  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.

Operator Algebras;Dynamical Systems;Group Theory