Finiteness properties of stabilisers of oligomorphic actions

TL;DR

This paper investigates the finiteness properties of stabilisers in oligomorphic group actions, revealing obstructions for certain groups and applications to wreath products and Brin-Thompson groups.

Contribution

It establishes new obstructions to finiteness properties of stabilisers in oligomorphic actions for a broad class of groups, including linear and virtual cohomological dimension groups.

Findings

Existence of finite subsets with non-{ ext{FP}_ ext{infty}} stabilisers in oligomorphic actions.

Obstructions to finiteness properties for permutational wreath products and twisted Brin-Thompson groups.

Improved criteria for finiteness properties of wreath and graph-wreath products.

Abstract

An action of a group on a set is oligomorphic if it has finitely many orbits of $n$-element subsets for all $n$. We prove that for a large class of groups (including all groups of finite virtual cohomological dimension and all countable linear groups), for any oligomorphic action of such a group on an infinite set there exists a finite subset whose stabiliser is not of type $\mathrm{FP}_\infty$. This leads to obstructions on finiteness properties for permutational wreath products and twisted Brin-Thompson groups. We also prove a version for actions on flag complexes, and discuss connections to the Boone-Higman conjecture. In the appendix, we improve on the criterion of Bartholdi-Cornulier-Kochloukova for finiteness properties of wreath products, and the criterion of Kropholler-Martino for finiteness properties of graph-wreath products.