Simply Intersecting Pair Maps in the Mapping Class Group

TL;DR

This paper studies simply intersecting pair maps within the Torelli group, providing topological characterizations, classifying their kernels under key homomorphisms, and analyzing the subgroup they generate.

Contribution

It offers a topological description of SIP-maps, classifies their kernels under Johnson and Birman-Craggs-Johnson homomorphisms, and examines the subgroup generated by SIP-maps.

Findings

SIP-maps' images under Johnson and Birman-Craggs-Johnson homomorphisms are characterized.

Certain SIP-maps are identified as lying in the kernels of these homomorphisms.

The subgroup generated by all SIP-maps is of infinite index in the Torelli group.

Abstract

The Torelli group, I(S_g), is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface. There are three types of elements that naturally arise in studying I(S_g): bounding pair maps, separating twists, and simply intersecting pair maps (SIP-maps). Historically the first two types of elements have been the focus of the literature on I(S_g), while SIP-maps have received relatively little attention until recently, due to an infinite presentation of I(S_g) introduced by Andrew Putman that uses all three types of elements. We will give a topological characterization of the image of an SIP-map under the Johnson homomorphism and Birman-Craggs-Johnson homomorphism. We will also classify which SIP-maps are in the kernel of these homomorphisms. Then we will look at the subgroup generated by all SIP-maps, SIP(S_g), and show it is an infinite index subgroup of I(S_g).