Counting and entropy for hyperbolic surface amalgams

TL;DR

This paper investigates the growth of closed geodesics and entropy in hyperbolic surface amalgams, revealing how geometric parameters influence entropy behavior and geodesic counts.

Contribution

It establishes the equivalence of topological and volume entropies in hyperbolic surface amalgams and analyzes their dependence on geometric data such as systole and pasting curve lengths.

Findings

Topological and volume entropies coincide in hyperbolic surface amalgams.

Entropy can grow exponentially with pasting length without a systole lower bound.

Bounds on the number of closed geodesics depend on systole and pasting curve lengths.

Abstract

This paper is about closed hyperbolic surface amalgams with a focus on the growth of the number of closed geodesics. As in the case of surfaces, we show that topological and volume entropies coincide, but we show stark differences in how they behave according to geometric data with upper and lower bounds on the number of closed geodesics which depend on the length of the systole and the length of the pasting curves. In particular, we show that the entropy can increase exponentially in terms of the pasting length in the absence of a lower bound on the systole.