Asymptotic resemblance relations on groups
Sh. Kalantari

TL;DR
This paper explores how asymptotic resemblance relations induced by coarse structures on groups behave, generalizes asymptotic dimensiongrad, and introduces set theoretic coupling, establishing conditions for asymptotic equivalence.
Contribution
It introduces a generalized asymptotic dimensiongrad for groups with coarse structures and defines set theoretic coupling, extending topological coupling concepts.
Findings
Asymptotic dimensiongrad is coarse invariant.
Set theoretic coupling implies asymptotic equivalence.
Generalization of topological coupling for groups.
Abstract
In this paper, we study properties of asymptotic resemblance relations induced by compatible coarse structures on groups. We generalize the notion of asymptotic dimensiongrad for groups with compatible coarse structures and show this notion is coarse invariant. We end by defining the notion of set theoretic coupling for groups with compatible coarse structures and showing this notion is the generalization of the notion of topological coupling for finitely generated groups. We show if two groups with compatible coarse structures admit a set theoretic coupling then they are asymptotic equivalent.
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.
