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Degree-one maps between hyperbolic 3-manifolds with the same volume limit


Author: Teruhiko Soma
Journal: Trans. Amer. Math. Soc. 353 (2001), 2753-2772
MSC (1991): Primary 57M99; Secondary 57M50
DOI: https://doi.org/10.1090/S0002-9947-01-02787-8
Published electronically: March 15, 2001
MathSciNet review: 1828472
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Abstract:

Suppose that $f_n:M_n\longrightarrow N_n$ $(n\in {\mathbf N})$ are degree-one maps between closed hyperbolic 3-manifolds with

\begin{displaymath}\lim_{n\rightarrow \infty} \operatorname{Vol} (M_n)=\lim_{n\rightarrow \infty}{\operatorname{Vol}}(N_n) <\infty. \end{displaymath}

Then, our main theorem, Theorem 2, shows that, for all but finitely many $n\in {\mathbf N}$, $f_n$ is homotopic to an isometry. A special case of our argument gives a new proof of Gromov-Thurston's rigidity theorem for hyperbolic 3-manifolds without invoking any ergodic theory. An example in §3 implies that, if the degree of these maps is greater than 1, the assertion corresponding to our theorem does not hold.


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Additional Information

Teruhiko Soma
Affiliation: Department of Mathematical Sciences, College of Science and Engineering, Tokyo Denki University, Hatoyama-machi, Saitama-ken 350-0394, Japan
Email: soma@r.dendai.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-01-02787-8
Keywords: Degree-one maps, hyperbolic $3$-manifolds, Gromov-Thurston's rigidity theorem
Received by editor(s): November 12, 1999
Received by editor(s) in revised form: July 10, 2000
Published electronically: March 15, 2001
Dedicated: Dedicated to Professor Shin’ichi Suzuki on his sixtieth birthday
Article copyright: © Copyright 2001 American Mathematical Society

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