Abstract

A notion of graph-wreath product of groups is introduced. We obtain sufficient conditions for these products to satisfy the topologically inspired finiteness condition type Fn. Under various additional assumptions we show that these conditions are necessary. Our results generalise results of Bartholdi, Cornulier and Kochloukova about wreath products. Graph-wreath products of groups include classical permutational wreath products and semidirect products of right-angled Artin groups by certain groups of automorphisms amongst others.

Highlights

  • In this paper we introduce a concept of graph-wreath product of groups which encompasses the notions of restricted wreath product and of right-angled Artin group as well as other graph products of groups

  • The finiteness conditions Fn are discussed in many articles, but we refer to the fundamental paper [4] of Bestvina and Brady for background material because that paper discusses a number of other ideas relevant to our paper

  • For any graph, ZΓ is the right-angled Artin group determined by Γ and (Z 2Z)Γ the right-angled Coxeter group

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Summary

Introduction

In this paper we introduce a concept of graph-wreath product of groups which encompasses the notions of restricted wreath product and of right-angled Artin group as well as other graph products of groups. Note that the property F2 is equivalent to finite presentability, and in the particular case n = 2, our result has already been proved for permutational wreath products by Cornulier. Both the results and the methods of proof in this paper have been influenced by the techniques of [11]. Wreath products The restricted wreath product A ≀ H of two groups A and H has base B the set of functions from H to A with finite support and head H It is the semidirect product B ⋊ H. A clique in such a graph consists of a finite set of vertices each pair of which are joined by an edge. For any graph, ZΓ is the right-angled Artin group determined by Γ and (Z 2Z)Γ the right-angled Coxeter group

The graph-wreath product and statement of results
Polyhedral Products and the Proofs of Theorems A and B
Proof of Theorem A
Proof of Theorem B
A little homological algebra and the Proof of Theorem C
Corollary
Proof of Theorem C
The proof of Theorem D
Houghton Groups
Full Text
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