Graph-wreath products and finiteness conditions

A notion of \emph{graph-wreath product} is introduced. We obtain sufficient conditions for these products to satisfy the topologically inspired finiteness condition type $\operatorname{F}_n$. Under various additional assumptions we show that these conditions are necessary. Our results generalise results of Cornulier about wreath products in case $n=2$. Graph-wreath products include classical permutational wreath products and semidirect products of right-angled Artin groups by groups of automorphisms amongst others.Comment: 12 page

On groups of type (FP)∞

AbstractLet G be a group. A ZG-module M is said to be of type (FP)∞ over ZG if and only if there is a projective resolution P∗ ↠M in which every Pi is finitely generated. We show that if G belongs to a large class of torsion-free groups, which includes torsion-free linear and soluble-by-finite groups, then every ZG-module of type (FP)∞ has finite projective dimension. We also prove that every soluble or linear group of type (FP)∞ is virtually of type (FP). The arguments apply to groups which admit hierarchical decompositions. We also make crucial use of a generalized theory of Tate cohomology recently developed by Mislin

Torsion-free covers of solvable minimax groups

We prove that every finitely generated solvable minimax group can be realized as a quotient of a torsion-free solvable minimax group. This result has an application to the investigation of random walks on finitely generated solvable minimax groups. Our methods also allow us to completely characterize the solvable minimax groups that are homomorphic images of torsion-free solvable minimax groups