Peripheral Poisson boundaries and jointly bi-harmonic functions

TL;DR

This paper characterizes jointly bi-harmonic functions on countable groups, explores the peripheral Poisson boundary of group von Neumann algebras, and resolves a conjecture on peripheral eigenvalues of symmetric Markov operators.

Contribution

It provides a complete description of the peripheral Poisson boundary and characterizes bi-harmonic functions, addressing open questions and conjectures in the field.

Findings

Characterization of jointly bi-harmonic functions on countable groups.

Complete description of the peripheral Poisson boundary for certain Markov operators.

Resolution of a conjecture on peripheral eigenvalues and eigenvectors.

Abstract

In this paper we answer a question of Kaimanovich by characterizing (jointly) bi-harmonic functions on countable, discrete groups with respect to a symmetric, generating measure. We also study the peripheral Poisson boundary of $L(\G)$ with respect to Markov operators arising from symmetric, generating probability measures on a countable, discrete group $\G$. We solve a recent conjecture of Bhat, Talwar and Kar regarding peripheral eigenvalues and their corresponding eigenvectors for such Markov operators, and provide a complete description of the peripheral Poisson boundary in the aforementioned scenario.