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2019 Local cut points and splittings of relatively hyperbolic groups
Matthew Haulmark
Algebr. Geom. Topol. 19(6): 2795-2836 (2019). DOI: 10.2140/agt.2019.19.2795

Abstract

We show that the existence of a nonparabolic local cut point in the Bowditch boundary (G,) of a relatively hyperbolic group (G,) 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.

Citation

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Matthew Haulmark. "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

Information

Received: 18 August 2017; Revised: 3 October 2018; Accepted: 23 January 2019; Published: 2019
First available in Project Euclid: 29 October 2019

zbMATH: 07142619
MathSciNet: MR4023329
Digital Object Identifier: 10.2140/agt.2019.19.2795

Subjects:
Primary: 20F65, 20F67

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.19 • No. 6 • 2019
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