Texas A&M University, Department of
Mathematics, 216 Milner Hall, 31st of March 2004, 3:00-4:00
Groups and Dynamcs Seminar
Double ergodicity
of the Poisson boundary and applications
Vadim Kaimanovich of University of
Rennes I
We prove that the Poisson boundary of any spread out non-degenerate
symmetric random walk on an arbitrary locally compact second countable
group G is doubly M^{sep}-ergodic with respect to the class M^{sep} of
separable coefficient Banach G-modules. The proof is direct and based
on an analogous property of the bilateral Bernoulli shift in the space
of increments of the random walk. As a corollary we obtain that any
locally compact sigma-compact group G admits a measure class preserving
action which is both amenable and doubly M^{sep}-ergodic. This
generalizes an earlier result of Burger and Monod obtained under the
assumption that G is compactly generated and allows one to dispose of
this assumption in numerous applications to the theory of bounded
cohomology.