Workshop on Geometric Structures,
Hitchin Components and Representation Varieties
Date: Oct. 20 ~ Oct. 24/ Place: Rm. 1114, KIAS, Seoul, Korea |

Title & Abstract | Home > Title & Abstract |

**Speaker:** Sasha Anan’in (Universidade de São Paulo)

**Title 1: **The hyperelliptic group

**Abstract 1: **The hyperelliptic group $H_n:=langle r_1,dots,r_nmid r_i^2=r_ndots r_2r_1=1rangle$ appears while studying relations between involutions in a Lie group. Dealing with discrete representations of $H_n$ on Poincar'e's disc provides an elementary proof of the Goldman-Toledo rigidity theorem. In some sense, every discrete representation of $H_n$ on Poincar'e's disc comes from that of $H_5$; moreover, to each Riemann surface $Sigma$ of genus $geqslant2$, one can naturally associate a couple of hyperelliptic surfaces that determine $Sigma$. A similar view in the context of the complex hyperbolic plane leads to a trivial disc bundle over a surface; again, the bundle comes from $H_5$.

**Title 2:** Towards compact $2$-ball quotients

**Abstract 2:** We show some promising approaches to constructing compact complex hyperbolic manifolds. They are based either on the construction of Dirichlet polyhedra, or on central extensions of plane triangle groups, or on hyperbolic spheres with conic points.

**Speaker: **Shinpei Baba (Heidelberg University)

**Title 1: **Neck-pinching of CP^1-structures

**Abstract 1: **We consider a diverging one-parameter family of marked CP^1-structures such that their holonomy representations converge in the PSL(2,C)-character variety. In this setting, we discuss about degeneration of CP^1-structures to noded CP^1-structures.

**Title 2: **Asymptotic Teichmüller rays

**Abstract 2: **We consider geodesic rays in the Teichmüller space with the Teichmüller metric. Every geodesic ray is given by a Riemann surface, its base point, and a measured foliation, its direction. Fixing a measured foliation, we discuss about conditions for two Teichmüller geodesics rays from difference base points to be (strongly) asymptotic. This is joint work with Subhjoy Gupta.

**Speaker:** Brian Patrick Collier (University of Illinois)

**Title: **Maximal SO(2,3) surface group representations and Labourie's conjecture

**Abstract:** The nonabelian Hodge correspondence provides a homeomorphism between the character variety of surface group representations into a real Lie group G and the moduli space of G-Higgs bundles. This homeomorphism however breaks the natural mapping class group action on the character variety. Generalizing techniques and conjectures of Labourie for Hitchin representations, we restore the mapping class group symmetry for all maximal SO(2, 3) = PSp(4, R) surface group representations. More precisely, we show that for each maximal SO(2, 3) representation there is a unique conformal structure in which the corresponding equivariant harmonic map to the symmetric space is a conformal immersion, or, equivalently, a minimal immersion. This is done by exploiting finite order fixed point properties of the associated maximal Higgs bundles.

**Speaker: **Carlos H. Grossi (Universidade de São Paulo)

**Title 1:** Coordinate-free aspects of hyperbolic geometries

**Abstract 1: **We plan to discuss a hermitian framework for studying several hyperbolic geometries. In practical applications, say, in the construction of complex hyperbolic manifolds, such framework seems to be more adequate than the traditional riemannian methods. It naturally provides an approach to study grassmannians of linear objects (geodesics, equidistant hypersurfaces, etc.) as well as to study geometric structures on ideal boundaries. So, we come to Schur functors; they are expected to yield interesting harmonic (pseudo-)isometric embeddings between the mentioned grassmannians and are related to invariants of hyperbolic manifolds. At the end, we exhibit a construction that can hopefully lead to new discrete invariants of discrete representations of surface groups.

**Title 2:** Disc bundles over surfaces and Kalashnikov

**Abstract 2:** First, we describe some general tools that help in constructing a disc bundle $M$ over a surface $Sigma$ (endowed with a given geometric structure) and in calculating its Euler number $eM$. We illustrate the use of these tools in the case of the complex hyperbolic plane with some examples, old and new. The relation $2(eM+chiSigma)=3tau$, where $tau$ is the Toledo invariant, holds in all our examples and can be conjecturally explained by the existence of a holomorphic section of the bundle.

**Speaker: **Yoshinobu Kamishima (Josai University)

**Title: **Survey on spherical CR-structures and related geometric structures

**Speaker:** Sean Lawton (George Mason University)

<Expository talks>

**Title:** Introduction to character varieties

**Abstract:** In the first talk we will discuss a general definition of a character variety, go over concrete examples, and discuss areas in mathematics where they naturally arise.

In the second talk we will focus in on a concrete family of character varieties as an example working to illustrate general results about these moduli spaces, and how non-commutative methods can be used to obtain them.

<Research talk>

**Title: **Topology of the moduli space of local systems over an open surface

**Abstract:** In this talk I will go over recent results concerning the topology of the moduli space of G-local systems over a surface with boundary, where G is a Lie group. In particular, we discuss results about the homotopy groups of character varieties of free groups.

**Speaker:** Bernhard Leeb (University of München)

**Title: **Finsler compactifications of symmetric spaces with applications to discrete groups

**Abstract: **We will study the geometry of symmetric spaces X=G/K of noncompact type from the CAT(0) perspective, in particular their asymptotic geometry and compactifications arising from natural Finsler metrics. We will then discuss aspects of the topological dynamics of the action of discrete subgroups $Gamma<G$ on these Finsler compactifications, namely the construction of domains of proper discontinuity and criteria for the cocompactness of the action on these domains. This leads to natural bordifications of the locally symmetric spaces $X/Gamma$ as orbifolds with corners by attaching $Ga$-quotients of suitable domains of proper discontinuity at infinity. In the case of Anosov subgroups, one obtains compactifications. This is joint work with Misha Kapovich.

**Speaker: **Arielle Leitner (Technion)

**Title: **Geometric Transitions of the Diagonal Cartan Subgroup in SL(n,R) and Generalized Cusps on Convex Projective Manifolds

**Abstract:** A geometric transition is a continuous path of geometries which abruptly changes type in the limit. We explore geometric transitions of the positive diagonal Cartan subgroup in SL(n,R). For n = 3, it turns out the diagonal Cartan subgroup has precisely 5 limits, and for n = 4, there are 15 limits, which give rise to generalized cusps on convex projective 3-manifolds. When n ≥ 7, there is a continuum of non conjugate limits of the Cartan subgroup, distinguished by projective invariants. To prove these results, we use some new techniques of working over the hyperreal numbers.

**Speaker:** Chris Manon (George Mason University)

**Title 1: **Newton-Okounkov bodies and Toric geometry of free group character varieties I

**Abstract 1: **Newton-Okounkov bodies are convex polyhedral invariants of commutative algebras which encode deep and interesting information about the associated algebraic varieties. We will describe how to use elements of representation theory to build a large combinatorial class of Newton-Okounkov bodies for free group character varieties of reductive groups. We describe how the associated Newton-Okounkov bodies are related to Outer Space, a simplicial space defined by Culler and Vogtmann to study the outer automorphism group of a free group.

**Title 2: **Newton-Okounkov bodies and Toric geometry of free group character varieties II

**Abstract 2: **Then we will describe two surprising constructions which emerge from the Newton-Okounkov theory in the $SL_2(C)$ case. Firstly, we show that each Newton-Okounkov body is the momentum image of a distinct integrable system in the $SL_2(C)$ free group character variety. Secondly, we use Newton-Okounkov bodies to construct special compactifications of character varieties which appear naturally in the theory of non-Abelian theta functions.

**
Speaker:** Xin Nie (KIAS)

**Title:**Meromorphic cubic differentials and convex projective structures

**Abstract:**Extending the Labourie-Loftin correspondence, we establish, on any punctured oriented surface, a one-to-one correspondence between convex projective structures with specific types of ends and punctured Riemann surface structures endowed with meromorphic cubic differentials whose poles are at the punctures. This generalizes previous results of Loftin, Benoist-Hulin and Dumas-Wolf.

**Speaker:**Ying Zhang (Soochow University)

**Title:**On Bowditch subsets of representation varieties of the one-hole torus group

**Abstract:**In the two hour lectures I will explain key ideas and techniques in Bowditch’s important work [Markoff triples and quasifuchsian groups, Proc. London Math. Soc., 77:3 (1998), 697--736] on subsets of PSL(2,C) representations of the one-hole torus group (a rank two free group) which satisfy the Bowditch conditions. I hope this will help arouse interests of people in attacking the connectedness conjecture of the Bowditch sets.