Publications (15)1.31 Total impact
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ABSTRACT: 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 nontrivial vertex stabilisers. The strategies are the same as in the classical case, with some technicalities arising from the presence of infinitevalence vertices. In particular, 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 automorphisms exist and that their are metrically characterized as minimal displaced points, showing in particular that the set of train tracks is closed. We also prove that relative train track maps exist in both the free group and free product case.12/2013;  [Show abstract] [Hide abstract]
ABSTRACT: Let the complexity 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 it to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which call the "stable complexity" of M. We study here the relation between the stable complexity of M and Gromov's simplicial volume M. It is immediate to show that M is smaller or equal than the stable complexity of M and it is natural to ask whether the two quantities coincide on aspherical manifolds with residually finite fundamental group. We show that this is not always the case: there is a constant C_n<1 such that M is smaller than C_n times the stable complexity for any hyperbolic manifold M of dimension at least 4. The question in dimension 3 is still open in general. We prove that the stable complexity equals M for any aspherical irreducible 3manifold M whose JSJ decomposition consists of Seifert pieces and/or hyperbolic pieces commensurable with the figureeight knot complement. The equality holds for all closed hyperbolic 3manifolds if a particular threedimensional version of the Ehrenpreis conjecture is true.01/2012;  [Show abstract] [Hide abstract]
ABSTRACT: It is wellknown that a point $T\in cv_N$ in the (unprojectivized) CullerVogtmann Outer space $cv_N$ is uniquely determined by its \emph{translation length function} $._T:F_N\to\mathbb R$. A subset $S$ of a free group $F_N$ is called \emph{spectrally rigid} if, whenever $T,T'\in cv_N$ are such that $g_T=g_{T'}$ for every $g\in S$ then $T=T'$ in $cv_N$. By contrast to the similar questions for the Teichm\"uller space, it is known that for $N\ge 2$ there does not exist a finite spectrally rigid subset of $F_N$. In this paper we prove that for $N\ge 3$ if $H\le Aut(F_N)$ is a subgroup that projects to an infinite normal subgroup in $Out(F_N)$ then the $H$orbit of an arbitrary nontrivial element $g\in F_N$ is spectrally rigid. We also establish a similar statement for $F_2=F(a,b)$, provided that $g\in F_2$ is not conjugate to a power of $[a,b]$.06/2011;  [Show abstract] [Hide abstract]
ABSTRACT: For a kflat F inside a locally compact CAT(0)space X, we identify various conditions that ensure that F bounds a (k+1)dimensional half flat in X. Our conditions are formulated in terms of the ultralimit of X. As applications, we obtain (1) constraints on the behavior of quasiisometries between tocally compact CAT(0)spaces, (2) constraints on the possible nonpositively curved Riemannian metrics supported by certain manifolds, and (3) a correspondence between metric splittings of a complete, simply connected, nonpositively curved Riemannian manifold and the metric splittings of its asymptotic cones. Furthermore, combining our results with the Ballmann, BurnsSpatzier rigidity theorem and the classical Mostow rigidity theorem, we also obtain (4) a new proof of Gromov's rigidity theorem for higher rank locally symmetric spaces. Comment: 21 pages. This article is a substantially improved version of our earlier preprint arXiv:0801.3636. It features more general results, with shorter, cleaner proofs. Applications remain the same12/2009; 
Article: The isometry group of Outer Space
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ABSTRACT: We prove analogues of Royden's Theorem for the Lipschitz metrics of Outer Space, namely that Isom(CV_n) is Out(F_n).12/2009; 
Article: Metric properties of Outer Space
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ABSTRACT: We define metrics on CullerVogtmann space, which are an analogue of the Teichmuller 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 nonsymmetric metric and, if they do not enter the thin part of Outer space, quasigeodesic for the symmetric metric.Publicacions Matematiques 03/2008; · 0.85 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Given a geodesic inside a simplyconnected, complete, nonpositively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a biLipschitz flat, then the original geodesic supports a nontrivial, orthogonal, parallel Jacobi field. As applications we obtain (1) constraints on the behavior of quasiisometries between complete, simply connected, NPCR manifolds, and (2) constraints on the NPCR metrics supported by certain manifolds, and (3) a correspondence between metric splittings of complete, simply connected NPCR manifolds, and metric splittings of its asymptotic cones. Furthermore, combining our results with the BallmannBurnsSpatzier rigidity theorem and the classic Mostow rigidity, we also obtain (4) a new proof of Gromov's rigidity theorem for higher rank locally symmetric spaces.02/2008;  [Show abstract] [Hide abstract]
ABSTRACT: We prove that, for a hyperbolic two bridge knot, infinitely many Dehn fillings are rigid in $SO_0(4,1)$. Here rigidity means that any discrete and faithful representation in $SO_0(4,1)$ is conjugate to the holonomy representation in $SO_0(3,1)$. We also show local rigidity for almost all Dehn fillings.01/2008;  [Show abstract] [Hide abstract]
ABSTRACT: We prove a compactness theorem for automorphisms of free groups. Namely, we show that the set of automorphisms keeping bounded the length of the uniform current is compact (up to conjugation.) This implies that the spectrum of the length of the images of the uniform current is discrete, answering to a conjecture of I. Kapovich.03/2006;  [Show abstract] [Hide abstract]
ABSTRACT: We prove a volumerigidity theorem for Fuchsian representations of fundamental groups of hyperbolic kmanifolds into Isom \mathbbHn\mathbb{H}^n. Namely, we show that if M is a complete hyperbolic kmanifold with finite volume, then the volume of any representation of π1(M) into isom \mathbbHn\mathbb{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 ‘kFuchsian’Geometriae Dedicata 01/2006; 117(1):111124. · 0.47 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: 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 di¤erent viewpoints. From one side we look at the combinatorial data of triangulations, and we develop an algorithmic method, which allows us to reduce 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 complete characterization of the choices of moduli defining global similarity structures on the torus (or on the Klein bottle).Adv. Geom. 01/2006; 6:397421.  [Show abstract] [Hide abstract]
ABSTRACT: We show that if G is a discrete subgroup of the group of the isometries of the hyperbolic kspace H^k, and if R is a representation of G into the group of the isometries of H^n, then any Requivariant map F from H^k to H^n extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of this fact, we obtain an extension of a result of Besson, Courtois and Gallot about the existence of volume nonincreasing, equivariant maps. Moreover, under an additional hypothesis, we show that the weak extension we obtain is actually a measurable Requivariant map from the boundary of H^k to the closure of H^n. We use this fact to obtain measurable versions of CannonThurstontype results for equivariant Peano curves.06/2004;  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we study the differences between algebraic and geometric solutions of hyperbolicity equations for ideally triangulated 3manifolds, and their relations with the variety of representations of the fundamental group of such manifolds into PSL(2, C). We show that the geometric solutions of compatibility equations form an open subset of the algebraic ones, and we prove uniqueness of the geometric solutions of hyperbolic Dehn filling equations. In the last section we study some examples, doing explicit calculations for three interesting manifolds.Topology and Its Applications  TOPOL APPL. 01/2004; 145(1).  [Show abstract] [Hide abstract]
ABSTRACT: Let W be a compact manifold and let \rho be a representation of its fundamental group into PSL(2,C). The volume of \rho is defined by taking any \rhoequivariant map from the universal cover of 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 the volume is welldefined in all cases. We prove here that the volume of a representation is always welldefined and depends only on the representation. 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(\rho)=vol(M) then \rho is discrete and faithful.06/2003;  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we study the difference between algebraic and geometric solutions of the hyperbolic Dehn filling equations for ideally triangulated 3manifolds. We show that any geometric solution is an algebraic one, and we prove the uniqueness of the geometric solutions. Then we do explicit calculations for three interesting examples. With the first two examples we see that not all algebraic solutions are geometric and that the algebraic solutions are not unique. The third example is a nonhyperbolic manifold that admits a positive, partially flat solution of the compatibility and completeness equations.06/2003;
Publication Stats
89  Citations  
1.31  Total Impact Points  
Top Journals
Institutions

2009

University of Bologna
 Department of Mathematics MAT
Bolonia, EmiliaRomagna, Italy


2006

Autonomous University of Barcelona
Cerdanyola del Vallès, Catalonia, Spain
