  In this paper, we study the growth of confined subgroups through boundary
actions of groups with contracting elements. We establish that the growth rate
of a confined subgroup is strictly greater than half of the ambient growth rate
in groups with purely exponential growth. Along the way, several results are
obtained on the Hopf decomposition for boundary actions of subgroups with
respect to conformal measures. In particular, we prove that confined subgroups
are conservative, and examples of subgroups with nontrivial Hopf decomposition
are constructed. We show a connection between Hopf decomposition and quotient
growth and provide a dichotomy on quotient growth of Schreier graphs for
subgroups in hyperbolic groups.
