Abstract

A commensuration of a group G is an isomorphism between finite-index subgroups of G. Equivalence classes of such maps form a group, whose importance first emerged in the work of Margulis on arithmetic lattices in semisimple Lie groups. Drawing motivation from this classical setting and from the study of mapping class groups of surfaces, I shall explain why, when N is at least 3, the group of automorphisms of the free group of rank N is its own abstract commensurator. Similar results hold for certain subgroups of Aut(F_N). A key element in the proofs is a non-abelian analogue of the Fundamental Theorem of Projective Geometry, in which projective subspaces are replaced by the free factors of a free group. This last result is the content of a long-running project with Mladen Bestvina, while the results on commensurators are the content of a similarly extended project with Ric Wade. If time allows, I shall discuss related open questions