Kazhdan groups have cost 1
arXiv:1810.11015
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.
16 pages, 2 figures. V2: Minor revisions. Added additional remarks and simulations. Accepted version, to appear in Invent. Math