Geometry & Topology

Commensurations of the Johnson kernel

Abstract

Let $K$ be the subgroup of the extended mapping class group, $Mod(S)$, 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 $Comm(K)≅Aut(K)≅Mod(S)$. More generally, we show that any injection of a finite index subgroup of $K$ into the Torelli group $ℐ$ of $S$ is induced by a homeomorphism. In particular, this proves that $K$ is co-Hopfian and is characteristic in $ℐ$. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of $ℐ$ into $ℐ$ is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes.

Article information

Source
Geom. Topol., Volume 8, Number 3 (2004), 1361-1384.

Dates
Revised: 25 October 2004
Accepted: 25 October 2004
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883470

Digital Object Identifier
doi:10.2140/gt.2004.8.1361

Mathematical Reviews number (MathSciNet)
MR2119299

Zentralblatt MATH identifier
1079.57017

Citation

Brendle, Tara E; Margalit, Dan. Commensurations of the Johnson kernel. Geom. Topol. 8 (2004), no. 3, 1361--1384. doi:10.2140/gt.2004.8.1361. https://projecteuclid.org/euclid.gt/1513883470

References

• Robert W Bell, Dan Margalit, Braid groups are almost co-Hopfian.
• Joan S Birman, Alex Lubotzky, John McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983) 1107–1120
• Daniel Biss, Benson Farb, ${K}_g$ is not finitely generated,\nl \arxivmath.GT/0405386
• Martin R Bridson, Karen Vogtmann, Automorphisms of automorphism groups of free groups, J. Algebra 229 (2000) 785–792
• Benson Farb, Lee Mosher, The geometry of surface-by-free groups, Geom. Funct. Anal. 12 (2002) 915–963
• Benson Farb, Automorphisms of the Torelli group, AMS sectional meeting, Ann Arbor, Michigan, March 1, 2002
• Benson Farb, Michael Handel, Commensurations of Out(${F}_n$). Preprint, personal communication, September 2004
• Benson Farb, Nikolai V Ivanov, The Torelli geometry and its applications.
• John L Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. 121 (1985) 215–249
• W J Harvey, Boundary structure of the modular group, from: “Riemann surfaces and related topics (Stony Brook, 1978)”, Ann. of Math. Stud. 97, Princeton Univ. Press, Princeton, N.J. (1981) 245–251
• Allen Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991) 189–194
• Elmas Irmak, Superinjective Simplicial Maps of Complexes of Curves and Injective Homomorphisms of Subgroups of Mapping Class Groups II.
• Elmas Irmak, Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups, Topology 43 (2004) 513–541
• Nikolai V Ivanov, Automorphisms of Teichmüller modular groups, from: “Topology and geometry–-Rohlin Seminar”, Lecture Notes in Math. 1346, Springer, Berlin (1988) 199–270
• Nikolai V Ivanov, Automorphism of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Notices (1997) 651–666
• Nikolai V Ivanov, John D McCarthy, On injective homomorphisms between Teichmüller modular groups. I, Invent. Math. 135 (1999) 425–486
• Dennis Johnson, The structure of the Torelli group. I. A finite set of generators for ${\mathcal I}$, Ann. of Math. 118 (1983) 423–442
• Dennis Johnson, The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves, Topology 24 (1985) 113–126
• Mustafa Korkmaz, Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology Appl. 95 (1999) 85–111
• Feng Luo, Automorphisms of the complex of curves, Topology 39 (2000) 283–298
• Howard Masur, Saul Schleimer, The Pants Complex Has Only One End.
• John D McCarthy, William R Vautaw, Automorphisms of Torelli groups.
• Darryl McCullough, Andy Miller, The genus $2$ Torelli group is not finitely generated, Topology Appl. 22 (1986) 43–49
• Shigeyuki Morita, Casson's invariant for homology $3$-spheres and characteristic classes of surface bundles. I, Topology 28 (1989) 305–323
• Gopal Prasad, Discrete subgroups isomorphic to lattices in semisimple Lie groups, Amer. J. Math. 98 (1976) 241–261
• Z Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank $1$ Lie groups. II, Geom. Funct. Anal. 7 (1997) 561–593