A generating set for the palindromic Torelli group

Fullarton, Neil J

doi:10.2140/agt.2015.15.3535

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A generating set for the palindromic Torelli group

Neil J Fullarton

Algebraic & Geometric Topology 15 (2015) 3535–3567

DOI: 10.2140/agt.2015.15.3535

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 $Π A_{n}$ 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 $Π A_{n}$ 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 $Π A_{n}$ consisting of those elements that act trivially on the abelianisation of $F_{n}$ , the palindromic Torelli group ${P ℐ}_{n}$ . The group ${P ℐ}_{n}$ 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 ${P ℐ}_{n}$ 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 $GL (n, ℤ)$ .

Keywords

automorphisms of free groups, palindromes, Torelli groups

Mathematical Subject Classification 2010

Primary: 20F65, 57M07, 57MXX

References

Publication

Received: 4 November 2014

Revised: 14 April 2015

Accepted: 19 April 2015

Published: 12 January 2016

Authors

Neil J Fullarton
Department of Mathematics Rice University MS 136 6100 Main Street Houston, TX 77005 USA

Volume 15, issue 6 (2015)