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15-19 Apr 2024 Autrans (France)
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TalksChristian El Emam - On a Bers Theorem for SL(3,C) Let S be a closed oriented surface, denote with Teich(S) its Teichmuller space, and with Hit_3(S) the Hitchin component of the SL(3,R)-character variety, namely the connected component containing Fuchsian representations.
The well-known Bers Theorem provides a biholomorphism between Teich(S)xTeich(\bar S) and an open subset of the SL(2,C)-character variety corresponding to quasi-Fuchsian representations.
In this talk, we present an analog result for SL(3,C). In fact, we introduce a notion of complex affine spheres in C^3, which extend the classical (real) affine spheres in R^3, and use them to prove that the natural inclusion of the diagonal of Hit_3(S)xHit_3(\bar S) inside the character variety of SL(3,C) extends to a unique local biholomorphism on a "big" open subset of Hit_3(S)xHit_3(\bar S) which in particular contains Hit_3(S)xTeich(\bar S) and Teich(S) x Hit_3(\bar S).
This is joint work with Nathaniel Sagman.
Xenia Flamm - Hilbert geometry over real closed fields Convex projective geometry is a rich subject and provides an important generalisation of Riemannian geometry. Convex projective surfaces arise as a geometric interpretation of Hitchin representations in SL(3,R). Their Hilbert metric encodes important information about the representation. Understanding degenerations of convex projective structures on a surface naturally leads to the study of the Hilbert geometry of subsets of the projective plane over a non-Archimedean real closed field F. The goal of this talk is to introduce the Hilbert metric (over F) and to describe the metric spaces associated to convex polygons in FP^2 endowed with the Hilbert metric. This is joint work with Anne Parreau.
Xiaolong Hans Han - The geometry of the Thurston norm, geodesic laminations and Lipschitz maps For closed hyperbolic 3-manifolds, Brock and Dunfield made a conjecture about the upper bound on the ratio of L2-norm to Thurston norm. We first talk about its proof assuming manifolds have bounded volume and describe some generic behavior. We then talk about the connection between the Thurston norm, best Lipschitz circle-valued maps, and maximal stretch laminations. We show that the distance between a level set and its translation is the reciprocal of the Lipschitz constant, bounded by the topological entropy of the pseudo-Anosov monodromy if M fibers.
Will Hide - Spectral gaps for random covers of hyperbolic surfaces Based on joint work with Michael Magee. We study the spectrum of the Laplacian for finite-area hyperbolic surfaces, focusing on the spectral gap, i.e. the smallest non-zero element of the spectrum. The spectral gap can be viewed as a measure of how highly connected a surface is, providing control over its diameter and Cheeger constant. It controls the rate of mixing of the geodesic flow and error terms in geodesic counting. For large genus compact surfaces, 1/4 is the asymptotically optimal spectral gap. We show that for any ε > 0, a uniformly random Riemannian cover of a fixed non-compact hyperbolic surface has no new Laplacian eigenvalues below 1/4 −ε with probability tending to 1 as n → ∞. As a consequence we prove the existence of a sequence of compact hyperbolic surfaces {X_i} with genera g_i → ∞ as i → ∞ and λ_1(X_i) → 1/4 , confirming a conjecture of Buser. I will discuss some ideas of the proofs as well as some further questions.
Ruojing Jiang - Minimal Surface Entropy of Hyperbolic Manifolds I will discuss the definition of minimal surface entropy and review the results for both closed hyperbolic manifolds of odd dimensions (n≥3) and hyperbolic 3-manifolds of finite volumes. On one hand, when the quantity is defined for a metric with sectional curvature no greater than -1, it reaches its minimum value if and only if the metric is hyperbolic. On the other hand, on a hyperbolic 3-manifold of finite volume, among all metrics with scalar curvature bounded from below by -6, the entropy reaches its maximum at the hyperbolic metric.
Filippo Mazzoli - Volume, entropy, and diameter in SO(p,q+1)-higher Teichmüller spaces.
The notion of \mathbb{H}^{p,q}-convex cocompact representations was introduced by Danciger, Guéritaud, and Kassel and provides a unifying framework for several interesting classes of discrete subgroups of the orthogonal groups SO(p,q+1), such as convex cocompact hyperbolic manifolds and maximal globally hyperbolic anti-de Sitter spacetimes of negative Euler characteristic. By recent works of Seppi-Smith-Toulisse and Beyrer-Kassel, we now know that any \mathbb{H}^{p,q}-representation of a group of cohomological dimension p admits a unique invariant maximal spacelike p-dimensional manifold inside the pseudo-Riemannian hyperbolic space \mathbb{H}^{p,q}, and that the space of \mathbb{H}^{p,q}-convex cocompact representations of a group \Gamma consists of a union of connected components of the associated SO(p,q+1)-character variety.
In this talk, I will describe a recent joint work with Gabriele Viaggi in which we provide various applications for the existence of invariant maximal spacelike submanifolds. These include a rigidity result for the pseudo-Riemannian critical exponent (which answers affirmatively to a question of Glorieux-Monclair), a comparison between entropy and volume, and several compactness and finiteness criteria in this framework.
Ognjen Tosic - Failure of strict plurisubharmonicity of energy functionals associated to surface group representations into Lie groups Given an irreducible representation $\rho:\pi_1(\Sigma_g)\to\mathrm{GL}(n,\mathbb{C})$, we define an energy functional $E_\rho$ on Teichmüller space of genus $g$ as follows: given a Riemann surface $X$ marked by $\Sigma_g$, let $h:\tilde{X}\to\mathrm{GL}(n,\mathbb{C})/\mathrm{U}(n)$ be the $\rho$-equivariant harmonic map from its universal cover $\tilde{X}$ to the symmetric space associated to $\mathrm{GL}(n,\mathbb{C})$. We then set $E_\rho(X)$ to be the total energy of $h$ over a single fundamental domain for the deck group action of $\pi_1(X)$ on $\tilde{X}$. By a result of Toledo, this energy map $E_\rho$ is plurisubharmonic, i.e. the Laplacian of the restriction of $E_\rho$ to any complex disk is non-negative. It is natural to ask, given a Beltrami differential $\mu$ on $X$ that represents a tangent direction in Teichmüller space, when does the Laplacian of $E_\rho$ vanish in the complex direction defined by $\mu$? In this talk, we relate this set of directions to two classical objects in non-abelian Hodge theory: the $\mathbb{C}^*$-action on the moduli space of Higgs bundles, and the Hitchin fibration.
Gabriele Viaggi - Geometry of hyperconvex representations
I will present some structural results on the geometry and topology of hyperconvex subgroups in PSL(d,C). This is a class of subgroups that share some similarities with classical Kleinian groups (discrete subgroups of PSL(2,C)). Our main focus is on the Hausdorff dimension of their limits sets which are fractal objects living in partial flag varieties of C^d. This is joint work with James Farre and Beatrice Pozzetti.
David Xu - Deformations of convex-cocompact representations into the isometry group of the infinite-dimensional hyperbolic space.
Convex-cocompact representations of hyperbolic groups into the isometry groups of hyperbolic spaces are a generalisation of quasi-Fuchsian representations in PSL(2,C). A classical result in this field, due to Marden and Thurston, states that convex-cocompact representations of finitely generated groups into PO(n,1) form an open subset of the space of representations (this is called the stability of convex-cocompact representations). In this talk, I will discuss the case of representations into the group of isometries of the infinite-dimensional hyperbolic space, where this stability property remains true. This allows the use of deformation techniques such as bending to produce other convex-cocompact representations.
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