New York Journal of Mathematics
Volume 13 (2007) 159-174

  

Wojciech Jaworski and C. Robinson Edward Raja

The Choquet-Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth


Published: June 21, 2007
Keywords: Choquet-Deny theorem, totally disconnected groups, polynomial growth, distal, random walks, Poisson boundary
Subject: 60B15, 43A05, 60J50, 22D05, 22D45

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 \bar{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:
  • The Choquet-Deny theorem holds for G.
  • The group of inner automorphisms of G acts distally on G.
  • Every inner automorphism of G is distal.
  • The contraction subgroup of every inner automorphism of G is trivial.
  • G is a SIN group.
We also show that for every probability measure μ 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 μ generates G.

Acknowledgements

The first author was supported by an NSERC Grant.


Author information

Wojciech Jaworski:
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
wjaworsk@math.carleton.ca

C. Robinson Edward Raja:
Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile Mysore Road, Bangalore 560 059, India
creraja@isibang.ac.in