A generating set for the palindromic Torelli group

TL;DR

This paper identifies a generating set for the palindromic Torelli group, a subgroup of automorphisms of free groups related to hyperelliptic mapping class groups, using a simplicial complex and adapting existing methods.

Contribution

It provides the first generating set for the palindromic Torelli group acting trivially on abelianisation, and derives a finite presentation for a related congruence subgroup.

Findings

Generated the palindromic Torelli group using a simplicial complex

Connected the group to finite presentations of congruence subgroups

Extended methods from Day-Putman to this new context

Abstract

A palindrome in a free group F_n is a word on some fixed free basis of F_n that reads the same backwards as forwards. The palindromic automorphism group \Pi A_n of the free group F_n consists of automorphisms that take each member of some fixed free basis of F_n to a palindrome; the group \Pi A_n has close connections with hyperelliptic mapping class groups, braid groups, congruence subgroups of GL(n,Z), and symmetric automorphisms of free groups. We obtain a generating set for the subgroup of \Pi A_n consisting of those elements acting trivially on the abelianisation of F_n, the palindromic Torelli group PI_n. The group PI_n is a free group analogue of the hyperelliptic Torelli subgroup of the mapping class group of an oriented surface. We obtain our generating set by constructing a simplicial complex on which PI_n acts in a nice manner, adapting a proof of Day-Putman. The generating set leads to a finite presentation of the principal level 2 congruence subgroup of GL(n,Z).