On the number of hyperbolic manifolds of complexity n

TL;DR

This paper estimates the growth rate of hyperbolic manifolds with boundary that admit a specific ideal triangulation, showing their count increases exponentially with the number of ideal triangles.

Contribution

It provides an asymptotic estimate for the number of hyperbolic manifolds with boundary based on their ideal triangulation complexity.

Findings

Number of such manifolds grows as exp(n log n + O(n))

Provides a quantitative measure of manifold complexity growth

Advances understanding of hyperbolic manifold enumeration

Abstract

We consider hyperbolic manifolds with boundary, which admit an ideal triangulation with n ideal triangles and one edge. We prove that the number of these manifolds is $\exp(n\ln(n)+O(n))$.