Unimodular random graphs with Property (T) have cost one

TL;DR

This paper extends the understanding of Property (T) groups and unimodular random graphs by providing a streamlined proof that such structures have cost one, and characterizes Property (T) in these contexts.

Contribution

It offers a simplified proof for cost one actions of Property (T) groups and extends the results to unimodular random graphs, including new characterizations.

Findings

Property (T) unimodular random graphs have cost one

Extended the Connes--Weiss and Glasner--Weiss theorems to unimodular random graphs

Provided a streamlined proof for groups with Property (T) admitting cost one actions

Abstract

Hutchcroft and Pete showed that countably infinite groups with Property (T) admit cost one actions, resolving a question of Gaboriau. We give a streamlined proof of their theorem, and extend it both to locally compact second countable groups and unimodular random graphs. We prove unimodular random graph analogues of the Connes--Weiss and Glasner--Weiss theorem characterising Property (T).