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Harmonic projection and hypercentral extensions

Antoine Gournay

TL;DR

The paper addresses how the Liouville property interacts with extensions by introducing a harmonic projection and proving that Liouville-ness can be preserved by [FC-]hypercentral extensions under suitable measures. The main approach shows that if $N\lhd \Gamma$ lies in the FC-hypercentre and $\nu$ generates $\Gamma/N$, there exist measures $\mu$ with $\phi_*(\mu)$ a lazy version of $\nu$ such that $\Gamma$ is $\mu$-Liouville iff $\Gamma/N$ is $\nu$-Liouville, formalized in the hypercentral and FC-hypercentre theorems. A key technical tool is the norm-1 harmonic projection $\pi$ onto the space of bounded $P$-harmonic functions, defined via convolution with a mean, along with kernel–image properties that enable transferring harmonicity through quotients. The work also explores generating-set stability and practical constructions to move Liouville between measures, contributing to a deeper understanding of random walks on groups and their harmonic functions.

Abstract

The Liouville property is a strong form of amenability, but contrary to amenability, it is not well-behaved under extensions. In this paper it is shown that, for some measures, the Liouville property is preserved by [FC-]hypercentral extensions. To this end a projection from $\ell^\infty$ onto the space of harmonic functions is introduced.

Harmonic projection and hypercentral extensions

TL;DR

The paper addresses how the Liouville property interacts with extensions by introducing a harmonic projection and proving that Liouville-ness can be preserved by [FC-]hypercentral extensions under suitable measures. The main approach shows that if lies in the FC-hypercentre and generates , there exist measures with a lazy version of such that is -Liouville iff is -Liouville, formalized in the hypercentral and FC-hypercentre theorems. A key technical tool is the norm-1 harmonic projection onto the space of bounded -harmonic functions, defined via convolution with a mean, along with kernel–image properties that enable transferring harmonicity through quotients. The work also explores generating-set stability and practical constructions to move Liouville between measures, contributing to a deeper understanding of random walks on groups and their harmonic functions.

Abstract

The Liouville property is a strong form of amenability, but contrary to amenability, it is not well-behaved under extensions. In this paper it is shown that, for some measures, the Liouville property is preserved by [FC-]hypercentral extensions. To this end a projection from onto the space of harmonic functions is introduced.
Paper Structure (4 sections, 12 theorems, 8 equations)

This paper contains 4 sections, 12 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Harmonic projection
  3. Central and hypercentral extensions
  4. Remarks on generating sets

Key Result

Theorem 1.1

Let $\Gamma$ be a countable group. Assume that $N\lhd \Gamma$ is contained in the FC-hypercentre of $\Gamma$. Let $\nu$ be a measure on $\Gamma/N$ whose support generate $\Gamma/N$. Then there are measures $\mu$ such that $\Gamma$ is $\mu$-Liouville if and only if $\Gamma/N$ is $\nu$-Liouville.

Theorems & Definitions (27)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.3
  • Corollary 3.4
  • proof
  • ...and 17 more