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The (co)homological dimension of homomorphism $\phi:G\to H$ is the maximal number $k$ such that the induced homomorphism is nonzero for some $H$-module. The following theorems are proven: THEOREM 1. For every homomorphism $\phi:G\to H$ of a geometrically finite group $G$ the homological dimension of $\phi$ equals the cohomological dimension, $hd(\phi)= cd(\phi)$. THEOREM 2. For every homomorphism $\phi:G\to H$ of geometrically finite groups $cd(\phi\times\phi)=2cd(\phi)$.

In this paper we study virtual rational Betti numbers of a nilpotent-by-abelian group $G$, where the abelianization $N/N'$ of its nilpotent part $N$ satisfies certain tameness property. More precisely, we prove that if $N/N'$ is $2(c(n-1)-1)$-tame as a $G/N$-module, $c$ the nilpotency class of $N$, then $\mathrm{vb}_j(G):=\sup_{M\in\mathcal{A}_G}\dim_\mathbb{Q} H_j(M,\mathbb{Q})$ is finite for all $0\leq j\leq n$, where $\mathcal{A}_G$ is the set of all finite index subgroups of $G$., Comment: 19 pages

In this paper we prove that the homological dimension of an elementary amenable group over an arbitrary commutative coefficient ring is either infinite or equal to the Hirsch length of the group. Established theory gives simple group theoretical criteria for finiteness of homological dimension and so we can infer complete information about this invariant for elementary amenable groups. Stammbach proved the special case of solvable groups over coefficient fields of characteristic zero in an important paper dating from 1970.

We revisit a recent paper of Fialowski and Iohara. They compute the homology of the Lie algebra $\mathfrak{gl}(\infty,R)$ for $R$ an associative unital algebra over a field of characteristic zero. We explain how to obtain essentially the same results by a completely different method.

We consider generalisations of Thompson's group $V$, denoted by $V_r(\Sigma)$, which also include the groups of Higman, Stein and Brin. It was shown by the authors in [20] that under some mild conditions these groups and centralisers of their finite subgroups are of type $\mathrm{F}_\infty$. Under more general conditions we show that the groups $V_r(\Sigma)$ are finitely generated and, under the mild conditions mentioned above, we see that they are finitely presented and give a recipe to find explicit presentations. For the centralisers of finite subgroups we find a suitable infinite presentation and then apply a general procedure to shorten this presentation. In the appendix, we give a proof of this general shortening procedure., Comment: 24 pages, no figures

We study the group $QV$, the self-maps of the infinite $2$-edge coloured binary tree which preserve the edge and colour relations at cofinitely many locations. We introduce related groups $QF$, $QT$, $\tilde{Q}T$, and $\tilde{Q}V$, prove that $QF$, $\tilde{Q}T$, and $\tilde{Q}V$ are of type $\mathrm{F}_\infty$, and calculate finite presentations for them. We calculate the normal subgroup structure and rational homology of all $5$ groups, the Bieri--Neumann--Strebel--Renz invariants of $QF$, and discuss the relationship of all $5$ groups with other generalisations of Thompson's groups., Comment: 31 pages; 14 figures; accepted version

We study the homological finiteness property FPm of permutational wreath products.

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., Comment: 14 pages

We consider generalisations of Thompson's group $V$, denoted $V_r(\Sigma)$, which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, $V_r(\Sigma)$ is the full automorphism group of a Cantor-algebra. Under some further minor restrictions, we prove that these groups are of type $\mathrm{F}_\infty$ and that this implies that also centralisers of finite subgroups are of type $\mathrm{F}_\infty$., Comment: 19 pages, 2 figures. Revised version. The original submission has now been split into two papers. The current submission contains the first one. The second part is being reworked and will be reposted soon independently. Lemma 4.8 was incorrect as stated and has since been rectified. The results of the paper are unchanged

We use Bieri-Strebel invariants to determine when a normal fibre product in the product of two finitely presented groups is finitely presented. We give conditions that imply and in some cases characterize the existence of such finitely presented fibre products., Comment: This version includes many changes and new results suggested by a referee

We develop new invariants similar to the Bieri-Strebel-Neumann-Renz invariants but in the category of Bredon modules (with respect to the class of the finite subgroups of G). We prove that for virtually soluble groups of type FP_{\infty} and finite extension of the Thompson group F the new invariants coincide with the classical ones.

We generalize a class of groups introduced by Herbert Abels to produce examples of virtually torsion free groups that have Bredon-finiteness length m-1 and classical finiteness length n-1 for all 0 < m <= n. The proof illustrates how Bredon-finiteness properties can be verified using geometric methods and a version of Brown's criterion due to Martin Fluch and the author., Comment: 17 pages, 2 figures, v2 more detailed

In this article we study the homology of nilpotent groups. In particular a certain vanishing result for the homology and cohomology of nilpotent groups is proved., Comment: 9 pages, Latex

The problem of homological stability helps us to catch the structure of group homology. We calculate homological stability of special orthogonal groups, and we also calculate the stability of orthogonal groups with determinant-twisted coefficients under a certain good situation. We also get some results about the structure of these homology., Comment: 12 pages

We define a family of groups that generalises Thompson's groups $T$ and $G$ and also those of Higman, Stein and Brin. For groups in this family we descrine centralisers of finite subgroups and show, that for a given finite subgroup $Q$, there are finitely many conjugacy classes of finite subgroups isomorphic to $Q$. We use this to show that whenever defined, the T versions of these groups have a slightly weaker property, quasi-$\underline{\operatorname F}_\infty$, to that of a group possessing a finite type model for the classifying space for proper actions ${\underline{E}}G$ if and only if they posses finite type models for the ordinary classifying space. We also generalise some well-known properties of ordinary cohomology to Bredon cohomology., Comment: 22 pages, to appear Groups, Geometry, and Dynamics

We use the properties of the refined Bloch group of a field to prove that H_3 of SL_2 of a global field is never finitely generated, and to calculate - up to some 2-torsion - H_3 of SL_2 of local fields with finite residue field of odd characteristic. We also give lower bounds for the 3-torsion in the H_3 of SL_2 of rings of S-integers., Comment: 42 pages. Details added and exposition improved thanks to referee. To appear, J. Pure Applied Algebra

We show that Brin's generalisations $2V$ and $3V$ of the Thompson-Higman group $V$ are of type $FP_\infty$. Our methods also give a new proof that both groups are finitely presented., Comment: 26 pages, 18 figures, revised version

We consider the general question of how the homological finiteness property left-FPn holding in a monoid influences, and conversely depends on, the property holding in the substructures of that monoid. In particular we show that left-FPn is inherited by the maximal subgroups in a completely simple minimal ideal, in the case that the minimal ideal has finitely many left ideals. For completely simple semigroups we prove the converse, and as a corollary show that a completely simple semigroup is of type left- and right-FPn if and only if it has finitely many left and right ideals and all of its maximal subgroups are of type FPn. Also, given an ideal of a monoid, we show that if the ideal has a two-sided identity element then the containing monoid is of type left-FPn if and only if the ideal is of type left-FPn., Comment: 25 pages

We study centralisers of finite order automorphisms of generalisations of Thompson's group F and conjugacy classes of finite subgroups in finite extensions of these groups. In particular, we show that centralisers of finite automorphisms in these generalised Thompson groups either admit a finite type classifying space or are not finitely generated. As an application we deduce results about the Bredon type of such finite extensions., Comment: 20 pages

We discuss properties of the complete Euler characteristic of a group G of type FP over the complex numbers and we relate it to the L2-Euler characteristic of the centralizers of the elements of G., Comment: To appear in: London Math. Society Lecture Note Series, Vol 358, 2009

Aimed at geometric applications, we prove the homology cobordism invariance of the $L^2$-betti numbers and $L^2$-signature defects associated to the class of amenable groups lying in Strebel's class $D(R)$, which includes some interesting infinite/finitenon-torsion-free groups. The proofs include the only prior known condition, that $\Gamma$ is a poly-torsion-free abelian group (or potentially a finite $p$-group.) We define a new commutator-type series which refines Harvey's torsion-free derived series of groups, using the localizations of groups and rings of Bousfield, Vogel, and Cohn. The series, called the local derived series, has versions for homology with arbitrary coefficients, and satisfies functoriality and an injectivity theorem. We combine these two new tools to give some applications to distinct homology cobordism types within the same simple homotopy type in higher dimensions, to concordance of knots in three manifolds, and to spherical space forms in dimension three., Comment: 33 pages

Kropholler's class of groups is the smallest class of groups which contains all finite groups and is closed under the following operator: whenever $G$ admits a finite-dimensional contractible $G$-CW-complex in which all stabilizer groups are in the class, then $G$ is itself in the class. Kropholler's class admits a hierarchical structure, i.e., a natural filtration indexed by the ordinals. For example, stage 0 of the hierarchy is the class of all finite groups, and stage 1 contains all groups of finite virtual cohomological dimension. We show that for each countable ordinal $\alpha$, there is a countable group that is in Kropholler's class which does not appear until the $\alpha+1$st stage of the hierarchy. Previously this was known only for $\alpha= 0$, 1 and 2. The groups that we construct contain torsion. We also review the construction of a torsion-free group that lies in the third stage of the hierarchy., Comment: 9 pages

We show that for soluble groups of type FPn, centralisers of finite subgroups need not be of type FPn., Comment: 14 pages

We show that any soluble group $G$ of type Bredon-$\FP_{\infty}$ with respect to the family of all virtually cyclic subgroups such that centralizers of infinite order elements are of type $\FP_{\infty}$ must be virtually cyclic. To prove this, we first reduce the problem to the case of polycyclic groups and then we show that a polycyclic-by-finite group with finitely many conjugacy classes of maximal virtually cyclic subgroups is virtually cyclic. Finally we discuss refinements of this result: we only impose the property Bredon-$\FP_n$ for some $n \leq 3$ and restrict to abelian-by-nilpotent, abelian-by-polycyclic or (nilpotent of class 2)-by-abelian groups., Comment: Corrected a mistake in Lemma 2.4 of the previous version, which had an effect on the results in Section 5 (the condition that all centralisers of infinite order elements are of type $FP_\infty$ was added)

We study actions of discrete groups on contractible topological spaces in which either (1) all stabilizers lie in the family of subgroups of prime power order or (2) all stabilizers lie in the family of finite subgroups. We compare the classifying spaces for actions with stabilizers in these two families, the Kropholler hierarchies build on these two families, and group cohomology relative to these two families., Comment: Version 2 contains a reference to related work of Yoav Segev, some other minor changes

Let F be the fundamental group of S, where S is a compact, connected, oriented surface with negative Euler characteristic and nonempty boundary. (1) The projective class of the chain \partial S in B_1(F) intersects the interior of a codimension one face of the unit ball in the stable commutator length pseudo-norm. (2) The unique homogeneous quasimorphism on F dual to this face (up to scale and elements of H^1) is the rotation quasimorphism associated to the action of F on the ideal boundary of the hyperbolic plane, coming from a hyperbolic structure on S. These facts follow from the fact that every homologically trivial 1-chain in S rationally cobounds an immersed surface with a sufficiently large multiple of the boundary. This is true even if S has no boundary., Comment: 19 pages, 5 figures; v.3 incorporates referees suggestions

If $G$ is a group, then we say that the functor $H^n(G,-)$ is finitary if it commutes with all filtered colimit systems of coefficient modules. We investigate groups with cohomology almost everywhere finitary; that is, groups with $n$th cohomology functors finitary for all sufficiently large $n$. We establish sufficient conditions for a group $G$ possessing a finite dimensional model for $e.g.$ to have cohomology almost everywhere finitary. We also prove a stronger result for the subclass of groups of finite virtual cohomological dimension, and use this to answer a question of Leary and Nucinkis. Finally, we show that if $G$ is a locally (polycyclic-by-finite) group, then $G$ has cohomology almost everywhere finitary if and only if $G$ has finite virtual cohomological dimension and the normalizer of every non-trivial finite subgroup of $G$ is finitely generated., Comment: 26 pages

For any group, there is a natural (pseudo-)norm on the vector space B1 of real (group) 1-boundaries, called the stable commutator length norm. This norm is closely related to, and can be thought of as a relative version of, the Gromov (pseudo)-norm on (ordinary) homology. We show that for a free group, the unit ball of this pseudo-norm is a rational polyhedron. It follows that stable commutator length in free groups takes on only rational values. Moreover every element of the commutator subgroup of a free group rationally bounds an injective map of a surface group. The proof of these facts yields an algorithm to compute stable commutator length in free groups. Using this algorithm, we answer a well-known question of Bavard in the negative, constructing explicit examples of elements in free groups whose stable commutator length is not a half-integer., Comment: 21 pages, 4 figures; version 2 incorporates referees' suggestions

Computations in the cohomology of finite groups., Comment: PhD thesis (Cambridge, 1990) with one extra chapter added 1992

For each subcomplex of the standard CW-structure on any torus, we compute the homology of a certain infinite cyclic regular covering space. In all cases when the homology is finitely generated, we also compute the cohomology ring. For aspherical subcomplexes of the torus our computation gives the homology and cohomology of Bestvina-Brady groups. We compute the cohomological dimension of each of these groups over any field and over any subring of the rationals., Comment: Submitted July 2006