Lines of minima in outer space

Abstract

We define lines of minima in the thick part of outer space for the free group $F_n$ with $n\ge 3$ generators. We show that these lines of minima are contracting for the Lipschitz metric. Every fully irreducible outer automorphism of $F_n$ defines such a line of minima. Now let $\Gamma$ be a subgroup of the outer automorphism group of $F_n$ which is not virtually abelian. We obtain that if $\Gamma$ contains at least one fully irreducible element, then for every $p\in (1,\infty )$ the second bounded cohomology group $H_b^2(\Gamma ,{\ell }^p(\Gamma \left)\right)$ is infinite-dimensional.