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 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 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 A n consisting of those elements that act 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 and Putman. The generating set leads to a finite presentation of the principal level 2 congruence subgroup of GL.n; Z/.