Document Type

Article

Publication Date

3-26-2024

Published In

Groups, Geometry, And Dynamics

Abstract

Bowditch characterized relative hyperbolicity in terms of group actions on fine hyperbolic graphs with finitely many edge orbits and finite edge stabilizers. In this paper, we define generalized fine actions on hyperbolic graphs, in which the peripheral subgroups are allowed to stabilize finite subgraphs rather than stabilizing a point. Generalized fine actions are useful for studying groups that act relatively geometrically on a CAT(0) cube complex, which were recently defined by the first two authors. Specifically, we show that any group acting relatively geometrically on a CAT(0) cube complex admits a generalized fine action on the one-skeleton of the cube complex. For generalized fine actions, we prove a criterion for relative quasiconvexity of subgroups that cocompactly stabilize quasiconvex subgraphs, generalizing a result of Martínez-Pedroza and Wise in the setting of fine hyperbolic graphs. As an application, we obtain a characterization of boundary separation in generalized fine graphs and use it to prove that Bowditch boundary points in relatively geometric actions are always separated by a hyperplane stabilizer.

Keywords

relatively geometric, Bowditch boundary, separation criteria, CAT(0) cube complexes

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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This work is freely available under a Creative Commons license.

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