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Fanny Kassel

Publications and source records attributed to Fanny Kassel.

At least 19 recordsLinked to original sources
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↗
arXiv2025

Limit cones of multi-Fuchsian representations

We study the set of normalized multi-lengths for representations of closed surface groups and free groups into $(\mathrm{PSL}_2\mathbf{R})^d$ whose projections to $\mathrm{PSL}_2\mathbf{R}$ are all convex cocompact. These multi-lengths define a convex cone in $\mathbf{R}^d_{\geq 0}$, called the limit cone. When $d=3$, we show the coexistence of different regimes: for some representations the limit cone has only a finite number of sides, which we can force to grow like the genus (or free rank); for other representations, extremal rays are dense in the boundary of the limit cone. We also give examples where the limit cone varies discontinuously with the representation.

math.GT↗
arXiv2025

A simultaneous Abels-Margulis-Soifer lemma

The Abels-Margulis-Soifer lemma states that if a semigroup $\Gamma$ acts strongly irreducibly by linear transformations on a finite-dimensional real vector space, then any element of $\Gamma$ can be multiplied by an element of some fixed finite subset of $\Gamma$ so that it becomes proximal (i.e. it acts on the corresponding projective space with an attracting fixed point and a repelling projective hyperplane) and even uniformly proximal (i.e. the distance between the attracting fixed point and the repelling projective hyperplane is uniformly bounded from below and the contraction towards the attracting fixed point is uniformly strong). We prove a version of this lemma simultaneously for linear representations of a semigroup $\Gamma$, acting on the corresponding projective spaces, and for representations of $\Gamma$ to isometry groups of (not necessarily proper) Gromov hyperbolic metric spaces, acting on the corresponding Gromov boundaries.

math.GR↗
arXiv2024

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 $\Gamma$ any discrete subgroup of $G$ acting properly discontinuously and cocompactly on some homogeneous space $G/H$ of $G$, then $\Gamma$ is quasi-isometrically embedded in $G$ and the action of $\Gamma$ 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↗
arXiv2024

Combination theorems in convex projective geometry

We prove a general combination theorem for discrete subgroups of $\mathrm{PGL}(n,\mathbb{R})$ preserving properly convex open subsets in the projective space $\mathbb{P}(\mathbb{R}^n)$, in the spirit of Klein and Maskit. We use it in particular to prove that a free product of two $(\mathbb{Z}$-)linear groups is again ($\mathbb{Z}$-)linear, and to construct Zariski-dense discrete subgroups of $\mathrm{PGL}(n,\mathbb{R})$ which are not lattices but contain a lattice of a smaller higher-rank simple Lie group. We also establish a version of our combination theorem for discrete groups that are convex cocompact in $\mathbb{P}(\mathbb{R}^n)$ in the sense of arXiv:1704.08711. In particular, we prove that a free product of two convex cocompact groups is convex cocompact, which implies that the free product of two Anosov groups is Anosov. We also prove a virtual amalgamation theorem over convex cocompact subgroups generalizing work of Baker-Cooper.

math.GR↗
arXiv2024

Discrete subgroups of semisimple Lie groups, beyond lattices

Discrete subgroups of SL(2,R) are well understood, and classified by the geometry of the corresponding hyperbolic surfaces. Discrete subgroups of higher-rank semisimple Lie groups, such as SL(n,R) for n>2, remain more mysterious. While lattices in this setting are rigid, there also exist more flexible, "thinner" discrete subgroups, which may have large and interesting deformation spaces, giving rise in particular to so-called higher Teichm\"uller theory. We survey recent progress in constructing and understanding such discrete subgroups from a geometric and dynamical viewpoint.

math.GR↗
arXiv2023

$\mathbb{H}^{p,q}$-convex cocompactness and higher higher Teichm\"uller spaces

For any integers $p\geq 2$ and $q\geq 1$, let $\mathbb{H}^{p,q}$ be the pseudo-Riemannian hyperbolic space of signature $(p,q)$. We prove that if $\Gamma$ is the fundamental group of a closed aspherical $p$-manifold, then the set of representations of $\Gamma$ to $\mathrm{PO}(p,q+1)$ which are convex cocompact in $\mathbb{H}^{p,q}$ is a union of connected components of $\mathrm{Hom}(\Gamma,\mathrm{PO}(p,q+1))$. More generally, we show that if $\Gamma$ is any finitely generated group with no infinite nilpotent normal subgroups and with virtual cohomological dimension $p$, then the set of injective and discrete representations of $\Gamma$ to $\mathrm{PO}(p,q+1)$ preserving a non-degenerate non-positive $(p-1)$-sphere in the boundary of $\mathbb{H}^{p,q}$ is a union of connected components of $\mathrm{Hom}(\Gamma,\mathrm{PO}(p,q+1))$. This gives new examples of higher-dimensional higher-rank Teichm\"uller spaces.

math.GT↗
arXiv2023

Convex cocompact actions in real projective geometry

We study a notion of convex cocompactness for discrete subgroups of the projective general linear group acting (not necessarily irreducibly) on real projective space, and give various characterizations. A convex cocompact group in this sense need not be word hyperbolic, but we show that it still has some of the good properties of classical convex cocompact subgroups in rank-one Lie groups. Extending our earlier work arXiv:1701.09136 from the context of projective orthogonal groups, we show that for word hyperbolic groups preserving a properly convex open set in projective space, the above general notion of convex cocompactness is equivalent to a stronger convex cocompactness condition studied by Crampon-Marquis, and also to the condition that the natural inclusion be a projective Anosov representation. We investigate examples.

math.GT↗
arXiv2022

Groupes de surface dans les réseaux des groupes de Lie semi-simples [d'après J. Kahn, V. Marković, U. Hamenstädt, F. Labourie et S. Mozes]

A cocompact lattice in a semisimple Lie group $G$ is a discrete subgroup $Γ$ such that the quotient $G/Γ$ is compact. Does such a lattice always contain a surface group, i.e. a subgroup isomorphic to the fundamental group of a compact hyperbolic surface? If so, does it contain surface subgroups close (in a precise quantitative sense) to Fuchsian subgroups of $G$, i.e to discrete subgroups of $G$ contained in a copy of $\operatorname{(P)SL}(2,\mathbf{R})$ in $G$? The case $G=\operatorname{PSL}(2,\mathbf{C})$ corresponds to a famous conjecture of Thurston on 3-dimensional hyperbolic manifolds, and the quantitative version of the case $G=\operatorname{PSL}(2,\mathbf{R}) \times \operatorname{PSL}(2,\mathbf{R})$ implies a conjecture of Ehrenpreis on pairs of compact hyperbolic surfaces; these two conjectures were proved by Kahn and Marković around ten years ago. Motivated by a question of Gromov, Hamenstädt solved the case that $G$ has real rank one, except for $G=\operatorname{SO}(2n,1)$. In a recent preprint (arXiv:1805.10189), Kahn, Labourie, and Mozes treat the case of a large class of semisimple Lie groups, including in particular all complex simple Lie groups; the surface groups they obtain are images of representations that are Anosov in the sense of Labourie. We present some of the ideas of their proof.

math.GR↗
arXiv2022

Eigenvalue gaps for hyperbolic groups and semigroups

Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the i-th and (i+1)-th Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index i. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We discuss the case of finitely generated semigroups, and propose a notion of Anosov representation in this setting.

math.DS↗
arXiv2021

Convex cocompactness for Coxeter groups

We investigate representations of Coxeter groups into $\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize which Coxeter groups admit such representations, and we fully describe the corresponding spaces of convex cocompact representations as reflection groups, in terms of the associated Cartan matrices. The Coxeter groups that appear include all infinite, word hyperbolic Coxeter groups; for such groups the representations as reflection groups that we describe are exactly the projective Anosov ones. We also obtain a large class of nonhyperbolic Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in $\mathbb{P}(\mathbb{R}^n)$ developed in arXiv:1704.08711.

math.GR↗
arXiv2020

Spectral analysis on pseudo-Riemannian locally symmetric spaces

We summarize recent results initiating spectral analysis on pseudo-Riemannian locally symmetric spaces $Γ\backslash G/H$, beyond the classical setting where $H$ is compact (e.g. theory of automorphic forms for arithmetic $Γ$) or $Γ$ is trivial (e.g. Plancherel-type formula for semisimple symmetric spaces).

math.SP↗
arXiv2019

Spectral analysis on standard locally homogeneous spaces

Let $X=G/H$ be a reductive homogeneous space with $H$ noncompact, endowed with a $G$-invariant pseudo-Riemannian structure. Let $L$ be a reductive subgroup of $G$ acting properly on $X$ and $\Gamma$ a torsion-free discrete subgroup of $L$. Under the assumption that the complexification $X_{\mathbb C}$ is $L_{\mathbb C}$-spherical, we prove an explicit correspondence between spectral analysis on the standard locally homogeneous space $X_{\Gamma}=\Gamma\backslash X$ and on $\Gamma\backslash L$ via branching laws for the restriction to $L$ of irreducible representations of $G$. In particular, we prove that the pseudo-Riemannian Laplacian on $X_{\Gamma}$ is essentially self-adjoint, and that it admits an infinite point spectrum when $X_{\Gamma}$ is compact or $\Gamma\subset L$ is arithmetic. The proof builds on structural results for invariant differential operators on spherical homogeneous spaces with overgroups.

math.RT↗
arXiv2019

Invariant differential operators on spherical homogeneous spaces with overgroups

We investigate the structure of the ring ${\mathbb D}_G(X)$ of $G$-invariant differential operators on a reductive spherical homogeneous space $X=G/H$ with an overgroup $\widetilde{G}$. We consider three natural subalgebras of ${\mathbb D}_G(X)$ which are polynomial algebras with explicit generators, namely the subalgebra ${\mathbb D}_{\widetilde{G}}(X)$ of $\widetilde{G}$-invariant differential operators on $X$ and two other subalgebras coming from the centers of the enveloping algebras of $\mathfrak g$ and $\mathfrak k$, where $K$ is a maximal proper subgroup of $G$ containing $H$. We show that in most cases ${\mathbb D}_G(X)$ is generated by any two of these three subalgebras, and analyze when this may fail. Moreover, we find explicit relations among the generators for each possible triple $(\widetilde{G},G,H)$, and describe "transfer maps" connecting eigenvalues for ${\mathbb D}_{\widetilde{G}}(X)$ and for the center $Z({\mathfrak g}_{\mathbb C})$ of the enveloping algebra of ${\mathbb g}_{\mathbb C}$.

math.RT↗
arXiv2019

Proper affine actions for right-angled Coxeter groups

For any right-angled Coxeter group $Γ$ on $k$ generators, we construct proper actions of $Γ$ on $\mathrm{O}(p,q+1)$ by right and left multiplication, and on the Lie algebra $\mathfrak{o}(p,q+1)$ by affine transformations, for some $p,q\in\mathbb N$ with $p+q+1=k$. As a consequence, any virtually special group admits proper affine actions on some $\mathbb R^n$: this includes e.g. surface groups, hyperbolic 3-manifold groups, examples of word hyperbolic groups of arbitrarily large virtual cohomological dimension, etc. We also study some examples in cohomological dimension two and four, for which the dimension of the affine space may be substantially reduced.

math.GT↗
arXiv2017

Convex cocompactness in pseudo-Riemannian hyperbolic spaces

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups PO(p,q) by considering their action on the associated pseudo-Riemannian hyperbolic space H^{p,q-1} in place of the Riemannian symmetric space. Following work of Barbot and Mérigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and the natural notion of convex cocompactness in this setting.

math.GT↗
arXiv2017

Maximally stretched laminations on geometrically finite hyperbolic manifolds

Let Gamma_0 be a discrete group. For a pair (j,rho) of representations of Gamma_0 into PO(n,1)=Isom(H^n) with j geometrically finite, we study the set of (j,rho)-equivariant Lipschitz maps from the real hyperbolic space H^n to itself that have minimal Lipschitz constant. Our main result is the existence of a geodesic lamination that is "maximally stretched" by all such maps when the minimal constant is at least 1. As an application, we generalize two-dimensional results and constructions of Thurston and extend his asymmetric metric on Teichmüller space to a geometrically finite setting and to higher dimension. Another application is to actions of discrete subgroups Gamma of PO(n,1)xPO(n,1) on PO(n,1) by right and left multiplication: we give a double properness criterion for such actions, and prove that for a large class of groups Gamma the action remains properly discontinuous after any small deformation of Gamma inside PO(n,1)xPO(n,1).

math.GT↗