## Algebraic & Geometric Topology

### A generating set for the palindromic Torelli group

Neil J Fullarton

#### Abstract

A palindrome in a free group $Fn$ is a word on some fixed free basis of $Fn$ that reads the same backwards as forwards. The palindromic automorphism group $ΠAn$ of the free group $Fn$ consists of automorphisms that take each member of some fixed free basis of $Fn$ to a palindrome; the group $ΠAn$ 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 $ΠAn$ consisting of those elements that act trivially on the abelianisation of $Fn$, 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 $GL(n, ℤ)$.

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3535-3567.

Dates
Revised: 14 April 2015
Accepted: 19 April 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841076

Digital Object Identifier
doi:10.2140/agt.2015.15.3535

Mathematical Reviews number (MathSciNet)
MR3450770

Zentralblatt MATH identifier
1368.20026

#### Citation

Fullarton, Neil J. A generating set for the palindromic Torelli group. Algebr. Geom. Topol. 15 (2015), no. 6, 3535--3567. doi:10.2140/agt.2015.15.3535. https://projecteuclid.org/euclid.agt/1510841076

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