The Brin-Thompson groups sV are of type F_\infty

TL;DR

This paper proves that the higher-dimensional Brin-Thompson groups sV are of type F__ for all natural numbers s, extending previous results for s up to 3 by analyzing a simplified associated space.

Contribution

It introduces a new approach by retracting the associated space to a more manageable subspace to establish the finiteness property for all s.

Findings

sV groups are of type F__ for all s

The retraction method simplifies the analysis of the group's properties

Extends previous results from s up to 3 to all natural s

Abstract

We prove that the Brin-Thompson groups sV, also called higher dimensional Thompson's groups, are of type F_\infty for all natural numbers s. This result was previously shown for s up to 3, by considering the action of sV on a naturally associated space. Our key step is to retract this space to a subspace sX which is easier to analyze.