Swarup’s open question on quasiconvexity from finite height and finite commensurator index

Determine whether every subgroup H of a hyperbolic group G that has finite height in G and finite index in its commensurator Comm_G(H) is quasiconvex in G.

Background

The paper studies algebraic properties of subgroup collections via the coset intersection complex and proves that several such properties are geometric (quasi-isometry invariant up to virtual isomorphism). In discussing height and width, the authors recall a longstanding problem attributed to Swarup concerning the relationship between finite height, commensurator index, and quasiconvexity in hyperbolic groups.

Their results show finite height and bounded packing are geometric, which they note can be seen as indirect evidence toward a positive resolution of Swarup’s question, but the question itself remains open.