Conformal dynamics at infinity for groups with contracting elements

TL;DR

This work builds a unified conformal-density theory at infinity for groups with contracting elements by introducing a convergence boundary and focusing on non-pinched Myrberg and conical points. It develops Patterson–Sullivan-type measures via shadows and a Shadow Lemma on a broad class of boundaries, culminating in a Hopf–Tsuji–Sullivan dichotomy that ties divergence of the Poincaré series to positivity/ergodicity of conformal measures on the Myrberg set. The framework relies on projection complexes (BBF) to model hyperbolic-like behavior through a quasi-tree of spaces, enabling embeddings of conical objects into Gromov boundaries and ensuring nice topological properties (Hausdorff, second countable) after quotienting by the finite-difference relation. The theory subsumes and connects classical boundaries (Gromov, Floyd, visual, Thurston, etc.) via horofunction boundaries and their quotients, and yields concrete applications to Poisson boundaries of random walks, co-growth of divergent groups, and measure-theoretic results for CAT(0) spaces and mapping class groups. Overall, the paper provides a versatile, boundary-focused toolkit for analyzing growth, ergodicity, and harmonic measures in a wide spectrum of geometric group actions with contracting dynamics.

Abstract

This paper develops a theory of conformal density at infinity for groups with contracting elements. We start by introducing a class of convergence boundary encompassing many known hyperbolic-like boundaries, on which a detailed study of conical points and Myrberg points is carried out. The basic theory of conformal density is then established on the convergence boundary, including the Sullivan shadow lemma and a Hopf--Tsuji--Sullivan dichotomy. This gives a unification of the theory of conformal density on the Gromov and Floyd boundary for (relatively) hyperbolic groups, the visual boundary for rank-1 CAT(0) groups, and Thurston boundary for mapping class groups. Besides that, the conformal density on the horofunction boundary provides a new important example of our general theory. Applications include the identification of Poisson boundary of random walks, the co-growth problem of divergent groups, measure theoretical results for CAT(0) groups and mapping class groups.

Figures (16)

• Figure 1: Assumption A (left) and Assumption B (right)
• Figure 2: Contracting property in Lemma \ref{['BigThree']}
• Figure 3: Admissible path
• Figure 4: Truncating a geodesic around $N_C(X_i)$.
• Figure 5: Schematic illustration of lifting standard paths

Theorems & Definitions (266)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.3: NS dynamics, see Lemma \ref{['SouthNorthLem']}
• Lemma 1.4: see Lemma \ref{['UniqueLimitSet']}
• Definition 1.5: Myrberg points
• Remark 1.6
• Lemma 1.7: see Lemma \ref{['VisualMetriconMyrberg']}
• Definition 1.8: Conical points, see Definition \ref{['ConicalDef1prime']}
• Lemma 1.9
• Proposition 1.10