Small optimal Margulis numbers force upper volume bounds

Abstract:

If $\lambda$ is a positive real number strictly less than $\operatorname {log}3$, 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 $\lambda <(\operatorname {log}3)/2$, such a $V_\lambda$ can be specified explicitly and is bounded above by \[ \lambda \bigg (6+\frac {880}{\operatorname {log}3-2\lambda } \operatorname {log}{1\over \operatorname {log}3-2\lambda }\bigg ),\] where $\operatorname {log}$ denotes the natural logarithm. These results imply that for $\lambda <\operatorname {log}3$, 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 $\lambda <(\operatorname {log}3)/2$.