AI 中文总结
该研究针对K3问题列表中的问题3.6，证明有限体积双曲3流形的有限覆盖共尾塔的归一化对数同调挠率极限不等于1/(6π)，通过构造理想直角多面体对应的棋盘流形的共尾塔实现。
AI 中文摘要
Baykur、Kirby和Ruberman的K3问题列表中的问题3.6询问，有限体积双曲3流形的有限覆盖共尾塔$M_0\longleftarrow M_1\longleftarrow M_2\longleftarrow\cdots$是否满足$\lim_{n\to\infty} \frac{\log|\operatorname{Tor} H_1(M_n;\mathbb Z)|}{\operatorname{vol}(M_n)} =\frac{1}{6\pi}$。我们给出否定答案：对每个理想直角多面体$P_0$，其对应的棋盘流形存在一个共尾塔，所有层均为$S^3$中的双曲链补，故每层的$\operatorname{Tor} H_1(M_n;\mathbb Z)=0$，归一化对数同调挠率恒为0。
英文摘要
Problem 3.6 in the $\mathrm{K3}$ problem list of Baykur, Kirby and Ruberman asks whether every cofinal tower \[ M_0\longleftarrow M_1\longleftarrow M_2\longleftarrow\cdots \] of finite covers of a finite-volume hyperbolic $3$-manifold satisfies \[ \lim_{n\to\infty} \frac{\log|\operatorname{Tor} H_1(M_n;\mathbb Z)|} {\operatorname{vol}(M_n)} =\frac{1}{6π}. \] We give a negative answer. For every ideal right-angled polyhedron $P_0$, the checkerboard manifold associated to $P_0$ admits a cofinal tower all of whose levels are hyperbolic link complements in $S^3$. Thus $\operatorname{Tor} H_1(M_n;\mathbb Z)=0$ at every level, and the normalized logarithmic homology torsion is identically zero.
Comments6 Pages,no figures
