Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2201.08312 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- Given a finitely generated subgroup $H$ of a finitely generated group $G$ and a non-principal ultrafilter $ω$, we consider a natural subspace, $Cone^ω_{G}(H)$, 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 $H^ω$. We show that the connectedness and convexity of $Cone^ω_{G}(H)$ 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 $Cone^ω_{G}(H)$ is connected. We then show that whether $H$ is strongly quasi-convex in $G$ is detected by a natural convexity property of $Cone^ω_{G}(H)$ in the asymptotic cone of $G$.