 

Wojciech Jaworski and C. Robinson Edward Raja
The ChoquetDeny theorem and distal properties of totally disconnected locally compact groups of polynomial growth


Published: 
June 21, 2007

Keywords: 
ChoquetDeny 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 ChoquetDeny 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 ChoquetDeny 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

