Gábor Pete: Kazhdan groups have cost 1

Speaker: Gábor Pete

Title: Kazhdan groups have cost 1

Abstract: A probabilistic definition of groups with Kazhdan's property (T), due to Glasner & Weiss (1997), is that on any Cayley graph G, for ergodic group-invariant random black-and-white colourings of the vertices, with the density of each colour bounded away from 0, the density of edges connecting black to white vertices remains bounded away from zero. Amenable groups and free groups do not have property (T), while SL_d(\Z) with d\geq 3 do.

The cost of a transitive graph is one half of the inf of the expected degree of invariant connected spanning subgraphs. Amenable transitive graphs and Cayley graphs of SL_d(\Z) with d\geq 3 have cost 1, while any Cayley graph of the free group on d generators has cost d, by Gaboriau (2000).

A question of Gaboriau aims to connect cost with the first L^2-Betti number of groups. For Kazhdan groups, the latter has been known to be 0 since Bekka & Valette (1997), which is equivalent to saying that the Wired and Free Spanning Forests coincide. Gaboriau's question then suggests that the cost of any Kazhdan Cayley graph should be 1.

This is what we prove, with Tom Hutchcroft (Cambridge), building on the work of Lyons & Schramm (1999) on invariant percolations.