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Local cut points and splittings of relatively hyperbolic groups

Matthew Haulmark

Algebraic & Geometric Topology 19 (2019) 2795–2836

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.

relatively hyperbolic groups, splittings, local cut points, JSJ splittings, relatively hyperbolic groups, ends of Spaces, group Boundaries
Mathematical Subject Classification 2010
Primary: 20F65, 20F67
Received: 18 August 2017
Revised: 3 October 2018
Accepted: 23 January 2019
Published: 20 October 2019
Matthew Haulmark
Department of Mathematics
Vanderbilt University
Nashville, TN
United States