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).

We show that for any natural number N there exists a right-angled Artin group A Γ for which Out(Aut(A Γ )) has order at least N . This is in contrast with the cases where A Γ is free or free abelian: for all n, Dyer-Formanek and Bridson-Vogtmann showed that Out(Aut(F n )) = 1, while Hua-Reiner showed |Out(Aut(Z n ))| ≤ 4. We also prove the analogous theorem for Out(Out(A Γ )). These theorems fit into a wider context of algebraic rigidity results in geometric group theory. We establish our results by giving explicit examples; one useful tool is a new class of graphs called austere graphs.

We prove that the moduli space of curves with level structures has an enormous amount of rational cohomology in its cohomological dimension. As an application, we prove that the coherent cohomological dimension of the moduli space of curves is at least g − 2. Well known conjectures of Looijenga would imply that this is sharp.

We construct the first known examples of infinite subgroups of the outer automorphism group of Out(A Γ ), for certain right-angled Artin groups A Γ . This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of Out(Out(A Γ )). This demonstrates a marked departure from the known behavior of Out(Out(A Γ )) when A Γ is free or free abelian, as in these cases Out(Out(A Γ )) has order at most 4. We also disprove a previous conjecture of the second author, producing new examples of finite order members of certain Out(Aut(A Γ )).

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