## Algebraic & Geometric Topology

### Local cut points and splittings of relatively hyperbolic groups

Matthew Haulmark

#### Abstract

We show that the existence of a nonparabolic local cut point in the Bowditch boundary $∂(G,ℙ)$ of a relatively hyperbolic group $(G,ℙ)$ 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 $Y$, then $Ends(X)$ is homeomorphic to $Ends(Y)$. Thus we propose an alternative definition of $Ends(G)$ which increases the class of spaces on which $G$ can act.

#### Article information

Source
Algebr. Geom. Topol., Volume 19, Number 6 (2019), 2795-2836.

Dates
Revised: 3 October 2018
Accepted: 23 January 2019
First available in Project Euclid: 29 October 2019

https://projecteuclid.org/euclid.agt/1572314542

Digital Object Identifier
doi:10.2140/agt.2019.19.2795

Mathematical Reviews number (MathSciNet)
MR4023329

Zentralblatt MATH identifier
07142619

#### Citation

Haulmark, Matthew. Local cut points and splittings of relatively hyperbolic groups. Algebr. Geom. Topol. 19 (2019), no. 6, 2795--2836. doi:10.2140/agt.2019.19.2795. https://projecteuclid.org/euclid.agt/1572314542

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