Central limit theorems for Green metrics on hyperbolic groups

Key Points

• The aim is to establish a counting central limit theorem for Green metrics on hyperbolic groups.
• Utilized finitely supported, admissible probability measures on a hyperbolic group.
• Ordered group elements according to Green metrics.
• Explored relationships to Anosov representations and group actions on hyperbolic metric spaces.
• The Green metrics satisfy a counting central limit theorem as expected.
• Results extend to other length functions associated with Anosov representations.
• Findings suggest broader implications for metrics in hyperbolic spaces.