The geometry of subgroup embeddings and asymptotic cones

TL;DR

This paper explores how the geometric properties of subgroup embeddings within finitely generated groups can be characterized by the structure of their associated asymptotic cones, revealing links between convexity, connectedness, and subgroup properties.

Contribution

It introduces a new framework connecting subgroup embedding properties with geometric features of asymptotic cones, including a generalized distortion function and convexity criteria.

Findings

Connectedness of $Cone^{ ext{ω}}_{G}(H)$ indicates subgroup distortion levels.

Convexity of $Cone^{ ext{ω}}_{G}(H)$ detects strong quasi-convexity of $H$ in $G$.

The generalized distortion function determines the connectedness of the asymptotic cone subspace.

Abstract

Given a finitely generated subgroup $H$ of a finitely generated group $G$ and a non-principal ultrafilter $\omega$, we consider a natural subspace, $Cone^{\omega}_{G}(H)$, of the asymptotic cone of $G$ corresponding to $H$. Informally, this subspace consists of the points of the asymptotic cone of $G$ represented by elements of the ultrapower $H^{\omega}$. We show that the connectedness and convexity of $Cone^{\omega}_{G}(H)$ detect natural properties of the embedding of $H$ in $G$. We begin by defining a generalization of the distortion function and show that this function determines whether $Cone^{\omega}_{G}(H)$ is connected. We then show that whether $H$ is strongly quasi-convex in $G$ is detected by a natural convexity property of $Cone^{\omega}_{G}(H)$ in the asymptotic cone of $G$.