[論文レビュー] Two Characterizations of Geometrically Infinite Actions on Gromov Hyperbolic Spaces

Chaodong Yang, Wenyuan Yang|arXiv (Cornell University)|Feb 23, 2026

Geometric and Algebraic Topology被引用数 0

ひとこと要約

The paper proves two new characterizations of geometrically infinite actions on proper Gromov hyperbolic spaces: (1) existence of escaping hyperbolic element sequences, and (2) uncountably many non-conical limit points.

ABSTRACT

We provide two new characterizations of geometrically infinite actions on Gromov hyperbolic spaces: one in terms of the existence of escaping geodesics, and the other via the presence of uncountably many non-conical limit points. These results extend corresponding theorems of Bonahon, Bishop, and Kapovich-Liu from the settings of Kleinian groups and pinched negatively curved manifolds to discrete groups acting properly on proper Gromov hyperbolic spaces.

研究の動機と目的

• Extend geometric finiteness characterizations from Kleinian groups and pinched manifolds to discrete groups acting properly on proper Gromov hyperbolic spaces.
• Establish two new criteria for geometric infiniteness: escaping sequences of hyperbolic elements and uncountably many non-conical limit points.
• Provide elementary hyperbolic-geometric proofs that do not rely on Margulis-type lemmas.
• Relate boundary dynamics to geometric finiteness in this broad setting.

提案手法

• Define escaping sequences of hyperbolic elements and show equivalence with geometric infiniteness (Theorem 1.1).
• Show that geometric infiniteness is equivalent to the presence of uncountably many non-conical limit points (Theorem 1.2).
• Utilize elementary hyperbolic geometry arguments, visual metrics, and Morse-type quasi-geodesic control without Margulis lemmas.
• Construct cusp-like geometric objects and use L-local quasi-geodesics to build required boundary dynamics.
• Employ convergence group action theory on the boundary to connect dynamics with action finiteness.

実験結果

リサーチクエスチョン

• RQ1 Can geometrically infinite actions on proper Gromov hyperbolic spaces be detected via escaping sequences of hyperbolic elements?
• RQ2 Does geometric infiniteness equivalently manifest as an uncountable set of non-conical limit points on the boundary?
• RQ3 How do boundary dynamics and cusp-like structures characterize geometric finiteness in this general setting?

主な発見

• An action is geometrically infinite if and only if there exists an escaping sequence of hyperbolic elements (Theorem 1.1).
• An action is geometrically infinite if and only if the set of non-conical limit points is uncountable (Theorem 1.2).
• The results extend Bonahon’s and Kapovich–Liu’s theorems from Kleinian and pinched manifolds to discrete groups acting properly on proper Gromov hyperbolic spaces.
• A purely hyperbolic-geometric approach is developed, avoiding reliance on Margulis-type lemmas.