Invariant measures on the space of horofunctions of a word hyperbolic group
Abstract
Description
We introduce a natural equivalence relation on the space $\sH_0$ of horofunctions of a word hyperbolic group that take the value 0 at the identity. We show that there are only finitely many ergodic measures that are invariant under this relation. This can be viewed as a discrete analog of the Bowen-Marcus theorem. Furthermore, if $η$ is such a measure and $G$ acts on a space $(X,μ)$ by p.m.p. transformations then $η\times μ$ is virtually ergodic with respect to a natural equivalence relation on $\sH_0\times X$. This is comparable to a special case of the Howe-Moore theorem. These results are applied to prove a new ergodic theorem for spherical averages in the case of a word hyperbolic group acting on a finite space.
37 pages. This new version corrects several typos including one in the statement of theorem 1.5
37 pages. This new version corrects several typos including one in the statement of theorem 1.5