Kazhdan groups have cost 1

TL;DR

This paper proves that all countably infinite groups with Kazhdan's property (T) have cost 1, resolving a significant open question in the field of geometric group theory.

Contribution

It establishes that Kazhdan groups have cost 1, advancing understanding of their measure-theoretic properties and answering a longstanding question.

Findings

Kazhdan groups have cost 1

Addresses a key open question in group theory

Progress towards fixed price 1 remains open

Abstract

We prove that every countably infinite group with Kazhdan's property (T) has cost 1, answering a well-known question of Gaboriau. It remains open if they have fixed price 1.