The geometry of subgroup embeddings and asymptotic cones

Abstract

Given a finitely generated subgroup $H$ of a finitely generated group $G$ and a non-principal ultrafilter $\omega$, we consider a natural subspace, $Con{e}_G^{\omega }(H)$, 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 $Con{e}_G^{\omega }(H)$ 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 $Con{e}_G^{\omega }(H)$ is connected. We then show that whether $H$ is strongly quasi-convex in $G$ is detected by a natural convexity property of $Con{e}_G^{\omega }(H)$ in the asymptotic cone of $G$.

20 pages, 4 figures, added a citation to a paper of Behrstock

References in corpus (1)

• On strongly quasiconvex subgroups