We provide a new and elegant approach to relative quasiconvexity for relatively hyperbolic groups in the context of Bowditch’s approach to relative hyperbolicity using cocompact actions on fine hyperbolic graphs. Our approach to quasiconvexity generalizes the other definitions in the literature that apply only for countable relatively hyperbolic groups. We also provide an elementary and self-contained proof that relatively quasiconvex subgroups are relatively hyperbolic.
"Relative quasiconvexity using fine hyperbolic graphs." Algebr. Geom. Topol. 11 (1) 477 - 501, 2011. https://doi.org/10.2140/agt.2011.11.477