[Paper Review] Characterisations of algebraic properties of groups in terms of harmonic functions

Matthew Tointon|arXiv (Cornell University)|Sep 22, 2014

Advanced Operator Algebra Research20 references3 citations

TL;DR

This paper establishes deep connections between algebraic properties of groups and the structure of their spaces of harmonic functions. Using harmonic function theory and Laplacian analysis, it proves that a group with a finitely supported symmetric measure has a finite-dimensional space of harmonic functions if and only if it is virtually cyclic, and extends this to results on positive harmonic functions, surjectivity of the Laplacian, and non-constant harmonic functions on vertex-transitive graphs.

ABSTRACT

We prove various results connecting structural or algebraic properties of graphs and groups to conditions on their spaces of harmonic functions. In particular: we show that a group with a finitely supported symmetric measure has a finite-dimensional space of harmonic functions if and only if it is virtually cyclic; we present a new proof of a result of V. Trofimov that an infinite vertex-transitive graph admits a non-constant harmonic function; we give a new proof of a result of T. Ceccherini-Silberstein, M. Coornaert and J. Dodziuk that the Laplacian on an infinite, connected, locally finite graph is surjective; and we show that the positive harmonic functions on a non-virtually nilpotent linear group span an infinite-dimensional space.

Motivation & Objective

• To characterize algebraic properties of groups using the dimensionality and structure of their harmonic function spaces.
• To establish connections between harmonic function theory and group-theoretic properties such as virtual cyclicity and virtual nilpotency.
• To provide new proofs of known results on harmonic functions and Laplacian surjectivity using novel duality and mean dimension techniques.
• To explore the role of amenability and stable finiteness in cellular automata and harmonic function theory.

Proposed method

• Uses the discrete Laplacian defined via a finitely supported symmetric generating probability measure on a group to define harmonic functions as functions in the kernel of the Laplacian.
• Applies mean dimension theory and duality between linear cellular automata and their transposes to relate properties of harmonic functions to group structure.
• Employs a Garden of Eden-type theorem for linear cellular automata to characterize surjectivity of the Laplacian on infinite, connected, locally finite graphs.
• Utilizes the concept of harmonic functions of polynomial growth and their finite-dimensionality to characterize virtually abelian and virtually cyclic groups.
• Applies results from random walk theory and spectral analysis on Cayley graphs to analyze harmonic function spaces.
• Leverages the equivalence between stable finiteness of group algebras and linear surjunctivity to reformulate Kaplansky’s conjecture in terms of harmonic functions and transposes of cellular automata.

Experimental results

Research questions

• RQ1 When is the space of harmonic functions on a group with a symmetric, finitely supported measure finite-dimensional?
• RQ2 What algebraic structure characterizes groups whose space of harmonic functions is finite-dimensional?
• RQ3 Does every infinite, connected, locally finite, vertex-transitive graph admit a non-constant harmonic function?
• RQ4 Is the Laplacian on an infinite, connected, locally finite graph always surjective?
• RQ5 What is the dimensionality of the space of positive harmonic functions on a non-virtually nilpotent linear group?

Key findings

• A group with a finitely supported symmetric measure has a finite-dimensional space of harmonic functions if and only if it is virtually cyclic.
• Every infinite, connected, locally finite, vertex-transitive graph admits a non-constant harmonic function, providing a new proof of a result by V. Trofimov.
• The Laplacian on any infinite, connected, locally finite graph is surjective, offering a new proof of a result by Ceccherini-Silberstein, Coornaert, and Dodziuk.
• The space of positive harmonic functions on a non-virtually nilpotent linear group is infinite-dimensional.
• The finite-dimensionality of the space of harmonic functions of polynomial growth of degree at most k characterizes virtually abelian groups.
• The equivalence between Kaplansky’s stable finiteness conjecture and the pre-injectivity of transpose cellular automata is established via harmonic function duality.