Hyperbolic volume, mod 2 homology, and k-freeness
Autor: | Guzman, Rosemary K., Shalen, Peter B. |
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Rok vydání: | 2020 |
Předmět: | |
Druh dokumentu: | Working Paper |
Popis: | We show that if $M$ is any closed, orientable hyperbolic $3$-manifold with ${\rm vol}\ M\le3.69$, we have ${\rm dim}\ H_1(M;{\bf F}_2)\le7$. This may be regarded as a qualitative improvement of a result due to Culler and Shalen, because the constant $3.69$ is greater than the ordinal corresponding to $\omega^2$ in the well-ordered set of finite volumes of hyperbolic $3$-manifolds. We also show that if ${\rm vol}\ M\le 3.77$, we have ${\rm dim}\ H_1(M;{\bf F}_2)\le10$. These results are applications of a new method for obtaining lower bounds for the volume of a closed, orientable hyperbolic $3$-manifold such that $\pi_1(M)$ is $k$-free for a given $k\ge4$. Among other applications we show that if $\pi_1(M)$ is $4$-free we have ${\rm vol}\ M>3.57$ (improving the lower bound of $3.44$ given by Culler and Shalen), and that if $\pi_1(M)$ is $5$-free we have ${\rm vol}\ M>3.77$. Comment: This is in effect a new paper, and is thus newly titled. Theorem 6.1 is stronger than the corresponding result in v1, and is proved by a different, simpler method. Certain numerical results, including one that qualitatively improves a result of Culler and Shalen, would have been impossible to prove with the methods of v1. The Dehn drilling arguments of the present version are entirely new. 80 pp |
Databáze: | arXiv |
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