On Property (FA) for wreath products

TL;DR

This paper characterizes when permutational wreath products possess Property (FA), linking it to properties of the factors, and explores the hereditary property and its relation to finitely presented groups.

Contribution

It provides a complete characterization of Property (FA) for permutational wreath products and examines hereditary Property (FA) in relation to finitely presented groups.

Findings

Wreath product A wr B has Property (FA) iff B has Property (FA) and A is finitely generated with finite abelianisation.

Many wreath products with hereditary Property (FA) are not quotients of finitely presented groups with the same property.

The paper establishes conditions under which wreath products have Property (FA) and explores limitations related to finite presentation.

Abstract

We characterize permutational wreath products with Property (FA). For instance, the standard wreath product A wr B of two nontrivial countable groups A,B, has Property (FA) if and only if B has Property (FA) and A is a finitely generated group with finite abelianisation. We also prove an analogous result for hereditary Property (FA). On the other hand, we prove that many wreath products with hereditary Property (FA) are not quotients of finitely presented groups with the same property.