#### Volume 15, issue 6 (2015)

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A generating set for the palindromic Torelli group

### Neil J Fullarton

Algebraic & Geometric Topology 15 (2015) 3535–3567
##### 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 $GL\left(n,ℤ\right)$, 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 ${\mathsc{P}\mathsc{ℐ}}_{n}$. The group ${\mathsc{P}\mathsc{ℐ}}_{n}$ 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 ${\mathsc{P}\mathsc{ℐ}}_{n}$ 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 $GL\left(n,ℤ\right)$.

##### Keywords
automorphisms of free groups, palindromes, Torelli groups
##### Mathematical Subject Classification 2010
Primary: 20F65, 57M07, 57MXX