Stefano Francaviglia

Metric properties of outer space

We define metrics on Culler-Vogtmann space, which are an analogue of the Thurston 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.

Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds

Let W be a compact manifold and let ρ be a representation of its fundamental group into PSL(2, C). Then the volume of ρ is defined by taking any ρ-equivariant map from the universal cover W~ to ℍ 3 and then by integrating the pullback of the hyperbolic volume form on a fundamental domain. It turns out that such a volume does not depend on the choice of the equivariant map. Dunfield extended this construction to the case of a noncompact (cusped) manifold M, but he did not prove that the volume is well defined in all cases. We prove here that the volume of a representation is always well defined and depends only on the representation. Moreover, we show that this volume can be easily computed by straightening any ideal triangulation of M. We show that the volume of a representation is bounded from above by the relative simplicial volume of M. Finally, we prove a rigidity theorem for representations of the fundamental group of a hyperbolic manifold. Namely, we prove that if M is hyperbolic and vol(ρ) = vol(M), then ρ is discrete and faithful.

The isometry group of outer space

Abstract We prove analogues of Royden’s Theorem for the Lipschitz metrics of Outer Space, namely that Isom ( CV n ) = Out ( F n ) .

Geodesic currents and length compactness for automorphisms of free groups

We prove a compactness theorem for automorphisms of free groups. Namely, we show that the set of automorphisms keeping the length of the uniform current bounded is compact (up to conjugation). This implies that the spectrum of the length of the images of the uniform current is discrete, proving a conjecture of I. Kapovich.

Branched projective structures with Fuchsian holonomy

We prove that if S is a closed compact surface of genus g 2, and if W 1.S/! PSL.2;C/ is a quasi-Fuchsian representation, then the space Mk; of branched projective structures on S with total branching order k and holonomy is connected, for k > 0. Equivalently, two branched projective structures with the same quasiFuchsian 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 Mk; for non-elementary representations . It is shown to be a smooth complex manifold modeled on Hurwitz spaces. 30F35, 57M20; 53A30, 14H15

The oriented graph of multi-graftings in the Fuchsian case

We prove the connectedness and compute the diameter of the oriented graph of multi-graftings associated to exotic CP1-structures on a compact surface S with a given holonomy representation of Fuchsian type.

Corrigendum to "Spectral rigidity of automorphic orbits in free groups"

Lemma 5.1 in our paper [6] says that every infinite normal subgroup of Out(F_N) contains a fully irreducible element; this lemma was substantively used in the proof of the main result, Theorem A in [6]. Our proof of Lemma 5.1 in [6] relied on a subgroup classification result of Handel-Mosher [9], originally stated in [9] for arbitrary subgroups H≤Out(F_N). It subsequently turned out (see p. 1 in [10]) that the proof of the Handel-Mosher theorem needs the assumption that H be finitely generated. Here we provide an alternative proof of Lemma 5.1 from [6], which uses the corrected version of the Handel-Mosher theorem and relies on the 0-acylindricity of the action of Out(F_N) on the free factor complex (due to Bestvina-Mann-Reynolds).

Similarity structures on the torus and the Klein bottle via triangulations

In this paper we study the possibility of defining a similarity structure on the torus and the Klein bottle using the combinatorial data of a triangulation. Given a choice of moduli for the triangles of a triangulation of a surface, the problem is to decide whether such moduli are compatible with a global similarity structure on the surface. We study this problem under two dierent viewpoints. From one side we look at the combi- natorial data of triangulations, and we develop an algorithmic method, which allows us to re- duce the general problem to a simpler one, which is easily solved. From the other side we study the problem more algebraically, looking at the properties of the holonomy, and we give a com- plete characterization of the choices of moduli defining global similarity structures on the torus (or on the Klein bottle).