Algebraic & Geometric Topology

A generating set for the palindromic Torelli group

Neil J Fullarton

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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, ).

Article information

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

Received: 4 November 2014
Revised: 14 April 2015
Accepted: 19 April 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 57M07: Topological methods in group theory 57MXX

automorphisms of free groups palindromes Torelli groups


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.

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