A generating set for the palindromic Torelli group

Fullarton, Neil J¹

¹Department of Mathematics, Rice University, MS 136, 6100 Main Street, Houston, TX 77005, USA

Algebraic and Geometric Topology, Tome 15 (2015) no. 6, pp. 3535-3567

Résumé

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).

Détail

DOI : 10.2140/agt.2015.15.3535

Classification : 20F65, 57M07, 57MXX
Keywords: automorphisms of free groups, palindromes, Torelli groups

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