Patterson-Sullivan theory for groups with a strongly contracting element

TL;DR

This work addresses subgroup growth questions for groups acting with contracting elements by extending Patterson-Sullivan theory to the horoboundary of spaces with contracting elements. It establishes lower and upper bounds on the growth rates $\omega(N,X)$ in terms of $\omega(G,X)$ and amenability of quotients, and develops a robust conformal-density framework (including ergodicity and almost-uniqueness on the reduced horoboundary) together with Shadow Principles. The results yield full-measure radial/contracting limit sets under divergence and provide a boundary-dynamics approach to growth problems in non-hyperbolic settings. Overall, the paper supplies new tools linking geometry of contracting elements, boundary dynamics, and subgroup growth phenomena with potential broad applications in non-positive curvature contexts.

Abstract

Using Patterson-Sullivan measures we investigate growth problems for groups acting on a metric space with a strongly contracting element.