Harmonic Functions on Compactly Generated Groups
Darren Creutz
TL;DR
This paper generalizes a known characterization of infinite finitely generated groups to all compactly generated groups, showing they admit nonconstant harmonic functions if and only if they are noncompact.
Contribution
It extends the equivalence between noncompactness and existence of nonconstant harmonic functions from finitely generated to all compactly generated groups.
Findings
Noncompact compactly generated groups admit nonconstant harmonic functions.
The characterization aligns with the known finitely generated case.
Harmonic functions serve as a criterion for noncompactness.
Abstract
A compactly generated group is noncompact if and only if it admits a nonconstant harmonic function (for some, equivalently for every, reasonable measure). This generalizes the known fact that a finitely generated group is infinite if and only if it admits a nonconstant harmonic function (for some, equivalently every, reasonable measure).
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.