Reducible normal generators for mapping class groups are abundant

TL;DR

This paper investigates conditions under which reducible mapping classes can normally generate the entire mapping class group, extending previous work on pseudo-Anosov elements and providing new criteria based on subsurface restrictions and translation lengths.

Contribution

It introduces a criterion for reducible mapping classes to normally generate the group, generalizing prior results focused on pseudo-Anosov classes, and explores their abundance.

Findings

Normal generators are abundant among reducible mapping classes.

A criterion based on invariant subsurface restrictions is established.

Asymptotic translation lengths characterize normal generation in this context.

Abstract

In this article, we study the normal generation of the mapping class group. We first show that a mapping class is a normal generator if its restriction on the invariant subsurface normally generates the (pure) mapping class group of the subsurface. As an application, we provided a criterion for reducible mapping classes to normally generate the mapping class groups in terms of its asymptotic translation lengths on Teichm\"uller spaces. This is an analogue to the work of Lanier-Margalit dealing with pseudo-Anosov normal generators.