Jacob Garcia: Characterizations of Stability via Morse Limit Sets

Jacob Garcia (Smith College)

Title: Characterizations of Stability via Morse Limit Sets

Abstract: An important example of Kleinian groups are the convex cocompact groups: every infinite order element of these groups is a loxodromic, and these groups are exactly the ones which admit Kleinian manifolds. A well known fact of convex cocompact groups is that they can be characterized exactly as the groups whose limit sets, on the visual boundary, are completely conical, or equivalently, completely horospherical. Convex cocompactness has been studied in the context of many non-hyperbolic spaces, such as mapping class groups, and has recently been generalized to the notion of subgroup stability. By using an analog of the visual boundary called the Morse boundary, a quasi-isometry invariant which "sees" hyperbolic directions for non-hyperbolic spaces, we show that subgroup stability is exactly classified by limit set conditions on the Morse Boundary which are analogous to the limit set conditions from the convex cocompact setting.