Degree-one maps between hyperbolic 3-manifolds with the same volume limit
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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 \[ \lim _{n\rightarrow \infty } \operatorname {Vol} (M_n)=\lim _{n\rightarrow \infty }{\operatorname {Vol}}(N_n) <\infty . \] 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.References
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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
- MR Author ID: 192547
- Email: soma@r.dendai.ac.jp
- Received by editor(s): November 12, 1999
- Received by editor(s) in revised form: July 10, 2000
- Published electronically: March 15, 2001
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1828472
Dedicated: Dedicated to Professor Shin’ichi Suzuki on his sixtieth birthday