A workshop on Teichmuller theory,
hyperbolicity and dynamics
talks and abstracts

June 24-28, 2019
IMPA

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Alexandre Ananin (ICMC-USP)
Hyperelliptic group

By definition, the so-called hyperelliptic group $G$ is simply the fundamental group of a hyperelliptic curve extended by its hyperelliptic involution. Equivalently, it is the group generated by $n$ involutions $r_1,\dots,r_n$ with the defining relation $r_1\dots r_n=1$, where $n$ is even. Denote by $T$ the usual Teichmuller space of all curves, i.e., the Hitchin component of representations of surface group in $PO(2,1)$, and by $H$, the Teichmuller space of hyperlliptic ones.

We show that to an arbitrary curve one can associate a couple of hyperelliptic ones so that $T$ is fibred twice over $H$ providing an embedding of $T$ into $H\times H$.

Also, we discuss the problem of existence of a Hitchin component of representations of $G$ in $PO(2,2)$.

Misha Belolipetsky (IMPA)
Arithmetic points in moduli spaces

Let Mg denote the moduli spaces of closed surfaces of genus g ™ 2, and let Ag be the set of points in Mg that correspond to arithmetic surfaces. We will discuss some results, conjectures, and intriguing observations about the distribution of the points from Ag in Mg.

Martin Bridgeman (Boston College)
Schwarzian derivatives, projective structures, and the Weil-Petersson gradient flow for renormalized volume.

We consider complex projective structures and their associated locally convex pleated surface. We relate their geometry in terms of the $L_2$ and $L_\infty$ norms the Schwarzian derivative. We show that these give a unifying approach that generalizes a number of well-known results for convex cocompact hyperbolic structures including bounds on the Lipschitz constant for the retract and the length of the bending lamination. We then use these bounds to study the Weil-Petersson gradient flow of renormalized volume on the space $CC(N)$ of convex cocompact hyperbolic structures on a compact manifold N with incompressible boundary. This leads to a proof of the conjecture that the renormalized volume has infimum given by one-half the simplicial volume of DN, the double of N.

Richard Canary (Ann Arbor)
Topological restrictions on Anosov representations

The theory of Anosov representations was introduced by Francois Labourie in his study of Hitchin representations. They have emerged as the natural analogue, for higher rank Lie groups, of Fuchsian representations, or more generally convex compact representations into rank one Lie groups. We will give a gentle introduction to Anosov representations, followed by a discussion of topological restrictions on the groups which admit Anosov representations into SL(d,R). For example, we will see characterizations of groups admitting Anosov representations into SL(3,R) and SL(4,R) and restrictions on the cohomological dimension for all values of d. (This is joint work with Kostas Tsouvalas.)

Sorin Dumitrescu (Nice)
Holomorphic Cartan geometries on simply connected manifolds

This talk deals with holomorphic Cartan geometries on compact complex manifols. The concept of holomorphic Cartan geometry encapsulates many interesting geometric structures including holomorphic Riemannian metrics, holomorphic affine connections or holomorphic projective connections. Conjecturaly, a compact complex simply connected manifold bearing a holomorphic Cartan geometry with model the complex homogeneous space $G/H$, must be biholomorphic to $G/H$. We present here some recent results going toward this direction. In particular, we show that compact complex simply connected manifolds do not admit holomorphic Riemannian metrics. We also show that compact complex simply connectd manifolds in Fujiki class $\mathcal C$ bearing holomorphic Cartan geometries of algebraic type are projective. Those results were obtain in a joint work with Indranil Biswas.

We also show that compact complex simply connected manifolds of algebraic dimension zero do not admit holomorphic Cartan geometries (this is a joint work with I. Biswas and B. McKay).

Philip Engel (University of Georgia)
Integral-affine structures and compactification of K3 moduli

I will discuss the construction of a geometrically meaningful compactification of the moduli space of degree 2 K3 surfaces. The key step is building a family of integral affine structures on the sphere, which represents a tropicalization of the universal family.

John Loftin (Rutgers)
Coordinates along neck pinches for moduli of convex real projective structures

It is well-known that the boundary of the Deligne-Mumford compactification of the moduli space of hyperbolic structures on a surface of genus at least 2 can be described in terms of Fenchel-Nielsen coordinates for a pants decomposition of the surface for which the necks to be pinched are boundaries of the pants. Along each loop which is pinched, the length parameter goes to zero, while the twist parameter disappears. These coordinates can be used to form orbifold coordinates for the Deligne-Mumford compactification. I will discuss the analogous degenerations for convex real projective structures, and how to extend a version of Goldman's Fenchel-Nielsen type coordinates to degenerations along necks. This involves some new classes of geometric limits, and involves a version of Goldman's interior parameters due to Zhang. I will also discuss how this relates to the complex structure on the moduli space as described via cubic differentials.

Alessia Mandini (PUC-Rio)
Hyperpolygons and parabolic Higgs bundles

Hyperpolygons spaces are a family of (finite dimensional, non-compact) hyperkaehler spaces, that can be obtained from coadjoint orbits by hyperkaehler reduction. Jointly with L. Godinho, we show that these space are diffeomorphic (in fact, symplectomorphic) to certain families of parabolic Higgs bundles. In this talk, I will describe this relation and use it to analyze the fixed points locus of certain natural involutions on the moduli space of parabolic Higgs bundles. I will show that each connected component of the fixed point locus of these involutions is identified with a moduli space of polygons in Minkowski 3-space.

This is based on

Gabriele Mondello (Rome)
Ergodic invariant measures on the space of geodesic currents

Asymptotics of many counting problems on a compact oriented hyperbolic surface S can be encoded into a locally finite measure m on the space C (S) of geodesic currents on S, which is invariant under the action of the mapping class group MCG(S) of S. In many cases such asymptotics is polynomial of degree d, and then the measure m will be d-homogeneous. The aim of this talk is to illustrate a result of classification of locally finite, MCG(S)-invariant ergodic measures on C (S) and of locally finite, MCG(S)-invariant, d-homogeneous, ergodic measures. Such result heavily relies on the classification of ergodic MCG(S)- invariant measures on the space ML (S) of measured laminations on S obtained by Lindenstrauss-Mirzakhani (and almost completely by Hamenstadt).

This is joint work with Viveka Erlandsson: Erlandsson V. and Mondello G., Ergodic invariant measures on the space of geodesic currents, https://arxiv.org/abs/1807.02144.

Dmitri Panov (King's College London)
Moduli spaces of spherical surfaces with conical singularities

A spherical surface with $n$ conical singularities is a surface $S$ with cone points $x_1, ... ,x_n$ and a metric $g$, such that $g$ has curvature $1$ on the complement $S$ to $(x_1, ... ,x_n)$ and has a conical singularity of angle $2\pi\theta_i$ at each $x_i$. Moduli spaces of spherical metrics with fixed angles are intriguing objects. Up to very recently the most basic questions about these spaces were open, in particular it was not known for which angles such spaces are non-empty, whether they can be disconnected, whether they project surjectively to the moduli space of curves with $n$ marked points. I'll speak about solutions of such questions, the talk is based on a joint work with Gabirele Mondello.

Misha Verbitsky (IMPA)
Hyperkahler structure on the space of quasi-Fuchsian representations

Let $G$ be the fundamental group of a compact Riemannian surface $S$. The space $X$ of irreducible $SL(2,\C)$-representations of $G$ is equipped with a hyperkahler structure, constructed by Hitchin. This hyperkaehler structure a priori depends on the complex structure of $S$. Interpreting $X$ as "higher Teichmuller space", V. Fock predicted that this hyperkahler structure is in fact independent from the choice of the complex structure on $S$. I would explain how this conjecture can be proved, using the functoriality of Kaledin-Feix construction of a hyperkahler structure in a neighbourhood of a Kahler manifold in its cotangent space.

Khadim War (IMPA)
Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points

We prove that for closed surfaces M with Riemannian metrics without conjugate points and genus than 2 the geodesic flow on the unit tangent bundle has a unique measure of maximal entropy. Furthermore, this measure is fully supported and the flow is mixing with respect to this measure. We formulate conditions under which this result extends to higher dimensions.

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