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

Wojciech Jaworski, Raja, C. R. E.|ArXiv.org|Feb 14, 2007

Geometric and Algebraic Topology30 references17 citations

TL;DR

This paper establishes equivalent characterizations for the validity of the Choquet-Deny theorem in compactly generated totally disconnected locally compact groups of polynomial growth and generalized $̅{FC}$-groups. It shows that the theorem holds if and only if inner automorphisms act distally, which is equivalent to trivial contraction subgroups and the SIN property, linking harmonic analysis with group structure and dynamics.

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 $μ$ 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 $G$.

Motivation & Objective

• To determine necessary and sufficient conditions for the Choquet-Deny theorem to hold in compactly generated totally disconnected locally compact groups of polynomial growth.
• To extend the analysis to a broader class of totally disconnected generalized $̅{FC}$-groups.
• To characterize the structure of Poisson boundaries for random walks on such groups.
• To clarify the relationship between distal group actions, contraction subgroups, and harmonic function theory in non-abelian settings.

Proposed method

• Analyzes the action of inner automorphisms on the group and defines distality via orbit closures not containing the identity.
• Introduces the contraction subgroup $ C(g) $ of an element $ g $, defined as $ \{ x \in G \mid \lim_{n \to \infty} g^n x g^{-n} = e \} $, and studies its triviality.
• Uses the equivalence between distal inner automorphisms and trivial contraction subgroups to derive structural conditions on the group.
• Applies results from topological dynamics and harmonic analysis, particularly the theory of Poisson boundaries and $ \mu $-boundaries.
• Leverages the fact that for metrizable, compactly generated, totally disconnected, polynomial-growth groups, distality of $ \operatorname{Inn}(G) $ implies the group is a SIN group.
• Constructs realizations of Poisson boundaries as homogeneous spaces $ G/H $, especially when the support of $ \mu $ generates $ G $.

Experimental results

Research questions

• RQ1 When does the Choquet-Deny theorem hold for compactly generated totally disconnected locally compact groups of polynomial growth?
• RQ2 What is the precise relationship between distal group actions and the structure of contraction subgroups in such groups?
• RQ3 Under what conditions is the Poisson boundary of a random walk on such a group a compact homogeneous space?
• RQ4 How do the properties of inner automorphisms (distality, trivial contraction subgroups) relate to the SIN property in these groups?
• RQ5 Can the $ \mu $-boundary be explicitly realized as a homogeneous space when $ \mu $ is adapted and the support generates $ G $?

Key findings

• For a metrizable, compactly generated, totally disconnected, locally compact group $ G $ of polynomial growth, the Choquet-Deny theorem holds if and only if the group of inner automorphisms acts distally on $ G $.
• The following five conditions are equivalent in such groups: (i) Choquet-Deny theorem holds; (ii) $ \operatorname{Inn}(G) $ acts distally; (iii) every inner automorphism is distal; (iv) the contraction subgroup of every $ g \in G $ is trivial; (v) $ G $ is a SIN group.
• For every probability measure $ \mu $ on such a group, the Poisson boundary is a homogeneous space of $ G $, and it is compact when $ \mu $ is adapted and its support generates $ G $.
• The $ \mu $-boundary can be realized as $ \mathbb{Z}_2^{\mathbb{Z}} / T $, where $ T $ is a closed $ \tau $-invariant subgroup, for the group $ G = \mathbb{Z}_2^{\mathbb{Z}} \rtimes_\tau \mathbb{Z} $, with $ \tau $ the shift automorphism.
• In the case of the group $ \mathbb{Z}_2^{\mathbb{Z}} \rtimes_\tau \mathbb{Z} $, the Choquet-Deny theorem fails because $ \operatorname{Inn}(G) $ does not act distally on the normal subgroup $ \mathbb{Z}_2^{\mathbb{Z}} \times \{0\} $, as $ C(\tau) $ is nontrivial.
• There exist adapted measures $ \mu_T $ such that the $ \mu_T $-boundary is isomorphic to $ \mathbb{Z}_2^{\mathbb{Z}} / T $ for each closed $ \tau $-invariant subgroup $ T $, and these are mutually non-isomorphic as $ G $-spaces.