ABSTRACT. We define metrics on Culler-Vogtmann space, which are an analogue of the Teichmüller metric and are constructed using stretching factors. In fact the metrics we study are related, one being a symmetrised version of the other. We investigate the basic properties of these metrics, showing the advantages and pathologies of both choices.We show how to compute stretching factors between marked metric graphs in an easy way and we discuss the behaviour of stretching factors under iterations of automorphisms.We study metric properties of folding paths, showing that they are geodesic for the non-symmetric metric and, if they do not enter the thin part of Outer space, quasi-geodesic for the symmetric metric.

In this paper we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of G-trees with possibly non-trivial vertex stabilisers. The strategies are the same as in the classical case, with some technicalities arising from the presence of infinite-valence vertices.We describe the Lipschitz metric and show how to compute it; we prove the existence of optimal maps; we describe geodesics represented by folding paths.We show that train tracks representative of irreducible (hence hyperbolic) automorphisms exist and that their are metrically characterized as minimal displaced points, showing in particular that the set of train tracks is closed (in particular answering to some questions raised in [22] concerning the axis bundle of irreducible automorphisms).Finally, we include a proof of the existence of simplicial train tracks map without using Perron-Frobenius theory.A direct corollary of this general viewpoint is an easy proof that relative train track maps exist in both the free group and free product case.

We prove that if S is a closed compact surface of genus g ≥ 2, and if ρ : π 1 (S) → PSL(2, C) is a quasi-Fuchsian representation, then the deformation space M k,ρ of branched projective structures on S with total branching order k and holonomy ρ is connected, as soon as k > 0. Equivalently, two branched projective structures with the same quasi-Fuchsian holonomy and the same number of branch points are related by a movement of branch points. In particular grafting annuli are obtained by moving branch points. In the appendix we give an explicit atlas for M k,ρ for non-elementary representations ρ. It is shown to be a smooth complex manifold modeled on Hurwitz spaces.

We prove a volume-rigidity theorem for fuchsian representations of fundamental groups of hyperbolic k-manifolds into Isom(H n ). Namely, we show that if M is a complete hyperbolic k-manifold with finite volume, then the volume of any representation of π 1 (M ) into Isom(H n ), 3 ≤ k ≤ n, is less than the volume of M , and the volume is maximal if and only if the representation is discrete, faithful and "k-fuchsian".

Let the Δ-complexity σ(M ) of a closed manifold M be the minimal number of simplices in a triangulation of M . Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M normalized by the covering degree, we can promote σ to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which we denote by σ∞(M ) and call the stable Δ-complexity of M .We study here the relation between the stable Δ-complexity σ∞(M ) of M and Gromov's simplicial volume M . It is immediate to show that M σ∞(M ) and it is natural to ask whether the two quantities coincide on aspherical manifolds with residually finite fundamental groups. We show that this is not always the case: there is a constant Cn < 1 such that M Cnσ∞(M ) for any hyperbolic manifold M of dimension n 4.The question in dimension 3 is still open in general. We prove that σ∞(M ) = M for any aspherical irreducible 3-manifold M whose JSJ decomposition consists of Seifert pieces and/or hyperbolic pieces commensurable with the figure-eight knot complement. The equality holds for all closed hyperbolic 3-manifolds if a particular three-dimensional version of the Ehrenpreis conjecture is true.