Work

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

Year: 2024

Type: article

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 groups of dynamical origin such as Röver–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 \textup{F}_{\infty} , answering a natural question left open by Matui.

Source: Groups Geometry and Dynamics

DOI: 10.4171/ggd/847

Author: Xin Li

Institution: University of Glasgow

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: European Commission

Awards: G434337789, G5768258237