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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.
Keywords: relatively hyperbolic groups, splittings, local cut points, JSJ splittings, relatively hyperbolic groups, ends of Spaces, group Boundaries
Haulmark, Matthew 1
@article{10_2140_agt_2019_19_2795,
author = {Haulmark, Matthew},
title = {Local cut points and splittings of relatively hyperbolic groups},
journal = {Algebraic and Geometric Topology},
pages = {2795--2836},
publisher = {mathdoc},
volume = {19},
number = {6},
year = {2019},
doi = {10.2140/agt.2019.19.2795},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.2795/}
}
TY - JOUR AU - Haulmark, Matthew TI - Local cut points and splittings of relatively hyperbolic groups JO - Algebraic and Geometric Topology PY - 2019 SP - 2795 EP - 2836 VL - 19 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.2795/ DO - 10.2140/agt.2019.19.2795 ID - 10_2140_agt_2019_19_2795 ER -
%0 Journal Article %A Haulmark, Matthew %T Local cut points and splittings of relatively hyperbolic groups %J Algebraic and Geometric Topology %D 2019 %P 2795-2836 %V 19 %N 6 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2019.19.2795/ %R 10.2140/agt.2019.19.2795 %F 10_2140_agt_2019_19_2795
Haulmark, Matthew. Local cut points and splittings of relatively hyperbolic groups. Algebraic and Geometric Topology, Tome 19 (2019) no. 6, pp. 2795-2836. doi: 10.2140/agt.2019.19.2795
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