论文标题
同源等级与双曲线量的比率，ii
The ratio of homology rank to hyperbolic volume, II
论文作者
论文摘要
在温和的拓扑限制下，我们获得了有限量的可定位的双曲线$ 3 $歧管$ m $在其数量方面的mod $ p $同源性（对于任何prime $ p $）的尺寸的新线性上限。本文中该论点的一个令人惊讶的特征是它们需要应用四种颜色定理。 如果$ m $关闭，并且（a）$π_1（m）$没有亚组同构对基本组的基本组，则可以封闭的，可定向的$ 2、3 $或$ 4 $，或（b）$ p = 2 $和$ m $包含no（嵌入式）$ 2 $ $ 4 $ 2，3 $ 2，3 $ 2，3 $ 4 H_1（M; F_P）<157.763 \ CDOT \ text {vol}（m）$。如果$ m $具有一个或多个尖端，那么我们将获得非常相似的界限，假设$π_1（m）$对$ g = 2，\ dots $ g $ g $ g $ g $ g $ g $ g $ g $ g = 2，\ dots，8 $的基本群的基本组没有亚组同构。这些结果应与我们以前的论文$ \ yalogy \ rank \ rank \ to \双曲\卷，\ i $的结果进行比较，其中我们获得了一个系数的范围为$ 168 $而不是$ 158 $，而不是158美元，而无需限制表面亚组或不可压缩的表面。在以后的论文中，使用更多参与的论点，我们希望获得与本文有关的界限，而无需限制。 这些论点还为$ \ \ text {vol} \，m $等级提供了新的线性上限（具有恒定条款），假设$π_1（m）$是$ 9 $ - free，或$ m $是关闭的，而$π_1（m）$ 5 $ $ $ $ $ $ $ $ $。
Under mild topological restrictions, we obtain new linear upper bounds for the dimension of the mod $p$ homology (for any prime $p$) of a finite-volume orientable hyperbolic $3$ manifold $M$ in terms of its volume. A surprising feature of the arguments in the paper is that they require an application of the Four Color Theorem. If $M$ is closed, and either (a) $π_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $2, 3$ or $4$, or (b) $p = 2$, and $M$ contains no (embedded, two-sided) incompressible surface of genus $2, 3$ or $4$, then $\text{dim}\, H_1(M;F_p) < 157.763 \cdot \text{vol}(M)$. If $M$ has one or more cusps, we get a very similar bound assuming that $π_1(M)$ has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus $g$ for $g = 2, \dots,8$. These results should be compared with those of our previous paper $The\ ratio\ of\ homology\ rank\ to\ hyperbolic\ volume,\ I$, in which we obtained a bound with a coefficient in the range of $168$ instead of $158$, without a restriction on surface subgroups or incompressible surfaces. In a future paper, using a much more involved argument, we expect to obtain bounds close to those given by the present paper without such a restriction. The arguments also give new linear upper bounds (with constant terms) for the rank of $π_1(M)$ in terms of $\text{vol}\,M$, assuming that either $π_1(M)$ is $9$-free, or $M$ is closed and $π_1(M)$ is $5$-free.
