Seminar series
Date
Mon, 09 Feb 2015
15:45
Location
C6
Speaker
Martin Bridson
Organisation
Oxford

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 $3$. I shall begin by sketching the fruitful analogy between automorphism groups of free groups, mapping class groups, and arithmetic lattices, particularly $SL_n({\mathbb{Z}})$. In this analogy, the free factor complex, introduced by Hatcher and Vogtmann, appears as the natural analogue in the $Aut(F)$ setting of the spherical Tits building associated to $SL_n $ and of the curve complex associated to a mapping class group. If $n>2$, Tits' generalisation of the Fundamental Theorem of Projective Geometry (FTPG) assures us that the automorphism group of the building is $PGL_n({\mathbb{Q}})$. Ivanov proved an analogous theorem for the curve complex, and our theorem complements this. These theorems allow one to identify the abstract commensurators of $GL_n({\mathbb{Z}})$, mapping class groups, and $Out(F)$, as I shall explain.

Please contact us with feedback and comments about this page. Last updated on 03 Apr 2022 01:32.