Lines of minima in Outer space

TL;DR

This paper introduces lines of minima in Outer space for free groups, proves their contraction properties, and explores implications for the bounded cohomology of subgroups of the outer automorphism group.

Contribution

It defines lines of minima in Outer space, shows their contraction under the Lipschitz metric, and links fully irreducible automorphisms to infinite-dimensional bounded cohomology.

Findings

Lines of minima are contracting in Outer space.

Fully irreducible automorphisms define lines of minima.

Subgroups with a fully irreducible element have infinite-dimensional second bounded cohomology.

Abstract

We define lines of minima in the thick part of Outer space for the free group Fn with n>2 generators. We show that these lines of minima are contracting for the Lipschitz metric. Every fully irreducible outer automorphism of Fn defines such a line a minima. Now let G be a subgroup of the outer automorphism group of Fn which is not virtually abelian. We obtain as an immediate application that if G contains at least one fully irreducible element then for every p<1 the second bounded cohomology group with coefficients in lp(G) is infinite dimensional.