1. The Choquet-Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth
- Author
-
Jaworski, W. and Raja, C. R. E.
- Subjects
Mathematics - Probability ,Mathematics - Group Theory ,60B15 ,60J50 ,22D40 - Abstract
We obtain sufficient and necessary conditions for the Choquet-Deny theorem to hold in the class of compactly generated totally disconnected locally compact groups of polynomial growth, and in a larger class of totally disconnected generalized $\ov{FC}$-groups. The following conditions turn out to be equivalent when $G$ is a metrizable compactly generated totally disconnected locally compact group of polynomial growth: (i) the Choquet-Deny theorem holds for $G$; (ii) the group of inner automorphisms of $G$ acts distally on $G$; (iii) every inner automorphism of $G$ is distal; (iv) the contraction subgroup of every inner automorphism of $G$ is trivial; (v) $G$ is a SIN group. We also show that for every probability measure $\mu$ on a totally disconnected compactly generated locally compact second countable group of polynomial growth, the Poisson boundary is a homogeneous space of $G$, and that it is a compact homogeneous space when the support of $\mu$ generates $G$., Comment: 15 pages
- Published
- 2007