Leírás
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.