Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.


Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: Albert Fathi, François Laudenbach and Valentin Poénaru
Translated by Djun Kim and Dan Margalit
Title: Thurston’s work on surfaces
Additional book information: Mathematical Notes, 48, Princeton University Press, 2012, xiii+255 pp., ISBN 9780691147352, US $60.00

References [Enhancements On Off] (What's this?)

  • [1] John W. Aaber and Nathan Dunfield, Closed surface bundles of least volume, Algebr. Geom. Topol. 10 (2010), no. 4, 2315-2342. MR 2745673 (2012c:57030), https://doi.org/10.2140/agt.2010.10.2315
  • [2] William Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics, vol. 820, Springer, Berlin, 1980. MR 590044 (82a:32028)
  • [3] Ian Agol, The virtual haken conjecture, arXiv:1204.2810.
  • [4] -, Ideal triangulations of pseudo-Anosov mapping tori, Topology and geometry in dimension three, Contemp. Math., vol. 560, Amer. Math. Soc., Providence, RI, 2011, pp. 1-17. MR 2866919 (2012m:57026)
  • [5] Pierre Arnoux and Jean-Christophe Yoccoz, Construction de difféomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 75-78 (French, with English summary). MR 610152 (82b:57018)
  • [6] D. Asimov and J. Franks, Unremovable closed orbits, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 22-29. MR 730260 (86a:58083), https://doi.org/10.1007/BFb0061407
  • [7] Max Bauer, Examples of pseudo-Anosov homeomorphisms, Trans. Amer. Math. Soc. 330 (1992), no. 1, 333-359. MR 1094557 (92g:57025), https://doi.org/10.2307/2154168
  • [8] Max Bauer, An upper bound for the least dilatation, Trans. Amer. Math. Soc. 330 (1992), no. 1, 361-370. MR 1094556 (92g:57024), https://doi.org/10.2307/2154169
  • [9] Lipman Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978), no. 1-2, 73-98. MR 0477161 (57 #16704)
  • [10] M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms, Topology 34 (1995), no. 1, 109-140. MR 1308491 (96d:57014), https://doi.org/10.1016/0040-9383(94)E0009-9
  • [11] Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J., 1974. Annals of Mathematics Studies, No. 82. MR 0375281 (51 #11477)
  • [12] Joan S. Birman and Mark E. Kidwell, Fixed points of pseudo-Anosov diffeomorphisms of surfaces, Adv. in Math. 46 (1982), no. 2, 217-220. MR 679909 (84b:58090), https://doi.org/10.1016/0001-8708(82)90024-X
  • [13] Joan S. Birman, Alex Lubotzky, and John McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983), no. 4, 1107-1120. MR 726319 (85k:20126), https://doi.org/10.1215/S0012-7094-83-05046-9
  • [14] Francis Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), no. 1, 139-162. MR 931208 (90a:32025), https://doi.org/10.1007/BF01393996
  • [15] Philip Boyland, Notes on dynamics of surface homeomorphisms, Warwick, preprint.
  • [16] Peter Brinkmann, An implementation of the Bestvina-Handel algorithm for surface homeomorphisms, Experiment. Math. 9 (2000), no. 2, 235-240. MR 1780208 (2001e:57017)
  • [17] Peter Brinkmann, A note on pseudo-Anosov maps with small growth rate, Experiment. Math. 13 (2004), no. 1, 49-53. MR 2065567 (2005b:37075)
  • [18] John Cantwell and Lawrence Conlon, Handel-Miller theory and finite depth foliations, arXiv:1006.4525.
  • [19] Andrew J. Casson and Steven A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988. MR 964685 (89k:57025)
  • [20] Jin-Hwan Cho and Ji-Young Ham, The minimal dilatation of a genus-two surface, Experiment. Math. 17 (2008), no. 3, 257-267. MR 2455699 (2009i:37096)
  • [21] François Dahmani, Vincent Guirardel, and Denis Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, arXiv:1111.7048v3.
  • [22] André de Carvalho, Toby Hall, and Rupert Venzke, On period minimal pseudo-Anosov braids, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1771-1776. MR 2470836 (2009m:37119), https://doi.org/10.1090/S0002-9939-08-09709-8
  • [23] Benson Farb (editor), Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics, vol. 74, American Mathematical Society, Providence, RI, 2006. MR 2251041 (2007e:57001)
  • [24] Benson Farb, Some problems on mapping class groups and moduli space, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, pp. 11-55. MR 2264130 (2007h:57018)
  • [25] Benson Farb, Christopher J. Leininger, and Dan Margalit, The lower central series and pseudo-Anosov dilatations, Amer. J. Math. 130 (2008), no. 3, 799-827. MR 2418928 (2009d:37072), https://doi.org/10.1353/ajm.0.0005
  • [26] Benson Farb, Christopher J. Leininger, and Dan Margalit, Small dilatation pseudo-Anosov homeomorphisms and 3-manifolds, Adv. Math. 228 (2011), no. 3, 1466-1502. MR 2824561 (2012f:37093), https://doi.org/10.1016/j.aim.2011.06.020
  • [27] Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125 (2012h:57032)
  • [28] Albert Fathi, Démonstration d'un théorème de Penner sur la composition des twists de Dehn, Bull. Soc. Math. France 120 (1992), no. 4, 467-484 (French, with English and French summaries). MR 1194272 (93j:57005)
  • [29] Sérgio R. Fenley, End periodic surface homeomorphisms and $ 3$-manifolds, Math. Z. 224 (1997), no. 1, 1-24. MR 1427700 (97m:57023), https://doi.org/10.1007/PL00004576
  • [30] John Franks and Michael Handel, Periodic points of Hamiltonian surface diffeomorphisms, Geom. Topol. 7 (2003), 713-756 (electronic). MR 2026545 (2004j:37101), https://doi.org/10.2140/gt.2003.7.713
  • [31] David Fried, Flow equivalence, hyperbolic systems and a new zeta function for flows, Comment. Math. Helv. 57 (1982), no. 2, 237-259. MR 684116 (84g:58083), https://doi.org/10.1007/BF02565860
  • [32] David Fried, Growth rate of surface homeomorphisms and flow equivalence, Ergodic Theory Dynam. Systems 5 (1985), no. 4, 539-563. MR 829857 (88f:58118), https://doi.org/10.1017/S0143385700003151
  • [33] Koji Fujiwara, Subgroups generated by two pseudo-Anosov elements in a mapping class group. II. Uniform bound on exponents, arXiv:0908.0995.
  • [34] Jean-Marc Gambaudo, Sebastian van Strien, and Charles Tresser, Vers un ordre de Sarkovskiĭ pour les plongements du disque préservant l'orientation, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 5, 291-294 (French, with English summary). MR 1042866 (91f:58072)
  • [35] Marlies Gerber and Anatole Katok, Smooth models of Thurston's pseudo-Anosov maps, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 1, 173-204. MR 672479 (84e:58056)
  • [36] Ji-Young Ham and Won Taek Song, The minimum dilatation of pseudo-Anosov 5-braids, Experiment. Math. 16 (2007), no. 2, 167-179. MR 2339273 (2008e:37043)
  • [37] Michael Handel, Global shadowing of pseudo-Anosov homeomorphisms, Ergodic Theory Dynam. Systems 5 (1985), no. 3, 373-377. MR 805836 (87e:58172), https://doi.org/10.1017/S0143385700003011
  • [38] Michael Handel, The forcing partial order on the three times punctured disk, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 593-610. MR 1452182 (98i:57026), https://doi.org/10.1017/S0143385797084940
  • [39] Michael Handel and Richard Miller, Handel-Miller theory and finite depth foliations, unpublished.
  • [40] Michael Handel and William P. Thurston, New proofs of some results of Nielsen, Adv. in Math. 56 (1985), no. 2, 173-191. MR 788938 (87e:57015), https://doi.org/10.1016/0001-8708(85)90028-3
  • [41] John L. Harer, The cohomology of the moduli space of curves, Theory of moduli (Montecatini Terme, 1985) Lecture Notes in Math., vol. 1337, Springer, Berlin, 1988, pp. 138-221. MR 963064 (90a:32026), https://doi.org/10.1007/BFb0082808
  • [42] Eriko Hironaka, Small dilatation mapping classes coming from the simplest hyperbolic braid, Algebr. Geom. Topol. 10 (2010), no. 4, 2041-2060. MR 2728483 (2012e:57033), https://doi.org/10.2140/agt.2010.10.2041
  • [43] Eriko Hironaka and Eiko Kin, A family of pseudo-Anosov braids with small dilatation, Algebr. Geom. Topol. 6 (2006), 699-738 (electronic). MR 2240913 (2008h:57027), https://doi.org/10.2140/agt.2006.6.699
  • [44] Nikolai V. Ivanov, Nielsen numbers of mappings of surfaces, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 122 (1982), 56-65, 163-164 (Russian, with English summary). Studies in topology, IV. MR 661465 (83h:55001)
  • [45] Nikolai V. Ivanov, Coefficients of expansion of pseudo-Anosov homeomorphisms, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988), no. Issled. Topol. 6, 111-116, 191. MR 964259 (89i:32047)
  • [46] Nikolai V. Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, vol. 115, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by E. J. F. Primrose and revised by the author. MR 1195787 (93k:57031)
  • [47] Nikolai V. Ivanov, Mapping class groups, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 523-633. MR 1886678 (2003h:57022)
  • [48] Nikolai V. Ivanov, Fifteen problems about the mapping class groups, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, pp. 71-80. MR 2264532 (2008b:57003)
  • [49] B. Kolev, Periodic orbits of period $ 3$ in the disc, Nonlinearity 7 (1994), no. 3, 1067-1071. MR 1275541 (95b:58120)
  • [50] Irwin Kra, On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981), no. 3-4, 231-270. MR 611385 (82m:32019), https://doi.org/10.1007/BF02392465
  • [51] Erwan Lanneau and Jean-Luc Thiffeault, On the minimum dilatation of braids on punctured discs, Geom. Dedicata 152 (2011), 165-182. Supplementary material available online. MR 2795241 (2012h:57035), https://doi.org/10.1007/s10711-010-9551-2
  • [52] Erwan Lanneau and Jean-Luc Thiffeault, On the minimum dilatation of pseudo-Anosov homeromorphisms on surfaces of small genus, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 1, 105-144 (English, with English and French summaries). MR 2828128 (2012e:37070), https://doi.org/10.5802/aif.2599
  • [53] Christopher J. Leininger, On groups generated by two positive multi-twists: Teichmüller curves and Lehmer's number, Geom. Topol. 8 (2004), 1301-1359 (electronic). MR 2119298 (2005j:57002), https://doi.org/10.2140/gt.2004.8.1301
  • [54] Darren D. Long and Ulrich Oertel, Hyperbolic surface bundles over the circle, Progress in knot theory and related topics, Travaux en Cours, vol. 56, Hermann, Paris, 1997, pp. 121-142. MR 1603150 (98m:57022)
  • [55] Jérôme Los, On the forcing relation for surface homeomorphisms, Inst. Hautes Études Sci. Publ. Math. 85 (1997), 5-61. MR 1471865 (98h:58151)
  • [56] Joseph Maher, Random walks on the mapping class group, Duke Math. J. 156 (2011), no. 3, 429-468. MR 2772067 (2012j:37069), https://doi.org/10.1215/00127094-2010-216
  • [57] Johanna Mangahas, A recipe for short word pseudo-anosovs, To appear.
  • [58] Howard Masur, Ergodic actions of the mapping class group, Proc. Amer. Math. Soc. 94 (1985), no. 3, 455-459. MR 787893 (86i:32044), https://doi.org/10.2307/2045234
  • [59] Shigenori Matsumoto, Topological entropy and Thurston's norm of atoroidal surface bundles over the circle, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 3, 763-778. MR 927609 (89c:57011)
  • [60] John McCarthy, A ``Tits-alternative'' for subgroups of surface mapping class groups, Trans. Amer. Math. Soc. 291 (1985), no. 2, 583-612. MR 800253 (87f:57011), https://doi.org/10.2307/2000100
  • [61] Curtis T. McMullen, Polynomial invariants for fibered 3-manifolds and Teichmüller geodesics for foliations, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 4, 519-560 (English, with English and French summaries). MR 1832823 (2002d:57015), https://doi.org/10.1016/S0012-9593(00)00121-X
  • [62] D. B. McReynolds, The congruence subgroup problem for braid groups: Thurston's proof, arXiv:0901.4663.
  • [63] Richard T. Miller, Geodesic laminations from Nielsen's viewpoint, Adv. in Math. 45 (1982), no. 2, 189-212. MR 664623 (83j:57004), https://doi.org/10.1016/S0001-8708(82)80003-0
  • [64] Hiroyuki Minakawa, Examples of pseudo-Anosov homeomorphisms with small dilatations, J. Math. Sci. Univ. Tokyo 13 (2006), no. 2, 95-111. MR 2277516 (2007m:37073)
  • [65] Lee Mosher, MSRI course on Mapping Class Groups, Lecture notes (Fall 2007, available at http://andromeda.rutgers.edu/~mosher).
  • [66] Jakob Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen, Acta Math. 50 (1927), no. 1, 189-358 (German). MR 1555256, https://doi.org/10.1007/BF02421324
  • [67] Jakob Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. II, Acta Math. 53 (1929), no. 1, 1-76. MR 1555290
  • [68] Jakob Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen. III, Acta Math. 58 (1932), no. 1, 87-167 (German). MR 1555345, https://doi.org/10.1007/BF02547775
  • [69] Jakob Nielsen, Surface transformation classes of algebraically finite type, Danske Vid. Selsk. Math.-Phys. Medd. 21 (1944), no. 2, 89. MR 0015791 (7,469c)
  • [70] Jean-Pierre Otal, Le spectre marqué des longueurs des surfaces à courbure négative, Ann. of Math. (2) 131 (1990), no. 1, 151-162 (French). MR 1038361 (91c:58026), https://doi.org/10.2307/1971511
  • [71] Robert C. Penner, A construction of pseudo-Anosov homeomorphisms, Trans. Amer. Math. Soc. 310 (1988), no. 1, 179-197. MR 930079 (89k:57026), https://doi.org/10.2307/2001116
  • [72] R. C. Penner, Bounds on least dilatations, Proc. Amer. Math. Soc. 113 (1991), no. 2, 443-450. MR 1068128 (91m:57010), https://doi.org/10.2307/2048530
  • [73] Gérard Rauzy, Échanges d'intervalles et transformations induites, Acta Arith. 34 (1979), no. 4, 315-328 (French). MR 543205 (82m:10076)
  • [74] Igor Rivin, Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms, Duke Math. J. 142 (2008), no. 2, 353-379. MR 2401624 (2009m:20077), https://doi.org/10.1215/00127094-2008-009
  • [75] Hongbin Sun, A transcendental invariant of pseudo-anosov maps, arXiv:1209.2613.
  • [76] William P. Thurston, A discussion on geometrization, Lecture, Harvard University, available at http://www.youtube.com/watch?v=Qzxk8VLqGcI. Video recording by Danny Calegari.
  • [77] -, Entropy in dimension one, unpublished.
  • [78] -, Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, arXiv:math.GT/9801045, 1986.
  • [79] William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357-381. MR 648524 (83h:57019), https://doi.org/10.1090/S0273-0979-1982-15003-0
  • [80] William P. Thurston, A norm for the homology of $ 3$-manifolds, Mem. Amer. Math. Soc. 59 (1986), no. 339, i-vi and 99-130. MR 823443 (88h:57014)
  • [81] William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417-431. MR 956596 (89k:57023), https://doi.org/10.1090/S0273-0979-1988-15685-6
  • [82] Chia-Yen Tsai, The asymptotic behavior of least pseudo-Anosov dilatations, Geom. Topol. 13 (2009), no. 4, 2253-2278. MR 2507119 (2010d:37081), https://doi.org/10.2140/gt.2009.13.2253
  • [83] William A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2) 124 (1986), no. 3, 441-530. MR 866707 (88g:58153), https://doi.org/10.2307/2007091
  • [84] Kim Whittlesey, Normal all pseudo-Anosov subgroups of mapping class groups, Geom. Topol. 4 (2000), 293-307 (electronic). MR 1786168 (2001j:57022), https://doi.org/10.2140/gt.2000.4.293
  • [85] Daniel Wise, The structure of groups with a quasiconvex hierarchy, preprint.

Review Information:

Reviewer: Dan Margalit
Affiliation: School of Mathematics Georgia Institute of Technology
Journal: Bull. Amer. Math. Soc. 51 (2014), 151-161
MSC (2010): Primary 37E30, 57M50, 20F34
DOI: https://doi.org/10.1090/S0273-0979-2013-01419-8
Published electronically: June 10, 2013
Review copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
American Mathematical Society