We define lines of minima in the thick part of Outer space for the free group Fn with n ≥ 3 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 of minima. Now let Γ be a subgroup of the outer automorphism group of Fn which is not virtually abelian. We obtain that if Γ contains at least one fully irreducible element then for every p ∈ (1, ∞) the second bounded cohomology group H 2 b (Γ, ℓ p (Γ)) is infinite dimensional.
For an aspherical oriented 3-manifold M and a subsurface X of the boundary of M with empty or incompressible boundary, we use surgery to identify a graph whose vertices are disks with boundary in X and which is quasi-isometrically embedded in the curve graph of X.
For an oriented surface S of genus g ≥ 0 with m ≥ 0 punctures and 3g − 3 + m ≥ 2, let Q(S) and Q W P (S) be the moduli space of area one quadratic differentials and of quadratic differentials of unit norm for the Weil-Petersson metric, respectively. We show that there is a Borel subset E of Q(S) which is invariant under the Teichmüller flow Φ t T and of full measure for every invariant Borel probability measure, and there is a measurable map Λ : E → Q W P (S) which conjugates Φ t T |E into the Weil-Petersson flow Φ t W P . This conjugacy induces a continuous injection of the space of all Φ t T -invariant Borel probability measures on Q(S) into the space of all Φ t W P -invariant Borel probability measures on Q W P (S). The map Θ is not surjective, but its image contains the Lebesgue Liouville measure. A measure not in the image corresponds to a locally finite infinite invariant Borel measure on Q(S).
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