топология / Farb, Margalit, A primer on mapping class groups

Conversely, every measured foliation (F, ν) on S can be “pinched down” to a measured train track (τ, µ). For us, this will mean that for every measured foliation (F, ν) there is a measured train track (τ, µ) with the property that performing the above “rectangle gluing procedure ” to (τ, µ) yields (F, ν). The track (τ, µ), along with the rectangle decomposition, can be obtained from (F, ν) constructively. One way to do this is to first construct the Markov partition (see Section 14.6), and then to go from there to the train track. The reader might want to try to find the train trac k hidden in Figure 14.8.

We are now ready to present the steps needed to analyze an arbitrarily given element of Mod(S).

Step 1 (Finitely many tracks). There is a finite collection of train tracks τ1, . . . , τr on S with the property that every measured foliation on S is carried by one of the τi. To see why this is true, take a finite 4-valent graph on S so that S \ is a union of disks. Then create tracks τi by choosing, for each vertex of , one of the two possible “smoothings” illustrated in Figure 15.11. A choice of smoothing for each vertex of gives a train track. Making all possible choices gives us the desired collection of tracks, since, locally, a foliation is given by a set of parallel segments.

Figure 15.11 Two ways to smooth an intersection.

Step 2 (Measures on a track). From the above discussion, we see that the set of all measured foliations carried by a given track is given by a collection of variables, one for each edge, that represent positive real numbers and satisfy a finite number of linear equations, namely the sw itch conditions. Thus the set of projective classes of measured foliations carried on any fixed train track τ is given by a polyhedron P (τ ). We identify any two polyhedra whose train tracks differ by Whitehead moves—see the right hand

In general, a Whitehead move is the procedure of collapsing, or “uncollapsing”, an edge (resp. leaf) of a train track (resp. folitation ) connecting two switches (resp. singularities). Taking only the polyhedra P (τ ) for which the train track τ is maximal (complementary regions are disks and oncepunctured disks), one checks that each P (τ ) is a (6g −7 + 2n)-dimensional polyhedron.

Figure 15.12 Whithead moves on measured foliations (left) and measured train tracks (right). The action on measures is the natural one.

Step 3 (Gluing polyhedra to get a ball). Some foliations are carried on more than one train track. Identify points of the P (τi) that correspond to equivalent projective measured foliations, where the equivalence is generated by Whitehead moves—see the left hand side of Figure 15.12. The main claim to prove is that, once all such identifications are made, the resulting space is homeomorphic to an open ball of dimension 6g − 6 + 2n. By the above identification of train tracks with foliations , the resulting space MF = MF(S) is actually equivalent to the space of all equivalence classes of measured foliations. Since both spaces admit an R - action, the quotient space of projective equivalence classes is homeomorphic to a sphere of dimension 6g − 7 + 2n. This sphere represents the space PMF = PMF(S) of (Whitehead equivalence classes of) projective classes of (nonzero) measured foliations on S.

Step 4 (Topology of a closed ball). The union Teich(S) PMF can be naturally topologized so that it is homeomorphic to a closed ball of dimension 6g − 6 + 2n. Let π : RS≥0 → P (RS≥0) denote the projectivization map. The open sets on the union are the open sets of Teich(S) and open sets of the form

(Teich(S) ∩ π−1(U )) (PMF ∩ U ) where U is an open set of P (RS≥0).

Step 5 (Applying Brouwer). Each [φ] = f Mod(S) naturally acts as a homeomorphism on Teich(S) PMF. To check this, one can either use the action on the space of measured foliations:

φ(F, ν) = (φ(F), φ ν).

or the action on “train track coordinates”: f acts on a track (τ, µ) by

f (τ, µ) = (f (τ ), f (µ))

For the latter action, one must prove that the resulting track is carried by one of the train tracks comprising the parameterization of MF. As in Section 15.1, the action on a measured train track depends not just on the track, but on the measure as well.

Thus, we may apply the Brouwer Fixed Point Theorem to conclude that the action of f has a fixed point in Teich(S) PMF. This means that

either f fixes a point of Teich(S), or there exists (F, µ) MF so that f · (F, µ) = (F, λµ) for some λ R .

Step 6 (Analyzing the fixed point). Say that a measured foliation is arational if it does not contain any closed leaves. We have the following cases for the fixed point of f :

1.f · X = X where X Teich(S)

2.f · (F, µ) = (F, λµ), where F is not arational

3.f · (F, µ) = (F, λµ), where F is arational and λ = 1

4.f · (F, µ) = (F, λµ), where F is arational and λ > 1

We have already seen in Chapter 13 that in Case 1 f is finite order. In Case 3, we deduce that f permutes the finite collection of rectangles in some rectangle decomposition of S induced by F, and so f must again be finite order. In Case 2, the foliation has closed leaves that are fixe d up to isotopy by f , and so f is reducible. In Case 4, then, by building a Markov partition for f , one can find a unique measured foliation that is transverse t o F and projectively invariant by f . This last step is the most technically involved part of Thurston's proof.

Other points of view. There are many other points of view on this topic, some of which we have not even touched upon. For example, the theory of measured laminations gives yet another way to think about measured

foliations. The Bestvina–Handel algorithm uses the theory of train tracks to give an explicit algorithm for determining the Nielsen–T hurston type of any mapping class [14]. Ivanov proved a number of structural theorems about Mod(S) using the dynamics of the action on the Thurston boundary. The original approach of Nielsen is different still—it invo lves analyzing the action of a mapping class on ∂H2.

Bibliography

[1]William Abikoff. The real analytic theory of Teichmuller¨ space, volume 820 of Lecture Notes in Mathematics. Springer, Berlin, 1980.

[2]Lars Ahlfors and Lipman Bers. Riemann's mapping theorem for variable metrics. Ann. of Math. (2), 72:385–404, 1960.

[3]Lars V. Ahlfors. On quasiconformal mappings. J. Analyse Math., 3:1–58; correction, 207–208, 1954.

[4]M. A. Armstrong. The fundamental group of the orbit space of a discontinuous group. Proc. Cambridge Philos. Soc., 64:299–301, 1968.

[5]E. Artin. Theory of braids. Ann. of Math. (2), 48:101–126, 1947.

[6]M. F. Atiyah. The signature of fibre-bundles. In Global Analysis (Papers in Honor of K. Kodaira), pages 73–84. Univ. Tokyo Press, Tokyo, 1969.

[7]R. Baer. Kurventypen auf Fl¨achen. J. reine angew. Math., 156:231– 246, 1927.

[8]R. Baer. Isotopie von Kurven auf orientierbaren, geschlossenen Fl¨achen und ihr Zusammenhang mit der topologischen Deformation der Fl¨achen. J. reine angew. Math., 159:101–111, 1928.

[9]Walter L. Baily, Jr. On the moduli of Jacobian varieties. Ann. of Math. (2), 71:303–314, 1960.

[10]Hyman Bass and Alexander Lubotzky. Automorphisms of groups and of schemes of finite type. Israel J. Math., 44(1):1–22, 1983.

[11]Lipman Bers. Quasiconformal mappings and Teichm ¨uller's theorem. In Analytic functions, pages 89–119. Princeton Univ. Press, Princeton, N.J., 1960.

[12]Lipman Bers. An extremal problem for quasiconformal mappings and a theorem by Thurston. Acta Math., 141(1-2):73–98, 1978.

[13]Lipman Bers. An inequality for Riemann surfaces. In Differential geometry and complex analysis, pages 87–93. Springer, Berlin, 1985.

[14]M. Bestvina and M. Handel. Train-tracks for surface homeomorphisms. Topology, 34(1):109–140, 1995.

[15]Mladen Bestvina, Kenneth Bromberg, Koji Fujiwara, and Juan Suoto. Shearing coordinates and convexity of length functions on Teichmueller space. arXiv:0902.0829.

[16]Mladen Bestvina, Kai-Uwe Bux, and Dan Margalit. Dimension of the Torelli group for Out(Fn). Invent. Math., 170(1):1–32, 2007.

[17]Joan S. Birman. Abelian quotients of the mapping class group of a 2-manifold. Bull. Amer. Math. Soc., 76:147–150, 1970.

[18]Joan S. Birman. Errata: “Abelian quotients of the mappi ng class group of a 2-manifold”. Bull. Amer. Math. Soc., 77:479, 1971.

[19]Joan S. Birman. On Siegel's modular group. Math. Ann., 191:59–68, 1971.

[20]Joan S. Birman. Braids, links, and mapping class groups. Princeton University Press, Princeton, N.J., 1974. Annals of Mathematics Studies, No. 82.

[21]Joan S. Birman. Mapping class groups of surfaces. In Braids (Santa Cruz, CA, 1986), volume 78 of Contemp. Math., pages 13–43. Amer. Math. Soc., Providence, RI, 1988.

[22]Joan S. Birman and Hugh M. Hilden. On the mapping class groups of closed surfaces as covering spaces. In Advances in the theory of Riemann surfaces (Proc. Conf., Stony Brook, N.Y., 1969), pages 81– 115. Ann. of Math. Studies, No. 66. Princeton Univ. Press, Princeton, N.J., 1971.

[23]Joan S. Birman and Hugh M. Hilden. On isotopies of homeomorphisms of Riemann surfaces. Ann. of Math. (2), 97:424–439, 1973.

[24]Joan S. Birman, Alex Lubotzky, and John McCarthy. Abelian and solvable subgroups of the mapping class groups. Duke Math. J., 50(4):1107–1120, 1983.

[25]Joan S. Birman and Bronislaw Wajnryb. Presentations of the mapping class group. Errata: “ 3-fold branched coverings and the mapping class group of a surface” [ Geometry and topology (College Park, MD, 1983/84), 24–46, Lecture Notes in Math., 1167, Springer, Berlin, 1985]; and “A simple presentation of the mapping class group of an orientable surface” [Israel J. Math. 45 (1983), no. 2-3, 157–174]; by Wajnryb. Israel J. Math., 88(1-3):425–427, 1994.

[26]Søren Kjærgaard Boldsen. Different versions of mappin g class groups of surfaces.

[27]Francis Bonahon. The geometry of Teichm ¨uller space via geodesic currents. Invent. Math., 92(1):139–162, 1988.

[28]Tara E. Brendle and Benson Farb. Every mapping class group is generated by 6 involutions. J. Algebra, 278(1):187–198, 2004.

[29]Tara E. Brendle and Dan Margalit. Commensurations of the Johnson kernel. Geom. Topol., 8:1361–1384 (electronic), 2004.

[30]Tara E. Brendle and Dan Margalit. Addendum to: “Commens urations of the Johnson kernel” [Geom. Topol. 8 (2004), 1361–1384; mr2119299]. Geom. Topol., 12(1):97–101, 2008.

[31]Martin R. Bridson and Andr´e Haefliger. Metric spaces of nonpositive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.

[32]L. E. J. Brouwer. Beweis der invarianz des n-dimensionalen gebiets. Math. Ann., 71(3):305–313, 1911.

[33]Kenneth S. Brown. Presentations for groups acting on simplyconnected complexes. J. Pure Appl. Algebra, 32(1):1–10, 1984.

[34]Kenneth S. Brown. Cohomology of groups, volume 87 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original.

[35]Ronald Brown and Philip J. Higgins. The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action. University of Wales, Bangor, Maths Preprint 02.25.

[36]Heinrich Burkhardt. Grundz¨uge einer allgemeinen Systematik der hyperelliptischen Functionen I. Ordnung. Mathematische Annalen, 35:198–296, 1890.

[37]W. Burnside. Theory of groups of finite order . Dover Publications Inc., New York, 1955. 2d ed.

[38]Peter Buser. Geometry and spectra of compact Riemann surfaces, volume 106 of Progress in Mathematics. Birkh¨auser Boston Inc., Boston, MA, 1992.

[39]Danny Calegari. Foliations and the geometry of 3-manifolds. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2007.

[40]Andrew J. Casson and Steven A. Bleiler. Automorphisms of surfaces after Nielsen and Thurston, volume 9 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1988.

[41]Jean Cerf. Topologie de certains espaces de plongements. Bull. Soc. Math. France, 89:227–380, 1961.

[42]D. R. J. Chillingworth. Winding numbers on surfaces. II. Math. Ann., 199:131–153, 1972.

[43]Thomas Church and Benson Farb. Parametrized abel-jacobi maps, a question of johnson, and a homological stability conjecture for the torelli group. arXiv:1001.1114, 2010.

[44]John B. Conway. Functions of one complex variable, volume 11 of

Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1978.

[45]M. Dehn. Die Gruppe der Abbildungsklassen. Acta Math., 69(1):135–206, 1938. Das arithmetische Feld auf Fl¨achen.

[46]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.

[47]P. Deligne and D. Mumford. The irreducibility of the space of curves

´

of given genus. Inst. Hautes Etudes Sci. Publ. Math., (36):75–109, 1969.

[48]Clifford J. Earle and James Eells. A fibre bundle descrip tion of Teichm ¨uller theory. J. Differential Geometry, 3:19–43, 1969.

[49]James Eells, Jr. and J. H. Sampson. Harmonic mappings of Riemannian manifolds. Amer. J. Math., 86:109–160, 1964.

[50]V.A. Efremoviˇc. On the proximity geometry of Riemanni an manifolds. Uspekhi Math Nauk, 8:189, 1953.

[51]D. B. A. Epstein. Curves on 2-manifolds and isotopies. Acta Math., 115:83–107, 1966.

[52]Edward Fadell and Lee Neuwirth. Configuration spaces. Math. Scand., 10:111–118, 1962.

[53]Benson Farb and Nikolai V. Ivanov. The Torelli geometry and its applications: research announcement. Math. Res. Lett., 12(2-3):293– 301, 2005.

[54]A. Fathi, F. Laudenbach, and V. Po´enaru, editors. Travaux de Thurston sur les surfaces, volume 66 of Asterisque´. Soci´et´ Math´ematique de France, Paris, 1979. S´eminaire Orsay, Wi th an English summary.

[55]W. Fenchel. Estensioni di gruppi discontinui e trasformazioni periodiche delle superficie. Atti Accad. Naz Lincei. Rend. Cl. Sci. Fis. Mat. Nat.(8), 5:326–329, 1948.

[56]W. Fenchel. Remarks on finite groups of mapping classes. Mat. Tidsskr. B., 1950:90–95, 1950.

[57]J. Franks and E. Rykken. Pseudo-Anosov homeomorphisms with quadratic expansion. Proc. Amer. Math. Soc., 127(7):2183–2192, 1999.

[58]Robert Fricke and Felix Klein. Vorlesungen uber¨ die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II: Die funktionentheoretischen Ausfuhrungen¨ und die Andwendungen, volume 4 of Bibliotheca Mathematica Teubneriana, Bande¨ 3. Johnson Reprint Corp., New York, 1965.

[59]William Fulton. Algebraic curves. Advanced Book Classics. Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1989. An introduction to algebraic geometry, Notes written with the collaboration of Richard Weiss, Reprint of 1969 original.

[60]F. R. Gantmacher. The theory of matrices. Vols. 1, 2. Translated by K. A. Hirsch. Chelsea Publishing Co., New York, 1959.

[61]Sylvain Gervais. Presentation and central extensions of mapping class groups. Trans. Amer. Math. Soc., 348(8):3097–3132, 1996.

[62]Jane Gilman. On the Nielsen type and the classification f or the mapping class group. Adv. in Math., 40(1):68–96, 1981.

Norm. Sup. (4), 6:53–66, 1973.

[64]Edna K. Grossman. On the residual finiteness of certain m apping class groups. J. London Math. Soc. (2), 9:160–164, 1974/75.

[65]Hessam Hamidi-Tehrani. Groups generated by positive multi-twists and the fake lantern problem. Algebr. Geom. Topol., 2:1155–1178 (electronic), 2002.

[66]Mary-Elizabeth Hamstrom. Some global properties of the space of homeomorphisms on a disc with holes. Duke Math. J., 29:657–662, 1962.

[67]Mary-Elizabeth Hamstrom. The space of homeomorphisms on a torus. Illinois J. Math., 9:59–65, 1965.

[68]Mary-Elizabeth Hamstrom. Homotopy groups of the space of homeomorphisms on a 2-manifold. Illinois J. Math., 10:563–573, 1966.

[69]Michael Handel. Global shadowing of pseudo-Anosov homeomorphisms. Ergodic Theory Dynam. Systems, 5(3):373–377, 1985.

[70]Michael Handel and William P. Thurston. New proofs of some results of Nielsen. Adv. in Math., 56(2):173–191, 1985.

[71]John Harer. The second homology group of the mapping class group of an orientable surface. Invent. Math., 72(2):221–239, 1983.

[72]John Harer. The cohomology of the moduli space of curves. In Theory of moduli (Montecatini Terme, 1985), volume 1337 of Lecture Notes in Math., pages 138–221. Springer, Berlin, 1988.

[73]W. J. Harvey. Boundary structure of the modular group. In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume 97 of Ann. of Math. Stud., pages 245–251, Princeton, N.J., 1981. Princeton Univ. Press.