Small optimal Margulis numbers force upper volume bounds

Abstract

If λ \lambda is a positive real number strictly less than log ⁡ 3 \operatorname {log}3 , there is a positive number V λ V_\lambda such that every orientable hyperbolic 3 3 -manifold of volume greater than V λ V_\lambda admits λ \lambda as a Margulis number. If λ > ( log ⁡ 3 ) / 2 \lambda >(\operatorname {log}3)/2 , such a V λ V_\lambda can be specified explicitly and is bounded above by \[ λ ( 6 + 880 log ⁡ 3 − 2 λ log ⁡ 1 log ⁡ 3 − 2 λ ) , \lambda \bigg (6+\frac {880}{\operatorname {log}3-2\lambda } \operatorname {log}{1\over \operatorname {log}3-2\lambda }\bigg ), \] where log \operatorname {log} denotes the natural logarithm. These results imply that for λ > log ⁡ 3 \lambda >\operatorname {log}3 , an orientable hyperbolic 3 3 -manifold that does not have λ \lambda as a Margulis number has a rank- 2 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 λ > ( log ⁡ 3 ) / 2 \lambda >(\operatorname {log}3)/2 .