Separation and relative quasiconvexity criteria for relatively geometric actions (Q6594732)

There are many equivalent formulations of relatively hyperbolic groups. \textit{B. H. Bowditch}, in [Int. J. Algebra Comput. 22, No. 3, 1250016, 66 p. (2012; Zbl 1259.20052)], describes relative hyperbolicity in terms of an action on a fine hyperbolic graph with finitely many edge orbits and finite edge stabilizers. A natural example of such a graph is the coned-off Cayley graph of a relatively hyperbolic pair, defined by \textit{B. Farb} [Geom. Funct. Anal. 8, No. 5, 810-840 (1998; Zbl 0985.20027)]. However, as shown by certain examples (see in particular Example 3.5 of this paper), fineness and other important finiteness properties of the action of \((G, \mathcal{P})\) on the coned-off Cayley graph are not quasi-isometry invariants.\N\NIn the paper under review, the authors 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 defined by the first and the second author [J. Lond. Math. Soc., II. Ser. 105, No. 1, 691--708 (2022; Zbl 1521.20096)] (see also [ the first and the second author, Compos. Math. 156, No. 4, 862--867 (2020; Zbl 1481.20167)]).\N\NSpecifically, the authors 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, they prove a criterion for relative quasiconvexity of subgroups that cocompactly stabilize quasiconvex subgraphs, generalizing a result of \textit{E. Martínez-Pedroza} and \textit{D. T. Wise} [Algebr. Geom. Topol. 11, No. 1, 477--501 (2011; Zbl 1229.20038)] in the setting of fine hyperbolic graphs. As an application, they 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.