Geometric intersection in representations of mapping class groups of surfaces

TL;DR

This paper establishes a link between geometric intersection detection in representations of surface mapping class groups and the injectivity of those representations, with applications to the Johnson filtration and separating curves.

Contribution

It proves that detecting geometric intersection in any representation implies injectivity up to center, and explores this in the context of the Johnson filtration and separating curves.

Findings

Detection of geometric intersection implies injectivity up to center.

Application to Johnson filtration shows how geometric intersection detection works in that setting.

Analysis of separating simple closed curves enhances understanding of representation properties.

Abstract

We show that the detection of geometric intersection in an arbitrary representation of the mapping class group of surface implies the injectivity of that representation up to center, and vice versa. As an application, we discuss the geometric intersection in the Johnson filtration. Also, we further consider the problem of detecting the geometric intersection between separating simple closed curves in a representation.