Boundary classification and two-ended splittings of groups with isolated flats

Abstract

In this paper we provide a classification theorem for 1-dimensional boundaries of groups with isolated flats. Given a group $\Gamma$ acting geometrically on a $CAT(0)$ space $X$ with isolated flats and 1-dimensional boundary, we show that if $\Gamma$ does not split over a virtually cyclic subgroup, then $\partial X$ is homeomorphic to a circle, a Sierpinski carpet, or a Menger curve. This theorem generalizes a theorem of Kapovich-Kleiner, and resolves a question due to Kim Ruane. We also study the relationship between local cut points in $\partial X$ and splittings of $\Gamma$ over $2$-ended subgroups. In particular, we generalize a theorem of Bowditch by showing that the existence of a local point in $\partial X$ implies that $\Gamma$ splits over a $2$-ended subgroup.