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

Full-text PDF

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.

**1.**M. Boileau and S. Wang,*Non-zero degree maps and surface bundles over*, J. Differential Geom.**43**(1996), 789-806. MR**98g:57023****2.**N. Dunfield,*Cyclic surgery, degree of maps of character curves, and volume rigidity for hyperbolic manifolds*, Invent. Math.**136**(1999), 623-657. MR**2000d:57022****3.**M. Freedman, J. Hass and P. Scott,*Least area incompressible surfaces in -manifolds*, Invent. Math.**71**(1983), 609-647. MR**85e:57012****4.**M. Fujii and T. Soma,*Totally geodesic boundaries are dense in the moduli space*, J. Math. Soc. Japan**49**(1997), 589-601. MR**99b:57029****5.**W. Goldman,*Discontinuous groups and the Euler class*, Ph. D. Thesis, U.C. Berkeley (1980).**6.**C. Hayat-Legrand, S. Wang and H. Zieschang,*Degree-one maps onto lens spaces*, Pacific J. Math.**176**(1996), 19-32. MR**98b:57030****7.**C. Hayat-Legrand, S. Wang and H. Zieschang,*Minimal Seifert manifolds*, Math. Ann.**308**(1997), 673-700. MR**98i:57029****8.**J. Hempel,*-manifolds*, Ann. of Math. Studies 86, Princeton Univ. Press, Princeton N.J. (1976). MR**54:3702****9.**W. Jaco,*Lectures on three-manifold topology*, C.B.M.S. Regional Conf. Ser. in Math. no. 43, Amer. Math. Soc., Providence, R.I. (1980). MR**81k:57009****10.**R. Kirby,*Problems in low-dimensional topology*, Geometric Topology (W.H. Kazez ed.), AMS/IP Studies in Advanced Mathematics vol. 2, Part 2, Amer. Math. Soc. and International Press (1997), 35-473. MR**98f:57001****11.**C. McMullen,*Amenability, Poincaré series and quasiconformal maps*, Invent. Math.**97**(1989), 95-127. MR**90e:30048****12.**C. McMullen,*Iteration on Teichmüller space*, Invent. Math.**99**(1990), 425-454. MR**91a:57008****13.**G. Mostow,*Quasi-conformal mappings in -space and the rigidity of hyperbolic space forms*, Publ. Math. I.H.E.S.**34**(1968), 53-104. MR**38:4679****14.**R. Myers,*Simple knots in compact, orientable -manifolds*, Trans. Amer. Math. Soc.**273**(1982), 75-91. MR**83h:57018****15.**A. Reid, S. Wang and Q. Zhou,*Generalized Hopfian property, minimal Haken manifold, and J. Simon's conjecture for -manifold groups*, preprint.**16.**Y. Rong,*Degree one maps between geometric -manifolds*, Trans. Amer. Math. Soc.**322**(1992), 411-436. MR**92j:57007****17.**T. Soma,*Bounded cohomology of closed surfaces*, Topology**36**(1997), 1221-1246. MR**99a:57011****18.**T. Soma,*Non-zero degree maps to hyperbolic -manifolds*, J. Differential Geom.**49**(1998), 517-546. MR**2000b:57034****19.**T. Soma,*Sequences of degree-one maps between geometric -manifolds*, Math. Ann.**316**(2000), 733-742. MR**2001b:57039****20.**W. Thurston,*The geometry and topology of -manifolds*, Lecture Notes, Princeton Univ., Princeton (1978), http://www.msri.org/publications/gt3m/.**21.**W. Thurston,*Three dimensional manifolds, Kleinian groups and hyperbolic geometry*, Bull. Amer. Math. Soc.**6**(1982), 357-381. MR**83h:57019****22.**W. Thurston,*Hyperbolic structures on -manifolds I: Deformation of acylindrical manifolds*, Ann. of Math.**124**(1986), 203-246. MR**88g:57014****23.**H.C. Wang,*Topics on totally discontinuous groups*, In: Symmetric Spaces (W. Boothby and G. Weiss eds.) Pure and Appl. Math.**8**, Marcel Dekker, New York (1972), 459-487. MR**54:2879****24.**S. Wang and Q. Zhou,*Any -manifold -dominates at most finitely many geometric -manifolds*, preprint.**25.**A. Zastrow,*On the (non)-coincidence of Milnor-Thurston homology theory with singular homology theory*, Pacific J. Math.**186**(1998), 369-396. MR**2000a:55008**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
57M99,
57M50

Retrieve articles in all journals with MSC (1991): 57M99, 57M50

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