Convex co-compact hyperbolic manifolds contain a smallest non-empty
geodesically convex subset, called their convex core. The "pleating" of
the boundary of this convex core is recorded by a measured lamination,
called the bending lamination, and Thurston conjectured that convex
co-compact hyperbolic 3-manifolds are uniquely determined by their
bending lamination. We will describe a proof...
We use the Selberg zeta function to study the limit behavior of resonances of a degenerating family of Kleinian Schottky groups. We prove that, after a suitable rescaling, the Selberg zeta functions converge to the Ihara zeta function of a limiting finite graph associated with the relevant non-Archimedean Schottky group acting on the Berkovich projective line.
An application of our...
An Anosov representation of a hyperbolic group $\Gamma$ quasi-isometrically embeds $\Gamma$ into a semisimple Lie group in a way which imitates and generalizes the behavior of a convex cocompact group acting on a rank-1 symmetric space; it is unknown whether every linear hyperbolic group admits an Anosov representation. In this talk, I will discuss joint work with Sami Douba, Balthazar...
A quasi-Fuchsian representation of a surface group in $\mathrm{PSL}(2,\mathbb C)$ is a discrete and faithful representation that preserves a Jordan curve on the Riemann sphere. These classical objects have a very rich structure as they lie at the crossroad of several areas of mathematics such as complex dynamics, Teichmüller theory, and 3-dimensional hyperbolic geometry. The invariant Jordan...
I will discuss recent work with C. Leininger in which we produce purely pseudo-Anosov surface subgroups of mapping class groups, obtaining the first compact atoroidal surface bundles over surfaces. We do this by constructing a type-preserving representation of the figure eight knot group into the mapping class group of the thrice-punctured sphere.
We define a "drilling" of a hyperbolic group along a maximal infinite cyclic subgroup, which results in a relatively hyperbolic group pair. This procedure is inspired by drilling a hyperbolic 3-manifold along an embedded geodesic. We show that, under suitable circumstances, drilling a hyperbolic group with 2-sphere hyperbolic boundary results in a relatively hyperbolic group with 2-sphere...
We show that the pressure metric on quasi-Fuchsian space has finite diameter. The talk is based on joint work with E. Fioravanti, U. Hamenstädt and Y. Zhang.
Anosov representations form a stable and rich class of discrete subgroups of semisimple Lie groups that today is recognized as the correct higher rank generalization of rank 1 convex cocompact subgroups of Lie groups. In this talk, after surveying what is known on classes of hyperbolic groups admitting Anosov representations, I will discuss some joint work with Subhadip Dey on constructing new...
We discuss the problem of computing the limit cone of a positive representation of a surface group into a real-split, semi-simple Lie group $G$. This is a closed cone in the positive Weyl chamber recording the range of possible spectral behavior achieved by the representation; it is convex with non-empty interior, assuming the representation is Zariski-dense. To compute it, one needs a way to...
For a given hyperbolic 3-manifold $M$ with Fuchsian ends, it is a question of Canary and Taylor whether $M$ minimizes the Hausdorff dimension of the limit set among its convex co-compact deformations. In this talk we will show how this holds locally if $M$ is sufficiently symmetric. Based in work in progress with Sami Douba and Andrés Sambarino.
Roughly speaking, a complete real hyperbolic manifold is geometrically finite if its convex core is the union of a compact set and finitely many ends that are isometric to ends of manifolds with elementary parabolic holonomy. This notion admits many different characterizations, and has been generalized to much broader settings such as rank-one symmetric spaces, Hadamard manifolds, or even...
I will discuss recent joint work with Martin Bobb, in which we study surface subgroups of $\mathrm{PGL}(4,\mathbb R)$ that act convex cocompactly on $\mathbb{RP}^3$ as well as their degenerations. I will focus on surface subgroups of the general coaffine group (the stabilizer of a line in $\mathbb{R}^4$). If time permits, I will also construct Zariski dense "bending lines" as an application.
Can quasi-isometric embeddings of the free group (or surface groups) into $\mathrm{SL}_3( \mathbb R)$ be described? Can one get information about their deformations? I will present some examples and partial answers to these questions contained in joint work with L. Carvajales and P. Lessa.
I will explain how convex projective geometry over ordered
non-archimedean fields may be used to study large scale properties of
individual real Hilbert geometries and degenerations of convex
projective actions. For example, by studying the case of polytopes,
we obtain an explicit description of the asymptotic cones
of real polytopes.
This is joint work with Xenia Flamm.
The notion of a d-pleated surface is a higher rank generalization of (abstract) pleated surfaces in three dimensional hyperbolic space. We give a description of the global topology of the space of conjugacy classes of $d$-pleated surfaces. We also prove that every connected component of the character variety contains exactly one connected component of the space of $d$-pleated surfaces. This is...
The Euler class of a surface group representation into $\mathrm{Diff}(\mathbb S^1)$ satisfies the Milnor—Wood inequality, and representations with maximal Euler class are semi-conjugated to Fuchsian representations by a theorem of Matsumoto. In higher regularity, Ghys proved a stronger rigidity theorem: for $k\geq3$, a maximal circle action by diffeomorphisms of class $\mathrm{C}^k$ is...
