Small optimal Margulis numbers force upper volume bounds

TL;DR

This paper establishes explicit volume bounds for orientable hyperbolic 3-manifolds based on their Margulis numbers, showing that small Margulis numbers imply upper volume limits and bounded fundamental group rank.

Contribution

It provides explicit bounds on volumes and fundamental group ranks for hyperbolic 3-manifolds with small Margulis numbers, extending previous theoretical results.

Findings

Explicit volume bounds for manifolds with Margulis number less than log3

Bounded rank of fundamental groups for manifolds lacking certain Margulis numbers

Quantitative relationships between Margulis numbers and manifold volume

Abstract

If $\lambda$ is a positive real number strictly less than $\log3$, there is a positive number $V_\lambda$ such that every orientable hyperbolic 3-manifold of volume greater than $V_\lambda$ admits $\lambda$ as a Margulis number. If $\lambda<(\log3)/2$, such a $V_\lambda$ can be specified explicitly, and is bounded above by $$\lambda\bigg(6+\frac{880}{\log3-2\lambda}\log{1\over\log3-2\lambda}\bigg),$$ where $\log$ denotes the natural logarithm. These results imply that for $\lambda<\log3$, an orientable hyperbolic 3-manifold that does not have $\lambda$ as a Margulis number has a rank-2 subgroup of bounded index in its fundamental group, and in particular has a fundamental group of bounded rank. Again, the bounds in these corollaries can be made explicit if $\lambda<(\log3)/2$.