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Nicolas Tholozan

Publications and source records attributed to Nicolas Tholozan.

At least 19 recordsLinked to original sources
arXiv2026

Sharpness of proper and cocompact actions on reductive homogeneous spaces

We prove that if $G$ is any noncompact connected real reductive linear Lie group, and $Γ$ any discrete subgroup of $G$ acting properly discontinuously and cocompactly on some homogeneous space $G/H$ of $G$, then $Γ$ is quasi-isometrically embedded in $G$ and the action of $Γ$ on $G/H$ is sharp, i.e. satisfies a strong, quantitative form of proper discontinuity. For noncompact reductive $H$, this was known as the Sharpness Conjecture, with applications to spectral analysis on pseudo-Riemannian locally symmetric spaces developed in arXiv:1209.4075. For $G/H$ rational of real corank one, we use sharpness to fully characterize properly discontinuous and cocompact actions on $G/H$ in terms of Anosov representations. This enables us to show that in real corank one, acting properly discontinuously and cocompactly on $G/H$ is an open property, and also to prove that a number of homogeneous spaces do not admit compact quotients, such as $\mathrm{SL}(n+1,\mathbb{K})/\mathrm{SL}(n,\mathbb{K})$ for $n>1$ and $\mathbb{K}=\mathbb{R}$, $\mathbb{C}$, or the quaternions.

math.GR↗
arXiv2026

Positive representations over real closed fields

We develop the theory of $Θ$-positive representations from general Fuchsian groups to linear groups over real closed fields. Our definition, which does not assume the boundary map to be continuous, encompasses many generalizations of positive or Anosov representations that have been considered in the literature.

math.GT↗
arXiv2026

Compact quotients of homogeneous spaces and homotopy theory of sphere bundles

A reductive homogeneous space $G/H$ is always diffeomorphic to the normal bundle of an orbit of a maximal compact subgroup of $G$. We prove that if $G/H$ admits compact quotients, then the sphere bundle associated to this normal bundle is fiber-homotopically trivial. We deduce that many reductive homogeneous spaces do not admit compact quotients, such as the complex spheres $\mathrm{O}(n+1,\mathbb{C})/\mathrm{O}(n,\mathbb{C})$ for all $n \notin \{1,3,7\}$, or $\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(m,\mathbb{R})$ for all $n>m>1$, which solves conjectures of T. Kobayashi from the early 1990s. We also prove that if the pseudo-Riemannian hyperbolic space $\mathbf{H}^{p,q}$ of signature $(p,q)$ admits compact quotients, then $p$ must be divisible by at least $2^{\lfloor q/2\rfloor}$.

math.GT↗
arXiv2023

Gromov-Thurston manifolds and anti-de Sitter geometry

We consider hyperbolic and anti-de Sitter (AdS) structures on $M\times (0,1)$, where $M$ is a $d$-dimensional Gromov-Thurston manifold. If $M$ has cone angles greater than $2π$, we show that there exists a "quasifuchsian" (globally hyperbolic maximal) AdS manifold such that the future boundary of the convex core is isometric to $M$. When $M$ has cone angles less than $2π$, there exists a hyperbolic end with boundary a concave pleated surface isometric to $M$. Moreover, in both cases, if $M$ is a Gromov-Thurston manifold with $2k$ pieces (as defined below), the moduli space of quasifuchsian AdS structures (resp. hyperbolic ends) satisfying this condition contains a submanifold of dimension $2k-3$. When $d=3$, the moduli space of quasifuchsian AdS (resp. hyperbolic) manifolds diffeomorphic to $M\times (0,1)$ contains a submanifold of dimension $2k-2$, and extends up to a "Fuchsian" manifold, that is, an AdS (resp. hyperbolic) warped product of a closed hyperbolic manifold by~$\R$. We use this construction of quasifuchsian AdS manifolds to obtain new compact quotients of $Ø(2d,2)/\U(d,1)$. The construction uses an explicit correspondence between quasifuchsian $2d+1$-dimensional AdS manifolds and compact quotients of $Ø(2d,2)/\U(d,1)$ which we interpret as the space of timelike geodesic Killing fields of $\AdS^{2d+1}$.

math.DG↗
arXiv2023

Hausdorff dimension of limit sets for projective Anosov representations

We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in $\mathbf{P}(\mathbb{R}^{n}) \times \mathbf{P}({\mathbb{R}^{n}}^*)$ is bounded between two critical exponents associated respectively to a highest weight and a simple root.

math.DG↗
arXiv2023

Chern-Simons theory and cohomological invariants of representation varieties

We prove a general local rigidity theorem for pull-backs of homogeneous forms on reductive symmetric spaces under representations of discrete groups. One application of the theorem is that the volume of a closed manifold locally modelled on a reductive homogeneous space $G/H$ is constant under deformation of the $G/H$-structure. The proof elaborates on an argument given by Labourie for closed anti-de Sitter $3$-manifolds. The core of the work is a reinterpretation of old results of Cartan, Chevalley and Borel, showing that the algebra of $G$-invariant forms on $G/H$ is generated by ``Chern-Weil forms'' and ``Chern-Simons forms''.

math.GT↗
arXiv2023

Simple Anosov representations of closed surface groups

We introduce and study \emph{simple Anosov representations} of closed hyperbolic surface groups, analogous to Minsky's \emph{primitive stable representations} of free groups. We prove that the set of simple Anosov representations into $\mathrm{SL}(d,\mathbb{C})$ with $d \geqslant 4$ strictly contains the set of Anosov representations. As a consequence, we construct domains of discontinuity for the mapping class group action on character varieties which contain non-discrete representations.

math.GT↗
arXiv2023

Fiber bundles associated with Anosov representations

Anosov representations $ρ$ of a hyperbolic group $Γ$ into a semisimple Lie group $G$ are known to admit cocompact domains of discontinuity in flag varieties $G/Q$, endowing the compact quotient manifolds $M_ρ$ with a $(G,G/Q)$-structure. In general the topology of $M_ρ$ can be quite complicated. In this article, we consider the case when $Γ$ is the fundamental group of a closed (real or complex) hyperbolic manifold $N$ and $ρ$ is a deformation of a (twisted) lattice embedding $Γ\to \mathrm{Isom}(\mathbb H_\mathbb K) \to G$ through Anosov representations. We prove that, in this situation, $M_ρ$ is alway a smooth fiber bundle over $N$. Determining the topology of the fiber seems hard in general. The second part of the paper focuses on the special case when $N$ is a surface, $ρ$ a quasi-Hitchin representation into $\mathrm{Sp}(4,\mathbb C)$, and $M_ρ$ is modelled on the space of complex Lagrangians in $\mathbb C^4$. We show that, in this case, the fiber is homeomorphic to $\mathbb{CP}^2 \sharp \overline{\mathbb{CP}^2}$.

math.GT↗
arXiv2022

Equidistribution of Hodge loci II

Let $\mathbb V$ be a polarized variation of Hodge structure over a smooth complex quasi-projective variety $S$. In this paper, we give a complete description of the typical Hodge locus for such variations. We prove that it is either empty or equidistributed with respect to a natural differential form, \emph{the pull-push form}. In particular, it is always analytically dense when the pull-push form does not vanish. When the weight is $2$, the Hodge numbers are $(q,p,q)$ and the dimension of $S$ is least $rq$, we prove that the typical locus where the Picard rank is at least $r$ is equidistributed in $S$ with respect to the volume form $c_q^r$, where $c_q$ is the $q$\textsuperscript{th} Chern form of the Hodge bundle. We obtain also several equidistribution results of the typical locus in Shimura varieties: a criterion for the density of the typical Hodge loci of a variety in $\mathcal{A}_g$, equidistribution of certain families of CM points and equidistribution of Hecke translates of curves and surfaces in $\mathcal A_g$. These results are proved in the much broader context of dynamics on homogeneous spaces of Lie groups which are of independent interest. The pull-push form appear in this greater generality and we provide several tools to determine it and we compute it in many examples.

math.AG↗
arXiv2022

Residually finite non linear hyperbolic groups

We exhibit the first examples of residually finite non-linear Gromov hyperbolic groups. Our examples are constructed as amalgamated products of torsion-free cocompact lattices in the rank 1 Lie group $\mathrm{Sp}(d,1)$, $d\geq 2$ along maximal cyclic subgroups.

math.GR↗
arXiv2022

Linearity and indiscreteness of amalgamated products of hyperbolic groups

We discuss the linearity and discreteness of amalgamated products of linear word-hyperbolic groups. In particular, we prove that the double of an Anosov group along a maximal cyclic subgroup is always linear, and we construct examples of such groups which do not admit any discrete and faithful representation in rank 1. We also build new examples of non-linear word-hyperbolic groups, elaborating on a previous work of Canary--Stover--Tsouvalas.

math.GR↗
arXiv2021

Compact connected components in relative character varieties of punctured spheres

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an elliptic element. These results extend a recent work of Deroin and the first author, which treated the case of $\textrm{PU}(1,1) = \mathrm{PSL}(2,\mathbb{R})$. Our proof relies on the non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles. The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory.

math.GT↗
arXiv2019

The geometry of maximal representations of surface groups into SO(2,n)

In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest. We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank 2.

math.DG↗
arXiv2016

Volume and non-existence of compact Clifford-Klein forms

This article studies the volume of compact quotients of reductive homogeneous spaces. Let $G/H$ be a reductive homogeneous space and $Γ$ a discrete subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. We prove that the volume of $Γ\backslash G/H$ is the integral, over a certain homology class of $Γ$, of a $G$-invariant form on $G/K$ (where $K$ is a maximal compact subgroup of $G$). As a corollary, we obtain a large class of homogeneous spaces the compact quotients of which have rational volume. For instance, compact quotients of pseudo-Riemannian spaces of constant curvature $-1$ and odd dimension have rational volume. This contrasts with the Riemannian case. We also derive a new obstruction to the existence of compact Clifford--Klein forms for certain homogeneous spaces. In particular, we obtain that $\mathrm{SO}(p,q+1)/\mathrm{SO}(p,q)$ does not admit compact quotients when $p$ is odd, and that $\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(m,\mathbb{R})$ does not admit compact quotients when $m$ is even.

math.GT↗
arXiv2016

Super-maximal representations from fundamental groups of punctured surfaces to $\mathrm{PSL}(2,\mathbb{R})$

We study a particular class of representations from the fundamental groups of punctured spheres $Σ_{0,n}$ to the group $\text{PSL} (2,\mathbb R)$ (and their moduli spaces), that we call \emph{super-maximal}. Super-maximal representations are shown to be \emph{totally non hyperbolic}, in the sense that every simple closed curve is mapped to a non hyperbolic element. They are also shown to be \emph{geometrizable} (appart from the reducible super-maximal ones) in the following very strong sense : for any element of the Teichmüller space $\mathcal T_{0,n}$, there is a unique holomorphic equivariant map with values in the lower half-plane $\mathbb H^-$. In the relative character variety, the components of super-maximal representations are shown to be compact, and symplectomorphic (with respect to the Atiyah-Bott-Goldman symplectic structure) to the complex projective space of dimension $n-3$ equipped with a certain multiple of the Fubiny-Study form that we compute explicitly (this generalizes a result of Benedetto--Goldman for the sphere minus four points). Those are the unique compact components in relative character varieties of $\text{PSL}(2,\mathbb R)$. This latter fact will be proved in a companion paper.

math.GT↗
arXiv2015

The Volume of complete anti-de Sitter 3-manifolds

Up to a finite cover, closed anti-de Sitter $3$-manifolds are quotients of $\mathrm{SO}_0(2,1)$ by a discrete subgroup of $\mathrm{SO}_0(2,1) \times \mathrm{SO}_0(2,1)$ of the form \[j\times ρ(Γ)~,\] where $Γ$ is the fundamental group of a closed oriented surface, $j$ a Fuchsian representation and $ρ$ another representation which is "strictly dominated" by $j$. Here we prove that the volume of such a quotient is proportional to the sum of the Euler classes of $j$ and $ρ$. As a consequence, we obtain that this volume is constant under deformation of the anti-de Sitter structure. Our results extend to (not necessarily compact) quotients of $\mathrm{SO}_0(n,1)$ by a discrete subgroup of $\mathrm{SO}_0(n,1) \times \mathrm{SO}_0(n,1)$.

math.GT↗
arXiv2015

Dominating surface group representations by Fuchsian ones

We prove that a representation from the fundamental group of a closed surface of negative Euler characteristic with values in the isometry group of a Riemannian manifold of sectional curvature bounded by -1 can be dominated by a Fuchsian representation. Moreover, we prove that the domination can be made strict, unless the representation is discrete and faithful in restriction to an invariant totally geodesic 2-plane of curvature -1. When applied to representations into PSL(2,R) of non-extremal Euler class, our result is a step forward in understanding the space of closed anti-de Sitter 3-manifolds.

math.DG↗