Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

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

Author(s): Teruhiko Soma
Journal: Trans. Amer. Math. Soc. 353 (2001), 2753-2772.
MSC (1991): Primary 57M99; Secondary 57M50
Posted: March 15, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract | References | Similar articles | Additional information

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.

References:

1.
M. Boileau and S. Wang, Non-zero degree maps and surface bundles over $S^1$, 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 $3$-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, $3$-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 $n$-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 $3$-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 $3$-manifold groups, preprint.

16.
Y. Rong, Degree one maps between geometric $3$-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 $3$-manifolds, J. Differential Geom. 49 (1998), 517-546. MR 2000b:57034

19.
T. Soma, Sequences of degree-one maps between geometric $3$-manifolds, Math. Ann. 316 (2000), 733-742. MR 2001b:57039

20.
W. Thurston, The geometry and topology of $3$-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 $3$-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 $3$-manifold $1$-dominates at most finitely many geometric $3$-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

Similar Articles:

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: 10.1090/S0002-9947-01-02787-8
PII: S 0002-9947(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
Posted: March 15, 2001
Dedicated: Dedicated to Professor Shin'ichi Suzuki on his sixtieth birthday
Copyright of article: Copyright 2001, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google