Hopf Decomposition and Maximal Quotient Growth

We study the Hopf decomposition of the conformal measures on the Gromov boundary for proper actions on Gromov hyperbolic spaces. Our main results provide a dichotomy on the quotient growth of Schreier graphs associated to subgroups of hyperbolic groups, which is proved by exploiting a relation between the Hopf decomposition and quotient growth. We also construct an abundance of infinitely generated free subgroups in any statistically convex-cocompact action on hyperbolic spaces with nontrivial conservative and dissipative components. They could be thought of as counterexamples to the Ahlfors area conjecture for infinitely generated groups.