Commensurations of the Johnson kernel

Brendle, Tara E¹ ; Margalit, Dan²

¹ Department of Mathematics, Cornell University, 310 Malott Hall, Ithaca, New York 14853, USA
² Department of Mathematics, University of Utah, 155 South 1440 East, Salt Lake City, Utah 84112, USA

Geometry & topology, Tome 8 (2004) no. 3, pp. 1361-1384.

Résumé

Let $\mathcal{K}$ be the subgroup of the extended mapping class group, $Mod\left(S\right)$, generated by Dehn twists about separating curves. Assuming that $S$ is a closed, orientable surface of genus at least 4, we confirm a conjecture of Farb that $Comm\left(\mathcal{K}\right)\cong Aut\left(\mathcal{K}\right)\cong Mod(S)$. More generally, we show that any injection of a finite index subgroup of $\mathcal{K}$ into the Torelli group $\mathcal{ℐ}$ of $S$ is induced by a homeomorphism. In particular, this proves that $\mathcal{K}$ is co-Hopfian and is characteristic in $\mathcal{ℐ}$. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of $\mathcal{ℐ}$ into $\mathcal{ℐ}$ is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes.

Détail

DOI : 10.2140/gt.2004.8.1361

Keywords: Torelli group, mapping class group, Dehn twist

Bibliographie

[1] R W Bell, D Margalit, Braid groups and the co–Hopfian property, J. Algebra 303 (2006) 275

[2] J S Birman, A Lubotzky, J Mccarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983) 1107

[3] D Biss, B Farb, $\mathcal{K}_g$ is not finitely generated, Invent. Math. 163 (2006) 213

[4] M R Bridson, K Vogtmann, Automorphisms of automorphism groups of free groups, J. Algebra 229 (2000) 785

[5] B Farb, L Mosher, The geometry of surface-by-free groups, Geom. Funct. Anal. 12 (2002) 915

[6] B Farb, Automorphisms of the Torelli group, AMS sectional meeting, Ann Arbor, Michigan, March 1 (2002)

[7] B Farb, M Handel, Commensurations of $\mathrm{Out}(F_n)$, preprint, personall communication, September (2004)

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

[9] J L Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. $(2)$ 121 (1985) 215

[10] W J Harvey, Boundary structure of the modular group, from: "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. 97, Princeton Univ. Press (1981) 245

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

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

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

[14] N V Ivanov, Automorphisms of Teichmüller modular groups, from: "Topology and geometry—Rohlin Seminar", Lecture Notes in Math. 1346, Springer (1988) 199

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

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

[17] D Johnson, The structure of the Torelli group I: A finite set of generators for $\cal{I}$, Ann. of Math. $(2)$ 118 (1983) 423

[18] D Johnson, The structure of the Torelli group II: A characterization of the group generated by twists on bounding curves, Topology 24 (1985) 113

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

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

[21] H Masur, S Schleimer, The pants complex has only one end, from: "Spaces of Kleinian groups", London Math. Soc. Lecture Note Ser. 329, Cambridge Univ. Press (2006) 209

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

[23] D Mccullough, A Miller, The genus 2 Torelli group is not finitely generated, Topology Appl. 22 (1986) 43

[24] S Morita, Casson's invariant for homology 3–spheres and characteristic classes of surface bundles I, Topology 28 (1989) 305

[25] G Prasad, Discrete subgroups isomorphic to lattices in semisimple Lie groups, Amer. J. Math. 98 (1976) 241

[26] Z Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups II, Geom. Funct. Anal. 7 (1997) 561