ as an action of G on by translations.

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.

Balance in complete cohomology

We describe an algorithm to find the virtual cohomological dimension of the automorphism group of a right-angled Artin group. The algorithm works in the relative setting; in particular it also applies to untwisted automorphism groups and basis-conjugating automorphism groups. The main new tool is the construction of free abelian subgroups of certain Fouxe-Rabinovitch groups of rank equal to their virtual cohomological dimension, generalizing a result of Meucci in the setting of free groups.

The theory of discrete groups acting on finite dimensional Euclidean open balls by hyperbolic isometries was borne around the end of 19th century within the works of Fuchs, Klein and Poincaré. We develop the theory of discrete groups acting by hyperbolic isometries on the open unit ball of an infinite dimensional separable Hilbert space. We present our investigations on the geometry of limit sets at the sphere at infinity with an attempt to highlight the differences between the finite and infinite dimensional theories. We discuss the existence of fixed points of isometries and the classification of isometries. Various notions of discreteness that were equivalent in finite dimensions, no longer turn out to be in our setting. In this regard, the robust notion of strong discreteness is introduced and we study limit sets for properly discontinuous actions. We go on to prove a generalization of the Bishop-Jones formula for strongly discrete groups, equating the Hausdorff dimension of the radial limit set with the Poincaré exponent of the group. We end with a short discussion on conformal measures and their relation with Hausdorff and packing measures on the limit set.

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.

We construct free abelian subgroups of the group U(AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.

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.

On Farrell–Tate cohomology of SL2 over S-integers

Bestvina and Mess [BM] proved a remarkable formula for torsion free hyperbolic groups $$ \dim_L\partial\Gamma=cd_L\Gamma-1 $$ connecting the cohomological dimension of a group $\Gamma$ with the cohomological dimension of its boundary $\partial\Gamma$. In [Be] Bestvina introduced a notion of $\sZ$-structure on a discrete group and noticed that his formula holds true for all torsion free groups with $\sZ$-structure. Bestvina's notion of $\sZ$-structure can be extended to groups containing torsion by replacing the covering space action in the definition by the geometric action. Though the Bestvina-Mess formula trivially is not valid for groups with torsion, we show that it still holds in the following modified form: {\it The cohomological dimension of a $\sZ$-boundary of a group $\Gamma$ equals its global cohomological dimension for every PID $L$ as the coefficient group} $$ \dim_L\partial\Gamma=gcd_L(\partial\Gamma). $$ Using this formula we show that the cohomological dimension of the boundary $\dim_{L}\partial\Gamma$ is a quasi-isometry invariant of a group.