Abstract
We construct a geometric model for the mapping class group \(\mathcal{M}\mathcal{C}\mathcal{G}\) of a non-exceptional oriented surface S of genus g with k punctures and use it to show that the action of \(\mathcal{M}\mathcal{C}\mathcal{G}\) on the compact metrizable Hausdorff space of complete geodesic laminations for S is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every subgroup of \(\mathcal{M}\mathcal{C}\mathcal{G}\).
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References
Adams, S.: Reduction of cocycles with hyperbolic target. Ergodic Theory Dyn. Syst. 16, 1111–1145 (1996)
Anantharaman-Delaroche, C., Renault, J.: Amenable Groupoids. Monographie 36 de l‘Enseignement Math. Genève (2000)
Baum, P., Connes, A., Higson, N.: Classifying space for proper G-actions and K-theory of group C *-algebras. Contemp. Math. 167, 241–291 (1994)
Bonahon, F.: Geodesic laminations on surfaces. Contemp. Math. 269, 1–37 (1997)
Bowditch, B.: Tight geodesics in the curve complex. Invent. Math. 171, 281–300 (2008)
Bridson, M., Haefliger, A.: Metric Spaces of Non-Positive Curvature. Springer Grundlehren, vol. 319. Springer, Berlin (1999)
Burger, M., Monod, N.: Bounded cohomology of lattices in higher rank Lie groups. J. Eur. Math. Soc. 1, 199–235 (1999)
Burger, M., Monod, N.: Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal. 12, 219–280 (2002)
Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Birkhäuser, Boston (1992)
Canary, R., Epstein, D., Green, P.: Notes on notes of Thurston. In: Epstein, D. (ed.) Analytical and Geometric Aspects of Hyperbolic Space. Lond. Math. Soc. Lect. Notes, vol. 111. Cambridge University Press, Cambridge (1987)
Casson, A., Bleiler, S.: Automorphisms of Surfaces after Nielsen and Thurston. Cambridge University Press, Cambridge (1988)
Farb, B., Lubotzky, A., Minsky, Y.: Rank one phenomena for mapping class groups. Duke Math. J. 106, 581–597 (2001)
Fathi, A., Laudenbach, F., Poénaru, V.: Travaux de Thurston sur les surfaces. Astérisque 66–67 (1991), 286 p.
Hamenstädt, U.: Train tracks and the Gromov boundary of the complex of curves. In: Minsky, Y., Sakuma, M., Series, C. (eds.) Spaces of Kleinian Groups. Lond. Math. Soc. Lect. Notes, vol. 329, pp. 187–207. Cambridge University Press, Cambridge (2006)
Hamenstädt, U.: Bounded cohomology and isometry groups of hyperbolic spaces. J. Eur. Math. Soc. 10, 315–349 (2008)
Higson, N.: Biinvariant K-theory and the Novikov conjecture. Geom. Funct. Anal. 10, 563–581 (2000)
Higson, N., Roe, J.: Amenable group actions and the Novikov conjecture. J. Reine Angew. Math. 519, 143–153 (2000)
Hubbard, J., Masur, H.: Quadratic differentials and foliations. Acta Math. 142, 221–274 (1979)
Ivanov, N.V.: Mapping class groups. In: Daverman, R.J., Sher, R.B. (eds.) Handbook of Geometric Topology (Chapter 12), pp. 523–633. Elsevier Science, North-Holland, Amsterdam (2002)
Kaimanovich, V.: Double ergodicity of the Poisson boundary and applications to bounded cohomology. Geom. Funct. Anal. 13, 852–861 (2003)
Kaimanovich, V.: Boundary amenability of hyperbolic spaces. Contemp. Math. 347, 83–114 (2004)
Kaimanovich, V., Masur, H.: The Poisson boundary of the mapping class group. Invent. Math. 125, 221–264 (1996)
Kerckhoff, S., Masur, H., Smillie, J.: Ergodicity of billiard flows and quadratic differentials. Ann. Math. 124, 293–311 (1986)
Kida, Y.: The mapping class group from the viewpoint of measure equivalence theory. Mem. Am. Math. Soc. arXiv:math.GR/0512230 (to appear)
Klarreich, E.: The boundary at infinity of the curve complex and the relative Teichmüller space. Unpublished Manuscript, Ann Arbor (1999)
Masur, H.: Interval exchange transformations and measured foliations. Ann. Math. 115, 169–201 (1982)
Masur, H., Minsky, Y.: Geometry of the complex of curves I: Hyperbolicity. Invent. Math. 138, 103–149 (1999)
McCarthy, J., Papadopoulos, A.: Dynamics on Thurston’s sphere of projective measured foliations. Comment. Math. Helv. 64, 133–166 (1989)
Mineyev, I., Monod, N., Shalom, Y.: Ideal bicombings for hyperbolic groups and applications. Topology 43, 1319–1344 (2004)
Mislin, G., Valette, A.: Proper Group Actions and the Baum–Connes Conjecture. Birkhäuser, Basel (2003)
Monod, N., Shalom, Y.: Cocycle superrigidity and bounded cohomology for negatively curved spaces. J. Differ. Geom. 67, 395–456 (2004)
Mosher, L.: Train track expansions of measured foliations. Unpublished Manuscript (2003)
Penner, R., Harer, J.: Combinatorics of Train Tracks. Ann. Math. Stud., vol. 125. Princeton University Press, Princeton (1992)
Schmidt, K., Walters, P.: Mildly mixing actions of locally compact groups. Proc. Lond. Math. Soc. 45, 506–518 (1982)
Thurston, W.: Three-dimensional geometry and topology. Unpublished Manuscript (1979)
Veech, W.: The Teichmüller geodesic flow. Ann. Math. 124, 441–530 (1986)
Yu, G.: The coarse Baum–Connes conjecture for groups with finite asymptotic dimension. Ann. Math. 147, 325–355 (1998)
Yu, G.: The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math. 139, 201–240 (2000)
Zhu, X., Bonahon, F.: The metric space of geodesic laminations on a surface I. Geom. Topol. 8, 539–564 (2004)
Zimmer, R.: Ergodic Theory and Semisimple Groups. Birkhäuser, Boston (1984)
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Hamenstädt, U. Geometry of the mapping class groups I: Boundary amenability. Invent. math. 175, 545–609 (2009). https://doi.org/10.1007/s00222-008-0158-2
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DOI: https://doi.org/10.1007/s00222-008-0158-2