arXiv subjects

Olivier Guichard

Publications and source records attributed to Olivier Guichard.

16 recordsLinked to original sources

arXiv2026

Generalizing Lusztig's total positivity II : geometric properties

Positive structures in Lie groups with respect to a subset $Θ$ of the set of positive roots provide a generalization of Lusztig's total positivity in split real Lie groups to the setting of general real semisimple Lie groups. In [GW25] Lie groups G admitting a positive structure were classified and many key properties of the unipotent positive semigroups were established. In this article we focus on the positive semigroup in G. We establish key geometric properties of elements in the positive semigroup. Further we introduce corresponding positive and non-negative parts of flag varieties and determine the topology of the non-negative parts of flag varieties in many cases. For symplectic flag varieties we provide explicit descriptions of the positive and non-negative flag varieties.

math.DG↗

arXiv2025

Positivity and representations of surface groups

In arXiv:1802.02833 Guichard and Wienhard introduced the notion of $Θ$-positivity, a generalization of Lusztig's total positivity to real Lie groups that are not necessarily split. Based on this notion, we introduce in this paper $Θ$-positive representations of surface groups. We prove that $Θ$-positive representations are $Θ$-Anosov. This implies that $Θ$-positive representations are discrete and faithful and that the set of $Θ$-positive representations is open in the representation variety. We show that the set of $Θ$-positive representations is closed within the set of representations that do not virtually factor through a parabolic subgroup. From this we deduce that for any simple Lie group $\mathsf G$ admitting a $Θ$-positive structure there exist components consisting of $Θ$-positive representations. More precisely we prove that the components parametrized using Higgs bundles methods in arXiv:2101.09377 consist of $Θ$-positive representations.

math.DG↗

arXiv2025

Exotic subgroups of hyperbolic groups

Gromov Hyperbolic groups have remarkable finiteness properties;for example those that are torsion-free are fundamental groups of finitecomplexes whose universal cover iscontractible (property~$F$). In this talk we will show thattheir subgroups can have exotic finiteness properties:there are hyperbolic groups containing finitely generated subgroups withintermediate finiteness properties (Llosa Isenrich and Py); there are hyperbolic groups containing finitely generated subgroups having theproperty~$F$ but which are not themselves hyperbolic (Italiano, Martelli, and Migliorini). This answersold questions about hyperbolic groups and their subgroups. The two mentioned results come from constructions of fibrations,the first in complex geometry and the second in hyperbolic geometry.We will describe the main points of these constructions.

math.GR↗

arXiv2024

Positivity, cross-ratios and the Collar Lemma

We prove that $Θ$-positive representations of fundamental groups of surfaces (possibly cusped or of infinite type) satisfy a collar lemma, and their associated cross-ratios are positive. As a consequence we deduce that $Θ$-positive representations form closed subsets of the representation variety.

math.DG↗

arXiv2024

Generalizing Lusztig's total positivity

We introduce the notion of $Θ$-positivity in real simple Lie groups. This notion at the same time generalizes Lusztig's total positivity in split real Lie groups and invariant orders in Lie groups of Hermitian type. We show that there are four families of Lie groups which admit $Θ$-positive structures, and investigate basic properties of $Θ$-positivity.

math.DG↗

arXiv2022

Parametrizing spaces of positive representations

Using Lusztig's total positivity in split real Lie groups V. Fock and A. Goncharov have introduced spaces of positive (framed) representations. For general semisimple Lie groups a generalization of Lusztig's total positivity was recently introduced by O. Guichard and A. Wienhard. They also introduced the associated space of positive representations. Here we consider the corresponding spaces of positive framed representations of the fundamental group of a punctured surface. We give several parametrizations of the spaces of framed positive representations. Using these parametrizations, we describe their topology and their homotopy type. We show that the number of connected components of the space of framed positive representations agrees with the number of connected components of the space of positive representations, and determine this number for simple Lie groups. Along the way, we also parametrize, for an arbitrary semisimple Lie group, the space of representations of the fundamental group of a punctured surface which are transverse with respect to a fixed ideal triangulation of the surface.

math.DG↗

arXiv2022

Noncommutative coordinates for symplectic representations

We introduce coordinates on the spaces of framed and decorated representations of the fundamental group of a surface with nonempty boundary into the symplectic group $Sp(2n,\mathbf R)$. These coordinates provide a noncommutative generalization of the parametrizations of the spaces of representations into $SL(2,\mathbf R)$ or $PSL(2,\mathbf R)$ given by Thurston, Penner, Kashaev, and Fock-Goncharov. On the space of decorated symplectic representations the coordinates give a geometric realization of the noncommutative cluster-like structures introduced by Berenstein-Retakh. The locus of positive coordinates maps to the space of framed maximal representations. We use this to determine an explicit homeomorphism between the space of framed maximal representations and a quotient by the group $O(n)$. This allows us to describe the homotopy type and, when $n=2$, to give an exact description of the singularities. Along the way, we establish a complete classification of pairs of nondegenerate quadratic forms.

math.DG↗

arXiv2020

Groupes convexes-cocompacts en rang supérieur

The convex-cocompact subgroups are central in hyperbolic geometry and more generally in negative curvature. Labourie introduced in 2005 the notion of 'Anosov' subgroup which proves progressively to be the right generalizations of convex-cocompact groups, especially after the works of Kapovich, Leeb and Porti. This exposé will review different caracteriations of those groups, emphasizing the parallel (or difference) with the negative curvature, and will give their basic properties.

math.GR↗

arXiv2018

Positivity and higher Teichmüller theory

We introduce $Θ$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $Θ$-positivity generalizes at the same time Lusztig's total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of Lie groups, SO(p,q) for p<q, and a family of exceptional Lie groups, which admit a $Θ$-positive structure. We describe key aspects of $Θ$-positivity and make a connection with representations of surface groups and higher Teichmüller theory.

math.DG↗

arXiv2016

Anosov representations and proper actions

We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of reductive groups.

math.GR↗

arXiv2015

Tameness of Riemannian locally symmetric spaces arising from Anosov representations

We construct compactifications of Riemannian locally symmetric spaces arising as quotients by Anosov representations. These compactifications are modeled on generalized Satake compactifications and, in certain cases, on maximal Satake compactifications. We deduce that these Riemannian locally symmetric spaces are topologically tame, i.e. homeomorphic to the interior of a compact manifold with boundary. We also construct domains of discontinuity (not necessarily with a compact quotient) in a much more general setting.

math.GT↗

arXiv2015

Compactification of certain Clifford-Klein forms of reductive homogeneous spaces

We describe smooth compactifications of certain families of reductive homogeneous spaces such as group manifolds for classical Lie groups, or pseudo-Riemannian analogues of real hyperbolic spaces and their complex and quaternionic counterparts. We deduce compactifications of Clifford-Klein forms of these homogeneous spaces, namely quotients by discrete groups Gamma acting properly discontinuously, in the case that Gamma is word hyperbolic and acts via an Anosov representation. In particular, these Clifford-Klein forms are topologically tame.

math.GT↗

arXiv2012

Anosov representations: Domains of discontinuity and applications

The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface groups, in particular in the context of higher Teichmueller spaces, and for lattices in SO(1,n). In this article we extend the notion of Anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties. In particular, given an Anosov representation of $Γ$ into G we explicitly construct open subsets of compact G-spaces, on which $Γ$ acts properly discontinuously and with compact quotient. As a consequence we show that higher Teichmueller spaces parametrize locally homogeneous geometric structures on compact manifolds. We also obtain applications regarding (non-standard) compact Clifford-Klein forms and compactifications of locally symmetric spaces of infinite volume.

math.DG↗

arXiv2010

Topological Invariants of Anosov Representations

We define new topological invariants for Anosov representations and study them in detail for maximal representations of the fundamental group of a closed oriented surface into the symplectic group.

math.DG↗

arXiv2007

Convex Foliated Projective Structures and the Hitchin Component for PSL(4,R)

In this article we give a geometric interpretation of the Hitchin component for PSL(4,R) in the representation variety of a closed oriented surface of higher genus. We show that representations in the Hitchin component are precisely the holonomy representations of properly convex foliated projective structures on the unit tangent bundle of the surface. From this we also deduce a geometric description of the Hitchin component the symplectic group PSp(4,R).

math.DG↗

arXiv2007

Well displacing representations and orbit maps

We discuss in this article a property of action of groups by isometries called "well displacing". An action is said to be well displacing, if the displacement function is equivalent to the the displacement function for the action on the Cayley graph. We relate this property with the fact that orbit maps are quasi-isometric embeddings. We first describe countrexamples that shows this two notions are unrelated in general. On the other hand we explain that for a certain class of groups -- in particular hyperbolic groups -- these two properties are equivalent. In the course of our discussion, we introduce an intrinsic property of the group -- that we called the U-property -- which says quantitatively how the norm an element is controlled by the translation length of finitely many related conjugacy classes. This property play a central role in our discussion.

math.GT↗