Thursday February 5, 16:00, room D1.07

Martin Bridson (Oxford)

The symmetries of the free factor complex

Abstract: I shall discuss joint work with Mladen Bestvina in which we prove that the group of simplicial automorphisms of the complex of free factors for a free group $F$ is exactly $\Aut(F)$, provided that $F$ has rank at least three. The free factor complex was introduced by Hatcher and Vogtmann as the natural analogue in the free setting of the classical building that is the complex of summands for $\ZZ^n$: the inclusion of $\ZZ^n$ in $\QQ^n$ induces an isomorphism from the poset of summands to the spherical building for $\GL(n,\QQ)$. If $n > 2$ then Tits' generalisation of the Fundamental Theorem of Projective Geometry (FTPG) assures us that the automorphism group of this building is $\PGL(n,\QQ)$, which has index two in the abstract commensurator of $\GL(n,\ZZ) = \Out(\ZZ^n)$. In the free setting, $\Out(F)$ is its own commensurator (I shall discuss this) and our theorem appears as the appropriate analogue of the FTPG.