1Swarthmore College, Swarthmore, USA 2University of Illinois at Chicago, Chicago, USA 3Technion – Israel Institute of Technology, Haifa, Israel; Brandeis University, Waltham, USA
Groups, geometry, and dynamics, Tome 18 (2024) no. 2, pp. 649-676
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.
1
Swarthmore College, Swarthmore, USA
2
University of Illinois at Chicago, Chicago, USA
3
Technion – Israel Institute of Technology, Haifa, Israel; Brandeis University, Waltham, USA
Eduard Einstein; Daniel Groves; Thomas Ng. Separation and relative quasiconvexity criteria for relatively geometric actions. Groups, geometry, and dynamics, Tome 18 (2024) no. 2, pp. 649-676. doi: 10.4171/ggd/774
@article{10_4171_ggd_774,
author = {Eduard Einstein and Daniel Groves and Thomas Ng},
title = {Separation and relative quasiconvexity criteria for~relatively geometric actions},
journal = {Groups, geometry, and dynamics},
pages = {649--676},
year = {2024},
volume = {18},
number = {2},
doi = {10.4171/ggd/774},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/774/}
}
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AU - Daniel Groves
AU - Thomas Ng
TI - Separation and relative quasiconvexity criteria for relatively geometric actions
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