Unimodular random graphs with Property (T) have cost one
Grabowski, Łukasz, Héctor Jardón-Sánchez, Mellick, Sam · arXiv (Cornell University) · 2025
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).