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

Published electronically:
March 15, 2001

MathSciNet review:
1828472

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Abstract | References | Similar Articles | Additional Information

Suppose that are degree-one maps between closed hyperbolic 3-manifolds with

Then, our main theorem, Theorem 2, shows that, for all but finitely many , 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:
http://dx.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