nLab Torelli group

Idea

The mapping class group $MCG(\Sigma^2_g)$ – of the closed oriented surface $\Sigma^2_g$ of genus $= g \in \mathbb{N}$ – canonically acts on the ordinary homology $H_1(\Sigma^2_g;\mathbb{Z}) \simeq \mathbb{Z}^g \times \mathbb{Z}^g$ of the surface, through the defining action of the symplectic group with integer coefficients, $Sp_{2g}(\mathbb{Z}) \coloneqq Sp_{2g}(\mathbb{R}) \cap GL_{2g}(\mathbb{Z})$.

The kernel of this action hence the subgroup acting trivially on the degree=1 homology, is called the Torelli group $I_g$, thus making a short exact sequence of groups

$$1 \to I_g \longrightarrow MCG(\Sigma^2_g) \longrightarrow Sp_{2g}(\mathbb{Z}) \to 1 \mathrlap{\,.}$$

(reviewed by Morita 2007 §6, Farb & Margalit 2012 §6)

Related concepts

• congruence subgroup

References

The origin of the terminology “Torelli group” in this context is

• Dennis Johnson: The Structure of the Torelli Group I: A Finite Set of Generators for $\mathcal{J}$, Annals of Mathematics 118 3 (1983) 423-442 [doi:10.2307/2006977, jstor:2006977]

Review:

• Shigeyuki Morita: Introduction to mapping class groups of surfaces and related groups, in: Handbook of Teichmüller theory, Volume I, EMS (2007) 353-386 [doi:10.4171/029-1/8, pdf]

• Benson Farb, Dan Margalit: A primer on mapping class groups, Princeton Mathematical Series, Princeton University Press (2012) [ISBN:9780691147949, jstor:j.ctt7rkjw, pdf]

• Andrew Putman: The Torelli group and congruence subgroups of the mapping class group, in: Moduli Spaces of Riemann Surfaces, IAS/Park City Mathematics Series 20 (2013) [arXiv:1201.3946, doi:10.1090/pcms/020]