Commensurations of Out(F_n)

Let \Out(F_n) denote the outer automorphism group of the free group $F_n$ with $n>3$. We prove that for any finite index subgroup \Gamma<\Out(F_n), the group \Aut(\Gamma) is isomorphic to the normalizer of $\Gamma$ in \Out(F_n). We prove that $\Gamma$ is {\em co-Hopfian} : every injective homomorphism $\Gamma\to \Gamma$ is surjective. Finally, we prove that the abstract commensurator \Comm(\Out(F_n)) is isomorphic to \Out(F_n).Comment: Revised version, 43 pages. To appear in Publ. Math. IHE

Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II

Let BS(1,n)= . We prove that any finitely-generated group quasi-isometric to BS(1,n) is (up to finite groups) isomorphic to BS(1,n). We also show that any uniform group of quasisimilarities of the real line is bilipschitz conjugate to an affine group.Comment: 42 pages. see also http://www.math.uchicago.edu/~far

Erratum for "The degree theorem in higher rank"

The purpose of this erratum is to correct a mistake in the proof of Theorem 4.1 of our paper \cite{CF}.Comment: Minor revisions suggested by refere