Subgroups of the Torelli group generated by two symmetric bounding pair maps

TL;DR

This paper proves that certain pairs of bounding pair maps in the Torelli group generate a free group, supporting a conjecture about the algebraic structure of the Torelli group.

Contribution

It demonstrates that pairs of symmetric bounding pair maps invariant under an involution generate a free subgroup, advancing understanding of the Torelli group's subgroup structure.

Findings

Bounding pair maps generate a free group under specified conditions

Supports conjecture that pairs in the Torelli group either commute or generate free groups

Provides new insights into the algebraic structure of the Torelli group

Abstract

Let {a,b} and {c,d} be two pairs of bounding simple closed curves on an oriented surface which intersect nontrivialy. We prove that if these pairs are invariant under the action of an orientation reversing involution, then the corresponding bounding pair maps generate a free group. This supports the conjecture stated by C. Leininger and D. Margalit that any pair of elements of the Torelli group either commute or generate a free group.