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Joan Porti

Publications and source records attributed to Joan Porti.

At least 37 records · Page 2Linked to original sources
arXiv2017

Anosov subgroups: Dynamical and geometric characterizations

We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of rank one Lie groups to higher rank. Our main goal is to give several new equivalent characterizations for this important class of discrete subgroups. Our characterizations capture "rank one behavior" of Anosov subgroups and are direct generalizations of rank one equivalents to convex cocompactness. Along the way, we considerably simplify the original definition, avoiding the geodesic flow. We also show that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group.

math.GR↗
arXiv2017

Dynamics on flag manifolds: domains of proper discontinuity and cocompactness

For noncompact semisimple Lie groups $G$ we study the dynamics of the actions of their discrete subgroups $Γ<G$ on the associated partial flag manifolds $G/P$. Our study is based on the observation that they exhibit also in higher rank a certain form of convergence type dynamics. We identify geometrically domains of proper discontinuity in all partial flag manifolds. Under certain dynamical assumptions equivalent to the Anosov subgroup condition, we establish the cocompactness of the $Γ$-action on various domains of proper discontinuity, in particular on domains in the full flag manifold $G/B$. We show in the regular case (of $B$-Anosov subgroups) that the latter domains are always nonempty if if $G$ has (locally) at least one noncompact simple factor not of the type $A_1, B_2$ or $G_2$.

math.MG↗
arXiv2016

Hyperbolic polygons of minimal perimeter in punctured discs

We prove that, among the polygons in a punctured disc with fixed angles, the perimeter is minimized by the polygon with an inscribed horocycle centered at the puncture. We generalize this to a disc with a cone point and to an annulus with a geodesic boundary component and a complete end. Then we apply this result to describe the minimum of the spine systole on the moduli space of punctured surfaces.

math.GT↗
arXiv2016

Reidemeister torsion, hyperbolic three-manifolds, and character varieties

This is a survey on Reidemeister torsion for hyperbolic three-manifolds of finite volume. Torsions are viewed as topological invariants and also as functions on the variety of representations in $\operatorname{ SL}_2(\mathbb C)$. In both cases, the torsions may also be computed after composing with finite dimensional representations of $\operatorname{ SL}_2(\mathbb C)$. In addition the paper deals with the torsion of the adjoint representation as a function on the variety of $\operatorname{ PSL}_{n+1}(\mathbb C)$-characters, using that the first cohomology group with coefficients twisted by the adjoint is the tangent space to the variety of characters.

math.GT↗
arXiv2015

A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings

We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups Isom(X) as undistorted subgroups which are coarsely uniformly regular. We show furthermore that they must be word hyperbolic. We introduced this class of discrete subgroups in our earlier paper, in the context of symmetric spaces, where various equivalent geometric and dynamical characterizations of word hyperbolic Morse subgroups were established, including the Anosov subgroup property.

math.GR↗
arXiv2015

The SL(3,C)-character variety of the figure eight knot

We give explicit equations that describe the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C). This has five components of dimension 2, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irreducible representations. Of these, one is distinguished as it contains the curve of irreducible representations coming from $Sym^2:SL(2,C) \to SL(3,C)$. The other two components are induced by exceptional Dehn fillings of the figure eight knot. We also describe the action of the symmetry group of the figure eight knot on the character varieties.

math.GT↗
arXiv2015

Representations of knot groups into $\mathrm{SL}_n(\mathbf{C})$ and twisted Alexander polynomials

Let $Γ$ be the fundamental group of the exterior of a knot in the three-sphere. We study deformations of representations of $Γ$ into $\mathrm{SL}_n(\mathbf{C})$ which are the sum of two irreducible representations. For such representations we give a necessary condition, in terms of the twisted Alexander polynomial, for the existence of irreducible deformations. We also give a more restrictive sufficient condition for the existence of irreducible deformations. We also prove a duality theorem for twisted Alexander polynomials and we describe the local structure of the representation and character varieties.

math.GT↗
arXiv2014

Morse actions of discrete groups on symmetric space

We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex cocompactness. We also prove that our notions are equivalent to the notion of Anosov subgroup, for which we provide a closely related, but simplified and more accessible reformulation, avoiding the geodesic flow of the group. We show moreover that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group. A substantial part of the paper is devoted to the coarse geometry of these discrete subgroups. A key concept which emerges from our analysis is that of Morse quasigeodesics in higher rank symmetric spaces, generalizing the Morse property for quasigeodesics in Gromov hyperbolic spaces. It leads to the notion of Morse actions of word hyperbolic groups on symmetric spaces,i.e. actions for which the orbit maps are Morse quasiisometric embeddings, and thus provides a coarse geometric characterization for the class of subgroups considered in this paper. A basic result is a local-to-global principle for Morse quasigeodesics and actions. As an application of our techniques we show algorithmic recognizability of Morse actions and construct Morse "Schottky subgroups'' of higher rank semisimple Lie groups via arguments not based on Tits' ping-pong. Our argument is purely geometric and proceeds by constructing equivariant Morse quasiisometric embeddings of trees into higher rank symmetric spaces.

math.GR↗
arXiv2013

Twisted cohomology for hyperbolic three manifolds

For a complete hyperbolic three manifold M, we consider the representations of its fundamental group obtained by composing a lift of the holonomy with complex finite dimensional representations of SL(2,C). We prove a vanishing result for the cohomology of M with coefficients twisted by these representations, using techniques of Matsushima-Murakami. We give some applications to local rigidity.

math.GT↗
arXiv2013

Higher dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds

For an oriented finite volume hyperbolic 3-manifold M with a fixed spin structure η, we consider a sequence of invariants {τ_n(M; η)}. Roughly speaking, {τ_n(M; η)} is the Reidemeister torsion of M with respect to the representation given by the composition of the lift of the holonomy representation defined by η, and the n-dimensional, irreducible, complex representation of SL(2,C). In the present work, we focus on two aspects of this invariant: its asymptotic behavior and its relationship with the complex-length spectrum of the manifold. Concerning the former, we prove that for suitable spin structures, log(τ_n(M; η)) grows as -n^2 Vol(M)/4π, extending thus the result obtained by W. Mueller for the compact case. Concerning the latter, we prove that the sequence {τ_n(M; η)} determines the complex-length spectrum of the manifold up to complex conjugation.

math.GT↗
arXiv2011

Local coordinates for SL(n,C) character varieties of finite volume hyperbolic 3-manifolds

Given a finite volume hyperbolic 3-manifold, we compose a lift of the holonomy in SL(2,C) with the n-dimensional irreducible representation of SL(2,C) in SL(n,C). In this paper we give local coordinates of the SL(n,C)-character variety around this representation. As a corollary, this representation is isolated among all representations that are unipotent at the cusps.

math.GT↗
arXiv2011

The infinitesimal projective rigidity under Dehn filling

To a hyperbolic manifold one can associate a canonical projective structure and ask whether it can be deformed or not. In a cusped manifold, one can ask about the existence of deformations that are trivial on the boundary. We prove that if the canonical projective structure of a cusped manifold is infinitesimally projectively rigid relative to the boundary, then infinitely many Dehn fillings are projectively rigid. We analyze in more detail the figure eight knot and the Withehead link exteriors, for which we can give explicit infinite families of slopes with projectively rigid Dehn fillings.

math.GT↗