Realisation and dismantlability

Hensel, Sebastian¹ ; Osajda, Damian² ; Przytycki, Piotr³

¹ Department of Mathematics, University of Chicago, 5734 S University Ave., Chicago, IL 60637, USA
² Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland, and Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
³ McGill University, The Department of Mathematics and Statistics, Burnside Hall, Room 1005, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada, and Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland

Geometry & topology, Tome 18 (2014) no. 4, pp. 2079-2126.

Résumé

We prove that a finite group acting on an infinite graph with dismantling properties fixes a clique. We prove that in the flag complex spanned on such a graph the fixed point set is contractible. We study dismantling properties of the arc, disc and sphere graphs. We apply our theory to prove that any finite subgroup $H$ of the mapping class group of a surface with punctures, the handlebody group, or $Out\left(F_n\right)$ fixes a filling (respectively simple) clique in the appropriate graph. We deduce some realisation theorems, in particular the Nielsen realisation problem in the case of a nonempty set of punctures. We also prove that infinite $H$ have either empty or contractible fixed point sets in the corresponding complexes. Furthermore, we show that their spines are classifying spaces for proper actions for mapping class groups and $Out\left(F_n\right)$.

Détail

DOI : 10.2140/gt.2014.18.2079

Classification : 20F65
Keywords: arc complex, sphere complex, disc complex, Nielsen realisation, dismantlability

Bibliographie

[1] J A Barmak, E G Minian, Strong homotopy types, nerves and collapses, Discrete Comput. Geom. 47 (2012) 301

[2] B H Bowditch, D B A Epstein, Natural triangulations associated to a surface, Topology 27 (1988) 91

[3] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer, Berlin (1999)

[4] P Buser, Geometry and spectra of compact Riemann surfaces, Modern Birkhäuser Classics, Birkhäuser (2010)

[5] V Chepoi, D Osajda, Dismantlability of weakly systolic complexes and applications, to appear in Trans. Amer. Math. Soc. (2014)

[6] M Culler, Finite groups of outer automorphisms of a free group, from: "Contributions to group theory" (editors K I Appel, J G Ratcliffe, P E Schupp), Contemp. Math. 33, Amer. Math. Soc. (1984) 197

[7] B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)

[8] D Gabai, Convergence groups are Fuchsian groups, Ann. of Math. 136 (1992) 447

[9] U Hamenstädt, S Hensel, The geometry of the handlebody groups I: Distortion, J. Topol. Anal. 4 (2012) 71

[10] U Hamenstädt, S Hensel, Sphere systems, intersections and the geometry of $\mathrm{Out}(F_n)$, (2014)

[11] J L Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. 121 (1985) 215

[12] J L Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157

[13] A Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991) 189

[14] A Hatcher, Homological stability for automorphism groups of free groups, Comment. Math. Helv. 70 (1995) 39

[15] L Ji, S A Wolpert, A cofinite universal space for proper actions for mapping class groups, from: "In the tradition of Ahlfors–Bers, V" (editors M Bonk, J Gilman, H Masur, Y Minsky, M Wolf), Contemp. Math. 510, Amer. Math. Soc. (2010) 151

[16] R P Kent Iv, C J Leininger, S Schleimer, Trees and mapping class groups, J. Reine Angew. Math. 637 (2009) 1

[17] S P Kerckhoff, The Nielsen realization problem, Bull. Amer. Math. Soc. 2 (1980) 452

[18] S P Kerckhoff, The Nielsen realization problem, Ann. of Math. 117 (1983) 235

[19] D G Khramtsov, Finite groups of automorphisms of free groups, Mat. Zametki 38 (1985) 386, 476

[20] S Krstić, K Vogtmann, Equivariant outer space and automorphisms of free-by-finite groups, Comment. Math. Helv. 68 (1993) 216

[21] F Laudenbach, Topologie de la dimension trois: Homotopie et isotopie, Astérisque 12, Société Mathématique de France, Paris (1974)

[22] H Masur, Measured foliations and handlebodies, Ergodic Theory Dynam. Systems 6 (1986) 99

[23] D Mccullough, Virtually geometrically finite mapping class groups of $3$–manifolds, J. Differential Geom. 33 (1991) 1

[24] D Meintrup, T Schick, A model for the universal space for proper actions of a hyperbolic group, New York J. Math. 8 (2002) 1

[25] G Mislin, Classifying spaces for proper actions of mapping class groups, Münster J. Math. 3 (2010) 263

[26] J Nielsen, Abbildungsklassen endlicher Ordnung, Acta Math. 75 (1943) 23

[27] R C Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987) 299

[28] N Polat, Finite invariant simplices in infinite graphs, Period. Math. Hungar. 27 (1993) 125

[29] P Przytycki, J Schultens, Contractibility of the Kakimizu complex and symmetric Seifert surfaces, Trans. Amer. Math. Soc. 364 (2012) 1489

[30] S Schleimer, Notes on the complex of curves, (2005)

[31] P Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988) 1

[32] T White, Fixed points of finite groups of free group automorphisms, Proc. Amer. Math. Soc. 118 (1993) 681

[33] S A Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987) 275

[34] H Zieschang, Finite groups of mapping classes of surfaces, Lecture Notes in Mathematics 875, Springer, Berlin (1981)

[35] B Zimmermann, Über Abbildungsklassen von Henkelkörpern, Arch. Math. $($Basel$)$ 33 (1979/80) 379

[36] B Zimmermann, Über Homöomorphismen $n$–dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen, Comment. Math. Helv. 56 (1981) 474