A finiteness theorem for hyperbolic 3-manifolds

TL;DR

This paper establishes a finiteness result for closed hyperbolic 3-manifolds with bounded geometric and spectral properties, linking their fundamental group generators to their classification.

Contribution

It proves a finiteness theorem for hyperbolic 3-manifolds with bounded injectivity radius and Laplacian eigenvalue, extending understanding of their geometric and algebraic structure.

Findings

Finite number of such manifolds exist under given bounds

Application to classification of arithmetic hyperbolic 3-manifolds

Connection between geometric bounds and fundamental group generators

Abstract

We prove that there are only finitely many closed hyperbolic 3-manifolds with injectivity radius and first eigenvalue of the Laplacian bounded below whose fundamental groups can be generated by a given number of elements. An application to arithmetic manifolds is also given.