Quasi-isometric rigidity of subgroups

Abstract: A central theme in geometric group theory: what are the relations between the algebraic and geometric properties of a finitely generated group. Finitely generated groups with "equivalent" geometries are called quasi-isometric. Let $G$ and $H$ be quasi-isometric finitely generated groups and let $P$ be a subgroup of $G$. Is there a subgroup $Q$ (or a collection of subgroups) of $H$ whose left cosets coarsely reflect the geometry of the left cosets of $P$ in $G$? We explore sufficient conditions on the pair $(G,P)$ for a positive answer. In the talk, we introduce notions of quasi-isometric pairs, and quasi-isometrically characteristic collection of subgroups. Distinct classes of qi-characteristic collections of subgroups have been studied in the literature on quasi-isometric rigidity, we will describe some of them. The talk will focus on putting context to our main result and illustrate it with some applications: If $G$ and $H$ are finitely generated quasi-isometric groups and $P$ is a qi-characteristic collection of subgroups of $G$, then there is a collection of subgroups $Q$ of $H$ such that $(G, P)$ and $(H, Q)$ are quasi-isometric pairs. This is joint work with Jorge Luis Sanchez (UNAM, Mexico).

group theoryrings and algebras

Audience: researchers in the topic

Al@Bicocca take-away