A
palindrome in a free group
is
a word on some fixed free basis of
that reads the same backwards as forwards. The
palindromic automorphism group
of the free
group
consists of automorphisms that take each member of some fixed free basis of
to a palindrome;
the group
has close connections with hyperelliptic mapping class groups, braid groups, congruence
subgroups of
,
and symmetric automorphisms of free groups. We obtain a generating set for the subgroup
of
consisting of those elements that act trivially on the abelianisation of
, the
palindromicTorelli group .
The group
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
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
.
Keywords
automorphisms of free groups, palindromes, Torelli groups
