小的最优Margulis数迫使上体积界
Small optimal Margulis numbers force upper volume bounds
摘要：
若$λ$是严格小于$\log3$的正实数，则存在正数$V_λ$使得每一个体积大于$V_λ$的可定向双曲3-流形允许$λ$为Margulis数。如果是$λ<(\log3)/2$，则可以显式指定这样的$V_λ$，并且其上界为$$λ\bigg(6+\frac{880}{\log3-2λ}\log{1\over\log3-2λ}\bigg),$$，其中$\log$表示自然对数。这些结果意味着，对于$λ<\log3$，一个不以$λ$为Margulis数的可定向双曲3-流形在其基本群中有一个秩为2的有界指数子群，特别是有一个秩为有界的基本群.同样，如果是$λ<(\log3)/2$，则这些推论中的界限可以显式化。
英文摘要：
If $λ$ is a positive real number strictly less than $\log3$, there is a positive number $V_λ$ such that every orientable hyperbolic 3-manifold of volume greater than $V_λ$ admits $λ$ as a Margulis number. If $λ<(\log3)/2$, such a $V_λ$ can be specified explicitly, and is bounded above by $$λ\bigg(6+\frac{880}{\log3-2λ}\log{1\over\log3-2λ}\bigg),$$ where $\log$ denotes the natural logarithm. These results imply that for $λ<\log3$, an orientable hyperbolic 3-manifold that does not have $λ$ 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 $λ<(\log3)/2$.
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