When is group cohomology finitary?

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.

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.

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

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 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.

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.

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.

We use the formalism of tensor network states to investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low-energy excitations. In particular, we show that the matrix product state transfer matrix (MPS-TM)—a central object in the computation of static correlation functions—provides important information about the location and magnitude of the minima of the low-energy dispersion relation(s), and we present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM’s eigenspectrum and give several arguments for the close relation between the structure of the low-energy spectrum of the system and the form of the static correlation functions. Finally, we discuss how the MPS-TM connects to the exact quantum transfer matrix of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of the MPS, which allows one to reinterpret variational MPS techniques (such as the density matrix renormalization group) as an application of Wilson’s numerical renormalization group along the virtual (imaginary time) dimension of the system.

On Farrell–Tate cohomology of SL2 over S-integers

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 study periodic groups such that the normaliser of any finite non-trivial subgroup of such a group is almost layer-finite. The class of groups satisfying this condition is rather wide and includes the free Burnside groups of odd period which is greater than 665 and the groups constructed by A. Yu. Olshanskii.We consider the classical question: how the properties of the system of subgroups of a group influence on the properties of the group? We show that almost layer-finiteness is transferred on the group G from the normalisers of non-trivial finite subgroups of the group G if G is periodic conjugately biprimitively finite group with a strongly embedded subgroup.We study the structure of an infinite Sylow 2-subgroup in a periodic conjugately biprimitively finite group in the case that the normaliser of any finite non-trivial subgroup is almost layer-finite.The results of the paper can be useful in the study of the class of periodic conjugately biprimitively finite groups (periodic Shunkov groups).

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 \Gamma which contains G as a finite index normal subgroup, the virtual cohomological dimension vcd (\Gamma) of \Gamma equals \underline{\mathrm{gd}}(\Gamma) but such that the outer automorphism group Out (G) is virtually torsion-free, admits a cocompact model for \underline{E} Out (G) but nonetheless has vcd(Out (G))\le \underline{\mathrm{gd}} (Out (G))-r .