#### Volume 19, issue 6 (2019)

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Local cut points and splittings of relatively hyperbolic groups

### Matthew Haulmark

Algebraic & Geometric Topology 19 (2019) 2795–2836
##### Abstract

We show that the existence of a nonparabolic local cut point in the Bowditch boundary $\partial \left(G,ℙ\right)$ of a relatively hyperbolic group $\left(G,ℙ\right)$ implies that $G$ splits over a $2$–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 $1$–dimensional Bowditch boundaries of relatively hyperbolic groups which satisfy certain properties, such as no splittings over $2$–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 $X$ and $Y\phantom{\rule{0.3em}{0ex}}$, then $Ends\left(X\right)$ is homeomorphic to $Ends\left(Y\right)$. Thus we propose an alternative definition of $Ends\left(G\right)$ which increases the class of spaces on which $G$ can act.

##### Keywords
relatively hyperbolic groups, splittings, local cut points, JSJ splittings, relatively hyperbolic groups, ends of Spaces, group Boundaries
##### Mathematical Subject Classification 2010
Primary: 20F65, 20F67