Splittings of Relatively Hyperbolic Groups and Classifications of 1-dimensional Boundaries

Abstract

In the first part of this dissertation, we show that the existence of non-parabolic local cut point in the relative (or Bowditch) boundary, $\relbndry$, of a relatively hyperbolic group $(\Gamma,\bbp)$ implies that $\Gamma$ splits over a $2$-ended subgroup. As a consequence we classify the homeomorphism type of the Bowditch boundary for the special case when the Bowditch boundary $\relbndry$ is one-dimensional and has no global cut points. In the second part of this dissertation, We study local cut points in the boundary of CAT(0) groups with isolated flats. In particular the relationship between local cut points in $\bndry X$ and splittings of $\Gamma$ over $2$-ended subgroups. We generalize a theorem of Bowditch by showing that the existence of a local point in $\bndry X$ implies that $\Gamma$ splits over a $2$-ended subgroup. The first chapter can be thought of as an key step in the proof of this result. Additionally, we provide a classification theorem for 1-dimensional boundaries of groups with isolated flats. Namely, 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 $\bndry X$ 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.