Book Review
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3363152
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Book Information:
Authors:
Benson Farb and
Dan Margalit
Title:
A primer on mapping class groups
Additional book information:
Princeton Mathematical Series, Vol. 49,
Princeton University Press,
Princeton, NJ,
2012,
xiv+472 pp.,
ISBN 978-0-691-14794-9,
US $75.00.
L. Ahlfors and G. Weill, A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 13 (1962), 975–978. MR 148896, DOI 10.1090/S0002-9939-1962-0148896-1
Lars V. Ahlfors, Lectures on quasiconformal mappings, 2nd ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. MR 2241787, DOI 10.1090/ulect/038
R. Baer, Kurventypen auf Flächen, J. Reine Angew. Math. 156 (1927), 231–246.
R. Baer, Isotopie von Kurven auf orientierbaren, geschlossenen Flächen und ihr Zusammenhang mit der topologischen Deformation der Flächen, J. Reine Angew. Math. 159 (1928), 101–111.
Jason Behrstock, Cornelia Druţu, and Mark Sapir, Median structures on asymptotic cones and homomorphisms into mapping class groups, Proc. Lond. Math. Soc. (3) 102 (2011), no. 3, 503–554. MR 2783135, DOI 10.1112/plms/pdq025
Jason Behrstock, Cornelia Druţu, and Mark Sapir, Addendum: Median structures on asymptotic cones and homomorphisms into mapping class groups [MR2783135], Proc. Lond. Math. Soc. (3) 102 (2011), no. 3, 555–562. MR 2783136, DOI 10.1112/plms/pdr008
Jason Behrstock, Bruce Kleiner, Yair Minsky, and Lee Mosher, Geometry and rigidity of mapping class groups, Geom. Topol. 16 (2012), no. 2, 781–888. MR 2928983, DOI 10.2140/gt.2012.16.781
Jason A. Behrstock and Yair N. Minsky, Centroids and the rapid decay property in mapping class groups, J. Lond. Math. Soc. (2) 84 (2011), no. 3, 765–784. MR 2855801, DOI 10.1112/jlms/jdr027
Lipman Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978), no. 1-2, 73–98. MR 477161, DOI 10.1007/BF02545743
M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms, Topology 34 (1995), no. 1, 109–140. MR 1308491, DOI 10.1016/0040-9383(94)E0009-9
Mladen Bestvina, Kenneth Bromberg, and Koji Fujiwara, Constructing group actions on quasi-trees and applications to mapping class groups, arXiv:1006.1939.
Mladen Bestvina, Kai-Uwe Bux, and Dan Margalit, The dimension of the Torelli group, J. Amer. Math. Soc. 23 (2010), no. 1, 61–105. MR 2552249, DOI 10.1090/S0894-0347-09-00643-2
Joan S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213–238. MR 243519, DOI 10.1002/cpa.3160220206
Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
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, DOI 10.1215/S0012-7094-83-05046-9
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, DOI 10.1017/CBO9780511623912
Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), no. 1, 91–119. MR 830040, DOI 10.1007/BF01388734
Max Dehn, Papers on group theory and topology, Springer-Verlag, New York, 1987. Translated from the German and with introductions and an appendix by John Stillwell; With an appendix by Otto Schreier. MR 881797, DOI 10.1007/978-1-4612-4668-8
P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. MR 262240
C. J. Earle and J. Eells, The diffeomorphism group of a compact Riemann surface, Bull. Amer. Math. Soc. 73 (1967), 557–559. MR 212840, DOI 10.1090/S0002-9904-1967-11746-4
Clifford J. Earle and James Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43. MR 276999
D. B. A. Epstein, Curves on $2$-manifolds and isotopies, Acta Math. 115 (1966), 83–107. MR 214087, DOI 10.1007/BF02392203
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, DOI 10.1090/pspum/074/2264130
Travaux de Thurston sur les surfaces, Société Mathématique de France, Paris, 1991 (French). Séminaire Orsay; Reprint of Travaux de Thurston sur les surfaces, Soc. Math. France, Paris, 1979 [ MR0568308 (82m:57003)]; Astérisque No. 66-67 (1991) (1991). MR 1134426
W. Fenchel, Estensioni di gruppi discontinui e trasformazioni periodiche delle superficie, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 5 (1948), 326–329 (Italian). MR 29161
W. Fenchel, Remarks on finite groups of mapping classes, Mat. Tidsskr. B 1950 (1950), 90–95 (Danish). MR 38069
Werner Fenchel and Jakob Nielsen, Discontinuous groups of isometries in the hyperbolic plane, De Gruyter Studies in Mathematics, vol. 29, Walter de Gruyter & Co., Berlin, 2003. Edited and with a preface by Asmus L. Schmidt; Biography of the authors by Bent Fuglede. MR 1958350
Robert Fricke and Felix Klein, Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausführungen und die Andwendungen, Bibliotheca Mathematica Teubneriana, Bande 3, vol. 4, Johnson Reprint Corp., New York; B. G. Teubner Verlagsgesellschaft, Stuttgart, 1965 (German). MR 0183872
Ursula Hamenstädt, Geometry of the mapping class groups. I. Boundary amenability, Invent. Math. 175 (2009), no. 3, 545–609. MR 2471596, DOI 10.1007/s00222-008-0158-2
Michael Handel and William P. Thurston, New proofs of some results of Nielsen, Adv. in Math. 56 (1985), no. 2, 173–191. MR 788938, DOI 10.1016/0001-8708(85)90028-3
John Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983), no. 2, 221–239. MR 700769, DOI 10.1007/BF01389321
John L. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. (2) 121 (1985), no. 2, 215–249. MR 786348, DOI 10.2307/1971172
John L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986), no. 1, 157–176. MR 830043, DOI 10.1007/BF01388737
John Harer, The third homology group of the moduli space of curves, Duke Math. J. 63 (1991), no. 1, 25–55. MR 1106936, DOI 10.1215/S0012-7094-91-06302-7
W. J. Harvey, Boundary structure of the modular group, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 245–251. MR 624817
A. Hatcher and W. Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), no. 3, 221–237. MR 579573, DOI 10.1016/0040-9383(80)90009-9
Allen Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991), no. 2, 189–194. MR 1123262, DOI 10.1016/0166-8641(91)90050-V
Sebastian Hensel, Piotr Przytycki, and Richard C. H. Webb, Slim unicorns and uniform hyperbolicity for arc graphs and curve graphs, arXiv:1301.5577.
John Hamal Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1, Matrix Editions, Ithaca, NY, 2006. Teichmüller theory; With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra; With forewords by William Thurston and Clifford Earle. MR 2245223
Stephen P. Humphries, Generators for the mapping class group, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 44–47. MR 547453
N. V. Ivanov, Stabilization of the homology of Teichmüller modular groups, Algebra i Analiz 1 (1989), no. 3, 110–126 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 3, 675–691. MR 1015128
N. V. Ivanov, Complexes of curves and Teichmüller spaces, Mat. Zametki 49 (1991), no. 5, 54–61, 158 (Russian); English transl., Math. Notes 49 (1991), no. 5-6, 479–484. MR 1137173, DOI 10.1007/BF01142643
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, DOI 10.1090/mmono/115
Nikolai V. Ivanov, Mapping class groups, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 523–633. MR 1886678
Dennis L. Johnson, Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1979), no. 1, 119–125. MR 529227, DOI 10.1090/S0002-9939-1979-0529227-4
Dennis Johnson, The structure of the Torelli group. I. A finite set of generators for ${\cal I}$, Ann. of Math. (2) 118 (1983), no. 3, 423–442. MR 727699, DOI 10.2307/2006977
Dennis Johnson, The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves, Topology 24 (1985), no. 2, 113–126. MR 793178, DOI 10.1016/0040-9383(85)90049-7
Dennis Johnson, The structure of the Torelli group. III. The abelianization of $\scr T$, Topology 24 (1985), no. 2, 127–144. MR 793179, DOI 10.1016/0040-9383(85)90050-3
Linda Keen, Intrinsic moduli on Riemann surfaces, Ann. of Math. (2) 84 (1966), 404–420. MR 203000, DOI 10.2307/1970454
Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. MR 690845, DOI 10.2307/2007076
Yoshikata Kida, The mapping class group from the viewpoint of measure equivalence theory, Mem. Amer. Math. Soc. 196 (2008), no. 916, viii+190. MR 2458794, DOI 10.1090/memo/0916
Catherine Labruère and Luis Paris, Presentations for the punctured mapping class groups in terms of Artin groups, Algebr. Geom. Topol. 1 (2001), 73–114. MR 1805936, DOI 10.2140/agt.2001.1.73
W. B. R. Lickorish, A finite set of generators for the homeotopy group of a $2$-manifold, Proc. Cambridge Philos. Soc. 60 (1964), 769–778. MR 171269, DOI 10.1017/s030500410003824x
Ib Madsen and Michael Weiss, The stable moduli space of Riemann surfaces: Mumford’s conjecture, Ann. of Math. (2) 165 (2007), no. 3, 843–941. MR 2335797, DOI 10.4007/annals.2007.165.843
W. Mangler, Die Klassen von topologischen Abbildungen einer geschlossenen Fläche auf sich, Math. Z. 44 (1939), no. 1, 541–554 (German). MR 1545786, DOI 10.1007/BF01210672
Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149. MR 1714338, DOI 10.1007/s002220050343
H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal. 10 (2000), no. 4, 902–974. MR 1791145, DOI 10.1007/PL00001643
Howard Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623–635. MR 417456
John McCarthy, A “Tits-alternative” for subgroups of surface mapping class groups, Trans. Amer. Math. Soc. 291 (1985), no. 2, 583–612. MR 800253, DOI 10.1090/S0002-9947-1985-0800253-8
James McCool, Some finitely presented subgroups of the automorphism group of a free group, J. Algebra 35 (1975), 205–213. MR 396764, DOI 10.1016/0021-8693(75)90045-9
Geoffrey Mess, The Torelli groups for genus $2$ and $3$ surfaces, Topology 31 (1992), no. 4, 775–790. MR 1191379, DOI 10.1016/0040-9383(92)90008-6
Edward Y. Miller, The homology of the mapping class group, J. Differential Geom. 24 (1986), no. 1, 1–14. MR 857372
Richard T. Miller, Geodesic laminations from Nielsen’s viewpoint, Adv. in Math. 45 (1982), no. 2, 189–212. MR 664623, DOI 10.1016/S0001-8708(82)80003-0
Shigeyuki Morita, Characteristic classes of surface bundles, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 386–388. MR 752805, DOI 10.1090/S0273-0979-1984-15321-7
Shigeyuki Morita, Characteristic classes of surface bundles, Invent. Math. 90 (1987), no. 3, 551–577. MR 914849, DOI 10.1007/BF01389178
David Mumford, Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328. MR 717614
Jakob Nielsen, Jakob Nielsen: collected mathematical papers. Vol. 1, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1986. Edited and with a preface by Vagn Lundsgaard Hansen. MR 865335
Jakob Nielsen, Jakob Nielsen: collected mathematical papers. Vol. 2, Contemporary Mathematicians, Birkhäuser Boston, Inc., Boston, MA, 1986. Edited and with a preface by Vagn Lundsgaard Hansen. MR 865336
Jerome Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978), no. 3, 347–350. MR 494115, DOI 10.1090/S0002-9939-1978-0494115-8
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Jing Tao, Linearly bounded conjugator property for mapping class groups, Geom. Funct. Anal. 23 (2013), no. 1, 415–466. MR 3037904, DOI 10.1007/s00039-012-0206-3
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References
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Review Information:
Reviewer:
Mladen Bestvina
Affiliation:
University of Utah
Journal:
Bull. Amer. Math. Soc.
51 (2014), 691-700
DOI:
https://doi.org/10.1090/S0273-0979-2014-01454-5
Published electronically:
May 27, 2014
Review copyright:
© Copyright 2014
American Mathematical Society