The geometry of subgroup embeddings and asymptotic cones
Andy Jarnevic
TL;DR
This work تنes a bridge between subgroup embeddings and the geometry of asymptotic cones by introducing Cone^{\omega}_{G}(H) for a subgroup H in G and a generalized distortion function \mu^{G}_{H}. It establishes that the connectedness and path-connectedness of Cone^{\omega}_{G}(H) across all non-principal ultrafilters \omega are equivalent to algebraic and geometric constraints captured by \mu^{G}_{H} and related distortion notions, linking cone topology to subgroup distortion. A key contribution is the convexity characterization: H is strongly quasi-convex in G if and only if Cone^{\omega}_{G}(H) is strongly convex in Cone^{\omega}(G) for all \omega, with significant corollaries such as the presence of cut points in asymptotic cones and obstructions to algebraic laws in G. Overall, the paper provides a conceptual and technical framework to study subgroup embeddings via asymptotic cones, yielding new insights into the interplay between coarse geometry and algebraic properties of groups.
Abstract
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 $G$.