Geometry & Topology

Small generating sets for the Torelli group

Andrew Putman

Full-text: Open access


Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup g of the genus g mapping class group has a finite generating set whose size grows cubically with respect to g. Our main tool is a new space called the handle graph on which g acts cocompactly.

Article information

Geom. Topol., Volume 16, Number 1 (2012), 111-125.

Received: 24 June 2011
Revised: 11 August 2011
Accepted: 1 November 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F05: Generators, relations, and presentations
Secondary: 20F38: Other groups related to topology or analysis 57M07: Topological methods in group theory 57N05: Topology of $E^2$ , 2-manifolds

Torelli group mapping class group


Putman, Andrew. Small generating sets for the Torelli group. Geom. Topol. 16 (2012), no. 1, 111--125. doi:10.2140/gt.2012.16.111.

Export citation


  • J S Birman, On Siegel's modular group, Math. Ann. 191 (1971) 59–68
  • T Brendle, B Farb, personal communication
  • K S Brown, Presentations for groups acting on simply-connected complexes, J. Pure Appl. Algebra 32 (1984) 1–10
  • B Farb, Some problems on mapping class groups and moduli space, from: “Problems on mapping class groups and related topics”, Proc. Sympos. Pure Math. 74, Amer. Math. Soc., Providence, RI (2006) 11–55
  • B Farb, D Margalit, A Primer on Mapping Class Groups, to be published by Princeton University Press
  • R Hain, Fundamental groups of branched coverings and the Torelli group in genus 3, in preparation
  • A Hatcher, W Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980) 221–237
  • D Johnson, Conjugacy relations in subgroups of the mapping class group and a group-theoretic description of the Rochlin invariant, Math. Ann. 249 (1980) 243–263
  • D Johnson, The structure of the Torelli group. I. A finite set of generators for ${\cal I}$, Ann. of Math. $(2)$ 118 (1983) 423–442
  • D Johnson, A survey of the Torelli group, from: “Low-dimensional topology (San Francisco, Calif., 1981)”, Contemp. Math. 20, Amer. Math. Soc., Providence, RI (1983) 165–179
  • D Johnson, The structure of the Torelli group. III. The abelianization of $\mathcal{T}$, Topology 24 (1985) 127–144
  • D Margalit, A Hatcher, Generating the Torelli group, in preparation
  • D McCullough, A Miller, The genus $2$ Torelli group is not finitely generated, Topology Appl. 22 (1986) 43–49
  • G Mess, The Torelli groups for genus $2$ and $3$ surfaces, Topology 31 (1992) 775–790
  • J Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978) 347–350
  • A Putman, Cutting and pasting in the Torelli group, Geom. Topol. 11 (2007) 829–865
  • A Putman, A note on the connectivity of certain complexes associated to surfaces, Enseign. Math. $(2)$ 54 (2008) 287–301
  • A Putman, An infinite presentation of the Torelli group, Geom. Funct. Anal. 19 (2009) 591–643
  • J-P Serre, Trees, Springer, Berlin (1980) Translated from the French by John Stillwell
  • W Tomaszewski, A basis of Bachmuth type in the commutator subgroup of a free group, Canad. Math. Bull. 46 (2003) 299–303