We show that the existence of a nonparabolic local cut point in the Bowditch boundary of a relatively hyperbolic group implies that splits over a –ended subgroup. This theorem generalizes a theorem of Bowditch from the setting of hyperbolic groups to relatively hyperbolic groups. As a consequence we are able to generalize a theorem of Kapovich and Kleiner by classifying the homeomorphism type of –dimensional Bowditch boundaries of relatively hyperbolic groups which satisfy certain properties, such as no splittings over –ended subgroups and no peripheral splittings.
In order to prove the boundary classification result we require a notion of ends of a group which is more general than the standard notion. We show that if a finitely generated discrete group acts properly and cocompactly on two generalized Peano continua and , then is homeomorphic to . Thus we propose an alternative definition of which increases the class of spaces on which can act.
"Local cut points and splittings of relatively hyperbolic groups." Algebr. Geom. Topol. 19 (6) 2795 - 2836, 2019. https://doi.org/10.2140/agt.2019.19.2795