Unimodular random graphs with Property (T) have cost one
Group Theory
2025-09-23 v2 Dynamical Systems
Probability
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).
Cite
@article{arxiv.2506.19454,
title = {Unimodular random graphs with Property (T) have cost one},
author = {Łukasz Grabowski and Héctor Jardón-Sánchez and Sam Mellick},
journal= {arXiv preprint arXiv:2506.19454},
year = {2025}
}
Comments
34 pages. New version includes an appendix with a self-contained, streamlined proof for countable discrete groups