Left regular representations of Garside categories II. Finiteness properties of topological full groups

Abstract: We study topological full groups attached to groupoid models for left regular representations of Garside categories. Groups arising in this way include Thompson's group $V$ and many of its variations such as R\"over-Nekrashevych groups. Our key observation is that a Garside structure for the underlying small category induces a new Garside structure for a new small category of bisections, and that our topological full group coincides with the fundamental group of the enveloping groupoid of the new category. As a consequence, we solve the word problem and identify general criteria for establishing finiteness properties of our topological full groups. In particular, we show that topological full groups arising from products of shifts of finite type are of type ${\rm F}_{\infty}$, answering a natural question left open by Matui.