On simplicial maps of the complexes of curves of nonorientable surfaces

Irmak, Elmas¹

¹Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403, USA

Algebraic and Geometric Topology, Tome 14 (2014) no. 2, pp. 1153-1180

Résumé

Let N be a compact, connected, nonorientable surface of genus g with n boundary components and C(N) be the complex of curves of N. Suppose that g + n ≤ 3 or g + n ≥ 5. If λ: C(N) →C(N) is an injective simplicial map, then λ is induced by a homeomorphism of N. If (g,n)≠(1,2) and λ: C(N) →C(N) is a simplicial map that satisfies the connectivity property, then λ is induced by a homeomorphism of N.

Détail

DOI : 10.2140/agt.2014.14.1153

Classification : 57M99, 20F38
Keywords: mapping class groups, simplicial maps, nonorientable surfaces

Bibliographie

[1] F Atalan, Automorphisms of complexes of curves on odd genus nonorientable surfaces

[2] F Atalan, M Korkmaz, Automorphisms of complexes of curves on nonorientable surfaces, to appear in Groups, Geometry and Dynamics

[3] J Behrstock, D Margalit, Curve complexes and finite index subgroups of mapping class groups, Geom. Dedicata 118 (2006) 71

[4] R W Bell, D Margalit, Injections of Artin groups, Comment. Math. Helv. 82 (2007) 725

[5] T E Brendle, D Margalit, Commensurations of the Johnson kernel, Geom. Topol. 8 (2004) 1361

[6] B Farb, N V Ivanov, The Torelli geometry and its applications: Research announcement, Math. Res. Lett. 12 (2005) 293

[7] W J Harvey, Boundary structure of the modular group, from: "Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference" (editors I Kra, B Maskit), Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 245

[8] E Irmak, Injective simplicial maps of the complexes of curves on nonorientable surfaces

[9] E Irmak, Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups, Topology 43 (2004) 513

[10] E Irmak, Complexes of nonseparating curves and mapping class groups, Michigan Math. J. 54 (2006) 81

[11] E Irmak, Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups, II, Topology Appl. 153 (2006) 1309

[12] E Irmak, Injective simplicial maps of the complex of arcs on nonorientable surfaces, Algebr. Geom. Topol. 9 (2009) 2055

[13] E Irmak, Simplicial maps of the complexes of curves on nonorientable surfaces (2012)

[14] E Irmak, Superinjective simplicial maps of the complexes of curves on nonorientable surfaces, Turkish J. Math. 36 (2012) 407

[15] E Irmak, N V Ivanov, J D Mccarthy, Automorphisms of surface braid groups

[16] E Irmak, M Korkmaz, Automorphisms of the Hatcher–Thurston complex, Israel J. Math. 162 (2007) 183

[17] E Irmak, J D Mccarthy, Injective simplicial maps of the arc complex, Turkish J. Math. 34 (2010) 339

[18] N V Ivanov, Automorphism of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Notices (1997) 651

[19] N V Ivanov, J D Mccarthy, On injective homomorphisms between Teichmüller modular groups, I, Invent. Math. 135 (1999) 425

[20] Y Kida, Automorphisms of the Torelli complex and the complex of separating curves, J. Math. Soc. Japan 63 (2011) 363

[21] Y Kida, Injections of the complex of separating curves into the Torelli complex (2011)

[22] Y Kida, The co-Hopfian property of the Johnson kernel and the Torelli group, Osaka J. Math. 50 (2013) 309

[23] Y Kida, S Yamagata, Automorphisms of the Torelli complex for the one-holed genus two surface

[24] Y Kida, S Yamagata, Commensurators of surface braid groups, J. Math. Soc. Japan 63 (2011) 1391

[25] Y Kida, S Yamagata, The co-Hopfian property of surface braid groups (2013)

[26] M Korkmaz, Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology Appl. 95 (1999) 85

[27] F Luo, Automorphisms of the complex of curves, Topology 39 (2000) 283

[28] D Margalit, Automorphisms of the pants complex, Duke Math. J. 121 (2004) 457

[29] J D Mccarthy, W R Vautaw, Automorphisms of Torelli groups

[30] M Scharlemann, The complex of curves on nonorientable surfaces, J. London Math. Soc. 25 (1982) 171

[31] K J Shackleton, Combinatorial rigidity in curve complexes and mapping class groups, Pacific J. Math. 230 (2007) 217