Small optimal Margulis numbers force upper volume bounds

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 $\lambda<(\log3)/2$.