Graph-wreath products and finiteness conditions

TL;DR

This paper introduces graph-wreath products, establishing conditions under which they satisfy certain finiteness properties, generalizing classical wreath product results and encompassing various group constructions.

Contribution

It defines graph-wreath products and provides sufficient and necessary conditions for them to satisfy finiteness conditions, extending previous work on wreath products.

Findings

Identified sufficient conditions for graph-wreath products to satisfy type F_n.

Proved these conditions are necessary under additional assumptions.

Generalized classical wreath product results to a broader class of groups.

Abstract

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.