Growth rate of confined subgroups of a discrete group

Confined subgroups of a Lie group or of a discrete group are generalizations of a Fuchsian group G that gives rise to a hyperbolic surface M of bounded injectivity radius. When the surface M has finite volume, the growth rate of G equals the volume growth rate of the hyperbolic plane H^2. Gekhtman and Levit established an inequality between the growth rates of G and the ambient space for a general confined subgroup of a rank-1 Lie group. In this talk, I will explain an analogous result for a confined subgroup G of a hyperbolic group, rank-1 CAT(0) group, or the mapping class group. This is established by studying the single (as opposed to the double) boundary action of G. Joint work with Ilya Gekhtman, Wenyuan Yang, and Tianyi Zheng.