# Rigidity, the complex of free factors, and the commensurator of Aut(F)

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Martin Bridson, University of Oxford

I shall begin by reviewing what is known about the automorphisms of various complexes naturally associated to mapping class groups of surfaces, $Aut(F)$ and $Out(F)$, drawing analogies with the classical case of SL(n,Z). There are various rigidity results concerning the automorphism groups of these complexes, which can be viewed as analogues and extensions of the fundamental theorem of projective geometry. In some cases these lead to algebraic rigidity results describing the abstract commensurators of the groups and various of their natural subgroups (eg the Torelli groups). The talk will be directed towards outlining a proof of the following theorems: for $n\ge 3$ the action of $Aut(F_n)$ on the complex $FF_n$ of free factors gives an isomorphism $Aut(F_n)\to Aut(FF_n)$; and the abstract commensurator of $Aut(F_n)$ is $Aut(F_n)$. The first result is joint work with Mladen Bestvina and the second is with Ric Wade.