A generating set for the palindromic Torelli group

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 that act trivially on the abelianisation of F-n, the palindromic Torelli group PIn. The group PIn 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 PIn acts in a nice manner, adapting a proof of Day and Putman. The generating set leads to a finite presentation of the principal level 2 congruence subgroup of GL(n, Z).