The geometry of subgroup embeddings and asymptotic cones

Jarnevic, Andy

doi:10.2140/agt.2025.25.699

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ISSN (electronic): 1472-2739

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The geometry of subgroup embeddings and asymptotic cones

Andy Jarnevic

Algebraic & Geometric Topology 25 (2025) 699–719

DOI: 10.2140/agt.2025.25.699

Abstract

Given a finitely generated subgroup $H$ of a finitely generated group $G$ and a nonprincipal ultrafilter $ω$ , we consider a natural subspace, ${Cone}_{G}^{ω} (H)$ , 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 ${Cone}_{G}^{ω} (H)$ 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 ${Cone}_{G}^{ω} (H)$ is connected. We then show that whether $H$ is strongly quasiconvex in $G$ is detected by a natural convexity property of ${Cone}_{G}^{ω} (H)$ in the asymptotic cone of $G$ .

Keywords

asymptotic cones, convexity, distortion

Mathematical Subject Classification

Primary: 20F65

References

Publication

Received: 21 February 2022

Revised: 6 February 2023

Accepted: 18 September 2023

Published: 16 May 2025

Authors

Andy Jarnevic
Vanderbilt University Nashville, TN United States

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Volume 25, issue 2 (2025)