Subgroups of the Torelli group

Abstract

Let Mod(Sg) be the mapping class group of an orientable surface of genus g, Sg. The action of Mod(Sg) on the homology of Sg induces the well-known symplectic representation: Mod(Sg) ---> Sp(2g, Z). The kernel of this representation is called the Torelli group, I(Sg). We will study two subgroups of I(Sg). First we will look at the subgroup generated by all SIP-maps, SIP(Sg). We will show SIP(Sg) is not I(Sg) and is in fact an infinite index subgroup of I(Sg). We will also classify which SIP-maps are in the kernel of the Johnson homomorphism and Birman-Craggs-Johnson homomorphism. Then we will look at the symmetric Torelli group, SI(Sg). More specifically, we will investigate the group generated by Dehn twists about symmetric separating curves denoted H(Sg). We will show the well-known Birman-Craggs-Johnson homomorphism is not able to distinguish among SI(Sg), H(Sg), or K(Sg), where K(Sg) is the subgroup generated by Dehn twists about separating curves. Elements of H(Sg) act naturally on the symmetric separating curve complex, CH(S). We will show that when g > 4 Aut(CH(Sg)) = SMod^{+/-}(Sg) / < i > where SMod(Sg) is the symmetric mapping class group and i is a fixed hyperelliptic involution. Lastly we will give an algebraic characterization of Dehn twists about symmetric separating curves.