A generating set for the palindromic Torelli group

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 ΠA 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 ΠA 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 ΠA n consisting of those elements that act trivially on the abelianisation of F n , the palindromic Torelli group PI n . The group PI 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 PI n acts in a nice manner, adapting a proof of Day-Putman [11]. The generating set leads to a finite presentation of the principal level 2 congruence subgroup of GL(n, Z).

Palindromic automorphisms of free groups

“…Finally, in Section 6, we prove that P I n = IA n ∩ EΠA ′ n . This result strengthens a recent result of Fullarton [3].…”

“…Clearly, IA n ∩ EΠA ′ n ⊂ IA n ∩ ΠA n = P I n . In [3], Fullarton showed that P I n is normally generated in ΠA n by the automorphisms [µ 12 3 and hence they also lie in IA n ∩EΠA ′ n . Note that, IA n is a normal subgroup of Aut(F n ).…”

A normal generating set for the Torelli group of a non-orientable closed surface

“…On the other hand, Szepietowski [18] gave a finite generating set for Γ 2 (N g ), and then the first author and Sato [6] gave a minimal generating set for Γ 2 (N g ). Fullarton [4], the second author [8], Margalit and Putman gave a finite presentation for Γ 2 (n) independently. Therefore, we obtain a normal generating set for I(N g ) in Γ 2 (N g ).…”

Palindromic automorphisms of free nilpotent groups

“…In particular, he studied involutions and center of ΠA(F n ). In a recent paper [10], Fullarton obtained a generating set for the palindromic IA-automorphism group P I(F n ). This was obtained by constructing an action of P I(F n ) on a simplicial complex modelled on the complex of partial bases due to Day and Putman [7].…”

“…This was obtained by constructing an action of P I(F n ) on a simplicial complex modelled on the complex of partial bases due to Day and Putman [7]. The papers [11] and [10] indicates a deep connection between palindromic automorphisms of free groups and geometry. Recently, Bardakov, Gongopadhyay and Singh [3] investigated many algebraic properties of ΠA(F n ).…”