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A generating set for the palindromic Torelli group

Neil J Fullarton

Algebraic & Geometric Topology 15 (2015) 3535–3567

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, ), 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 Pn. The group Pn 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 Pn 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, ).

automorphisms of free groups, palindromes, Torelli groups
Mathematical Subject Classification 2010
Primary: 20F65, 57M07, 57MXX
Received: 4 November 2014
Revised: 14 April 2015
Accepted: 19 April 2015
Published: 12 January 2016
Neil J Fullarton
Department of Mathematics
Rice University
MS 136
6100 Main Street
Houston, TX 77005