Degreeone maps between hyperbolic 3manifolds with the same volume limit
Author:
Teruhiko Soma
Journal:
Trans. Amer. Math. Soc. 353 (2001), 27532772
MSC (1991):
Primary 57M99; Secondary 57M50
Published electronically:
March 15, 2001
MathSciNet review:
1828472
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Suppose that are degreeone maps between closed hyperbolic 3manifolds 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 GromovThurston's rigidity theorem for hyperbolic 3manifolds 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.
Michel
Boileau and Shicheng
Wang, Nonzero degree maps and surface bundles over
𝑆¹, J. Differential Geom. 43 (1996),
no. 4, 789–806. MR 1412685
(98g:57023)
 2.
Nathan
M. Dunfield, Cyclic surgery, degrees of maps of character curves,
and volume rigidity for hyperbolic manifolds, Invent. Math.
136 (1999), no. 3, 623–657. MR 1695208
(2000d:57022), http://dx.doi.org/10.1007/s002220050321
 3.
Michael
Freedman, Joel
Hass, and Peter
Scott, Least area incompressible surfaces in 3manifolds,
Invent. Math. 71 (1983), no. 3, 609–642. MR 695910
(85e:57012), http://dx.doi.org/10.1007/BF02095997
 4.
Michihiko
Fujii and Teruhiko
Soma, Totally geodesic boundaries are dense in the moduli
space, J. Math. Soc. Japan 49 (1997), no. 3,
589–601. MR 1452704
(99b:57029), http://dx.doi.org/10.2969/jmsj/04930589
 5.
W. Goldman, Discontinuous groups and the Euler class, Ph. D. Thesis, U.C. Berkeley (1980).
 6.
Claude
HayatLegrand, Shicheng
Wang, and Heiner
Zieschang, Degreeone maps onto lens spaces, Pacific J. Math.
176 (1996), no. 1, 19–32. MR 1433981
(98b:57030)
 7.
Claude
HayatLegrand, Shicheng
Wang, and Heiner
Zieschang, Minimal Seifert manifolds, Math. Ann.
308 (1997), no. 4, 673–700. MR 1464916
(98i:57029), http://dx.doi.org/10.1007/s002080050096
 8.
John
Hempel, 3Manifolds, Princeton University Press, Princeton, N.
J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86.
MR
0415619 (54 #3702)
 9.
William
Jaco, Lectures on threemanifold topology, CBMS Regional
Conference Series in Mathematics, vol. 43, American Mathematical
Society, Providence, R.I., 1980. MR 565450
(81k:57009)
 10.
William
H. Kazez (ed.), Geometric topology, AMS/IP Studies in Advanced
Mathematics, vol. 2, American Mathematical Society, Providence, RI;
International Press, Cambridge, MA, 1997. MR 1470749
(98f:57001)
 11.
Curt
McMullen, Amenability, Poincaré series and quasiconformal
maps, Invent. Math. 97 (1989), no. 1,
95–127. MR
999314 (90e:30048), http://dx.doi.org/10.1007/BF01850656
 12.
C.
McMullen, Iteration on Teichmüller space, Invent. Math.
99 (1990), no. 2, 425–454. MR 1031909
(91a:57008), http://dx.doi.org/10.1007/BF01234427
 13.
G.
D. Mostow, Quasiconformal mappings in 𝑛space and the
rigidity of hyperbolic space forms, Inst. Hautes Études Sci.
Publ. Math. 34 (1968), 53–104. MR 0236383
(38 #4679)
 14.
Robert
Myers, Simple knots in compact, orientable
3manifolds, Trans. Amer. Math. Soc.
273 (1982), no. 1,
75–91. MR
664030 (83h:57018), http://dx.doi.org/10.1090/S00029947198206640300
 15.
A. Reid, S. Wang and Q. Zhou, Generalized Hopfian property, minimal Haken manifold, and J. Simon's conjecture for manifold groups, preprint.
 16.
Yong
Wu Rong, Degree one maps between geometric
3manifolds, Trans. Amer. Math. Soc.
332 (1992), no. 1,
411–436. MR 1052909
(92j:57007), http://dx.doi.org/10.1090/S00029947199210529096
 17.
Teruhiko
Soma, Bounded cohomology of closed surfaces, Topology
36 (1997), no. 6, 1221–1246. MR 1452849
(99a:57011), http://dx.doi.org/10.1016/S00409383(97)000037
 18.
Teruhiko
Soma, Nonzero degree maps to hyperbolic 3manifolds, J.
Differential Geom. 49 (1998), no. 3, 517–546.
MR
1669645 (2000b:57034)
 19.
Teruhiko
Soma, Sequences of degreeone maps between geometric
3manifolds, Math. Ann. 316 (2000), no. 4,
733–742. MR 1758451
(2001b:57039), http://dx.doi.org/10.1007/s002080050352
 20.
W. Thurston, The geometry and topology of manifolds, Lecture Notes, Princeton Univ., Princeton (1978), http://www.msri.org/publications/gt3m/.
 21.
William
P. Thurston, Threedimensional manifolds, Kleinian
groups and hyperbolic geometry, Bull. Amer.
Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524
(83h:57019), http://dx.doi.org/10.1090/S027309791982150030
 22.
William
P. Thurston, Hyperbolic structures on 3manifolds. I. Deformation
of acylindrical manifolds, Ann. of Math. (2) 124
(1986), no. 2, 203–246. MR 855294
(88g:57014), http://dx.doi.org/10.2307/1971277
 23.
Hsien
Chung Wang, Topics on totally discontinuous groups, Symmetric
spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970),
Dekker, New York, 1972, pp. 459–487. Pure and Appl. Math., Vol.
8. MR
0414787 (54 #2879)
 24.
S. Wang and Q. Zhou, Any manifold dominates at most finitely many geometric manifolds, preprint.
 25.
Andreas
Zastrow, On the (non)coincidence of MilnorThurston homology
theory with singular homology theory, Pacific J. Math.
186 (1998), no. 2, 369–396. MR 1663826
(2000a:55008), http://dx.doi.org/10.2140/pjm.1998.186.369
 1.
 M. Boileau and S. Wang, Nonzero degree maps and surface bundles over , J. Differential Geom. 43 (1996), 789806. MR 98g:57023
 2.
 N. Dunfield, Cyclic surgery, degree of maps of character curves, and volume rigidity for hyperbolic manifolds, Invent. Math. 136 (1999), 623657. MR 2000d:57022
 3.
 M. Freedman, J. Hass and P. Scott, Least area incompressible surfaces in manifolds, Invent. Math. 71 (1983), 609647. MR 85e:57012
 4.
 M. Fujii and T. Soma, Totally geodesic boundaries are dense in the moduli space, J. Math. Soc. Japan 49 (1997), 589601. MR 99b:57029
 5.
 W. Goldman, Discontinuous groups and the Euler class, Ph. D. Thesis, U.C. Berkeley (1980).
 6.
 C. HayatLegrand, S. Wang and H. Zieschang, Degreeone maps onto lens spaces, Pacific J. Math. 176 (1996), 1932. MR 98b:57030
 7.
 C. HayatLegrand, S. Wang and H. Zieschang, Minimal Seifert manifolds, Math. Ann. 308 (1997), 673700. 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 threemanifold 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 lowdimensional topology, Geometric Topology (W.H. Kazez ed.), AMS/IP Studies in Advanced Mathematics vol. 2, Part 2, Amer. Math. Soc. and International Press (1997), 35473. MR 98f:57001
 11.
 C. McMullen, Amenability, Poincaré series and quasiconformal maps, Invent. Math. 97 (1989), 95127. MR 90e:30048
 12.
 C. McMullen, Iteration on Teichmüller space, Invent. Math. 99 (1990), 425454. MR 91a:57008
 13.
 G. Mostow, Quasiconformal mappings in space and the rigidity of hyperbolic space forms, Publ. Math. I.H.E.S. 34 (1968), 53104. MR 38:4679
 14.
 R. Myers, Simple knots in compact, orientable manifolds, Trans. Amer. Math. Soc. 273 (1982), 7591. 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), 411436. MR 92j:57007
 17.
 T. Soma, Bounded cohomology of closed surfaces, Topology 36 (1997), 12211246. MR 99a:57011
 18.
 T. Soma, Nonzero degree maps to hyperbolic manifolds, J. Differential Geom. 49 (1998), 517546. MR 2000b:57034
 19.
 T. Soma, Sequences of degreeone maps between geometric manifolds, Math. Ann. 316 (2000), 733742. 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), 357381. MR 83h:57019
 22.
 W. Thurston, Hyperbolic structures on manifolds I: Deformation of acylindrical manifolds, Ann. of Math. 124 (1986), 203246. 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), 459487. 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 MilnorThurston homology theory with singular homology theory, Pacific J. Math. 186 (1998), 369396. 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, Hatoyamamachi, Saitamaken 3500394, Japan
Email:
soma@r.dendai.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002994701027878
PII:
S 00029947(01)027878
Keywords:
Degreeone maps,
hyperbolic $3$manifolds,
GromovThurston'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
