# Schedule, Titles & Abstracts

##### Schedule

Time \ Day | Monday | Tuesday | Wednesday | Thursday | Friday |

10:00-10:50 | Guéritaud | Burger | Series | Falbel | |

10:50-11:15 | Coffee | Coffee | Coffee | Coffee | |

11:15-12:05 | Canary | Loftin | Pozzetti | Dumas | Deraux |

12:05-13:40 | Lunch | Lunch | Lunch | Lunch | Lunch |

13:40-14:30 | Zhang | Wentworth | Free afternoon | Drumm |
Schlenker |

14:40-15:30 | Goldman | McIntosh | Will | Brock | |

15:30-16:00 | Tea | Tea | Tea | Tea | |

16:00-16:50 | Kassel | Wolff | Sakuma | ||

18:00 | Dinner | Dinner | Dinner |

##### Titles and abstracts

Title: *Renormalized volume of hyperbolic 3-manifolds and the Weil-Petersson geometry of Teichmüller space*

Title: *Real Algebraic Methods in Higher Teichmueller Theory*

Abstract: For a finitely generated group H and a real algebraic semisimple group G we define RChar(H,G), the real spectrum compactification of the character variety Char(H,G) of H in G and relate it to the Parreau compactification PChar(H,G) obtained by projectivizing Weyl chamber valued length functions. As an application we deduce that the closure of a connected component C in PChar(H,G) is arcwise connected in a controlled way. We specialize then to the case where H is a compact surface group, G=SL(n,R) and C=Hit(n) the Hitchin component. We show that length functions in the Parreau boundary of Hit(n) all come from representations of H over non-archimedean real closed fields that are positive in the sense of Fock-Goncharov. We suggest that this characterizes completely such length functions.

This is joint work with Alessandra Iozzi, Anne Parreau and Beatrice Pozzetti.

Title: *Dynamics and the Hitchin component*

Abstract: Hitchin discovered a component of the space of representations of a surface group into PSL(n,R), which bears many resemblances to the Teichmuller spaceof Fuchsian representations of the surface group into PSL(2,R). Labourie introduced dynamical techniques to show that these Hitchin representations are discrete, faithful quasi-isometric embeddings. Sambarino associated Anosov flows to Hitchin representations whose periods record the spectral data of the representation. In this talk, we will see how to use these flows to attach and study dynamical quantities to Hitchin representations, e.g. entropies, Liouville currents and associated Liouville volumes. We will also discuss rigidity results and natural Riemannian metrics on the Hitchin component.

(These results are joint work with Francois Labourite and Andres Sambarino.)

Title: *A new non-arithmetic lattice in PU(3,1)*

Abstract: It is well known that there are non-arithmetic lattices in the isometry group of complex hyperbolic n-space, at least for small values of the dimension n. For a long time, the only known examples were obtained by a famous construction due do Deligne and Mostow(later reinterpreted by Thurston). I will explain how to check arithmeticity and commensurability relations in another family of lattices, constructed by Couwenberg, Heckman and Looijenga; this family turns out to produce non-arithmetic lattices that are new in the sense that they are not commensurable to any Deligne-Mostow example.

Title:* The Geometry of the Bidisk*

Abstract: The bidisk, $H^2 \times H^2$, is a rank $2$ geometry with many interesting properties. We consider the geometry of the bidisk under different metrics. We will be especially interested in bisectors of the bidisk, and define $n$-hyperbolae in $H^2$. We will show that bisectors are unique, and build some variants of bisectors, lunar hypersurfaces, that may be of use in understanding the action of some discrete groups.

This is joint work with Virginie Charette and Youngju Kim.

**David Dumas** (Illinois at Chicago)

Title: *Asymptotics of Hitchin's metric on Teichmüller space*

Abstract: In 1987, Hitchin constructed a Riemannian metric on the Teichmüller space of a compact surface that depends on the choice of a base point in this space. This metric arises naturally from the construction of the moduli space of Higgs bundles, and it is the restriction of a hyperkähler metric on the SL(2,C) character variety, however it is not invariant under the action of the mapping class group. I will discuss recent work with Andrew Neitzke showing that as one goes to infinity in a generic direction in Teichmüller space, Hitchin's metric is exponentially asymptotic to another metric that has a simple, explicit formula (the "semiflat approximation"). This behavior was predicted in a conjecture of Gaiotto-Moore-Neitzke, who also proposed a more general asymptotic formula for the hyperkähler metric on the complex character variety.

Title: *Flag structures on 3-manifolds*

Abstract: Path geometry and CR structures on real 3 manifolds were studied by E. Cartan. There is an interesting local geometry with curvature invariants and an interesting global geometry of those structures which are flat. The model spaces are closed orbits of SL(3,R) and SU(2,1) in a complex flag manifold. We will review these geometries and discuss a notion of FLAG STRUCTURE, which includes both geometries, its curvature invariants and the associated flat manifolds which may be thought as totally real embeddings into a flag manifold.

Title: *Dynamics of geometric structures and flat connections*

Abstract: We describe dynamical systems arising from the classification of locally homogeneous geometric structures and flat connections. Their classification mimics that of Riemann surfaces by the Riemann moduli space, the quotient of Teichmueller space by the mapping class group. However, unlike Riemann surfaces, the mapping class group actions on character varieties may be chaotic, leading to dynamic complexity. We discuss specific examples of these dynamics for some simple surfaces, where the relative character varieties appear as cubic surfaces in 3-space. Complicated dynamics seems to accompany complicated topology, and we interpret them in terms of geometric structures (singular hyperbolic structures). I will report on joint work with McShane-Stantchev-Tan, which in turn is based on ideas of Bowditch on the Markoff equation.

Title: *Affine manifolds and Coxeter groups*

Abstract: Any right-angled Coxeter group on N generators acts properly discontinuously on the real affine space of dimension N(N-1)/2; together with known embedding properties, this provides many new examples of complete affine manifolds. I will outline a proof of this result, obtained in joint work with J.Danciger and F.Kassel.

Title: Convex cocompactness for right-angled Coxeter groups

Abstract: We will describe the moduli space of representations of right-angled Coxeter groups W as reflection groups yielding convex cocompact projective structures. For Gromov hyperbolic W, these representations are so-called Anosov

representations, with strong dynamical properties.

This is joint work with J. Danciger and F. Guéritaud.****

**John Loftin** (Rutgers, Newark)

Title: *Coordinates along neck pinches for moduli of convex real projective structures*

Abstract: 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.

This is based on joint work with Tengren Zhang.

Title: *Moduli spaces of equivariant minimal discs*

Abstract: An equivariant minimal disc is a minimal immersion of the Poincare disc in a noncompact symmetric space which intertwines two actions of the fundamental group of a closed oriented hyperbolic surface. The action on the domain (Poincare disc) is Fuchsian while the action on the codomain (noncompact symmetric space) is via a reductive representation into its isometry group G. John Loftin and I have defined the moduli space of these so that it is naturally related to Teichmuller space and the character variety of G. We studied these moduli spaces for G=PU(2,1) and G=SO_0(4,1) by exploiting the relationship with Higgs bundle moduli, and now have a conceptually clear picture of what to expect for rank one symmetric spaces. I will describe what we discovered, focussing mostly on the real hyperbolic space case, and the questions this raises about convex cocompact representations (a.k.a Anosov embeddings).

**Beatrice Pozzetti** (Heidelberg)

Title: *Critical exponent and Hausdorff dimension for Anosov representations*

Abstract: Whenever $\Gamma$ is a convex cocompact subgroup of the group of isometries of the hyperbolic space, Patterson-Sullivan theory allows to relate the asymptotic growth rate of orbit points for the action of $\Gamma$ on $\mathbb H^n$ and the Hausdorff dimension of the limit set of $\Gamma$ in $\partial \mathbb H^n$. Anosov representations form a robust generalization of convex cocompactness for subgroups of higher rank Lie groups. However the relation between the Hausdorff dimension of their limit set and some orbit growth rate is much more elusive since, on the one hand, the action of $\Gamma$ on the boundary is not conformal, and, on the other, many different orbit growth functions can be considered. In my talk I'll report on joint work with A. Sambarino and A. Wienhard in which we find large classes of Anosov representations for which we can obtain such a relation.

Title: *Kleinian groups generated by two parabolic transformations*

Abstract: At a workshop held in Budapest in 2002, Ian Agol announced a classification of non-free Kleinian groups generated by two parabolic transformations, which generalizes the characterization, due to Colin Adams, of the hyperbolic 2-bridge link groups.

I will talk about my joint project with Hirotaka Akiyoshi, Ken’ichi Ohshika, and John Parker to give a full proof to the announcement. I will also explain a conjectural picture of the space of Kleinian groups generated by two parabolic transformations, which was found through discussion with Gaven Martin, and a possible approach towards the proof of the conjecture suggested by Gaven Martin and John Parker.

**Jean-Marc Schlenker** (Luxembourg)

Title: *Induced metrics on convex hulls of quasicircles*

Abstract: We consider the "universal" version of results and conjectures concerning the induced metrics on the boundary of the convex cores of quasifuchsian manifolds, as well as constant curvature surfaces in those manifolds. Given a quasisymmetric homeomorphism $u$ of the circle, there is a quasicircle $C$ in $\Bbb{C} P^1$ (resp. $\Bbb{R} P^1\times \Bbb{R} P^1$) such that the gluing

map between the two connected components of the boundary of the convex hull of $C$ in $H^3$ (resp. $AdS^3$) is determined by $u$. $C$ is conjectured to be unique. Similar results apply to the induced metrics and third fundamental forms on the boundary of convex domains in $H^3$ (resp. $AdS^3$) with constant curvature and asymptotic boundary a quasicircle.

Joint work with Francesco Bonsante, Jeff Danciger and Sara Maloni.

Title: *Geometry in non-discrete groups: Primitive stability and the Bowditch Q-condition are equivalent*

Abstract: There can be geometrical conditions on a group of hyperbolic isometries which may sometimes be of interest even when the group is not discrete. We discuss two different such conditions which pertain to the primitive elements in an SL(2,C) representation of the free group F_2. One is Minsky’s condition of primitive stability, and the other is the so-called BQ-condition introduced by Bowditch and generalised by Tan, Wong and Zhang. We prove these two conditions are equivalent, aided by an auxiliary condition which constrains the location of the axes of those primitive elements which are palindromic words. The same result has been announced independently by Jaijeong Lee and Binbin Xu.

Title: *Asymptotic structure of Higgs bundle moduli and pleated surfaces*

Abstract: The topic of this talk is the asymptotic structure of the SL(2, ℂ) character variety of a closed surface group. Recent work of Taubes, Mazzeo, et al., and Mochizuki describes the large scale behavior of solutions to the Hitchin equations in terms of certain limiting configurations. I will show how these correspond in a precise way, via harmonic maps, to Bonahon’s parametrization of pleated surfaces in hyperbolic 3-space by transverse and bending cocycles for a geodesic lamination. This gives a geometric interpretation of the asymptotic integrable system.

This is joint work with Andreas Ott, Jan Swoboda, and Mike Wolf.

Title: *Hilbert geometry without convexity*

Abstract: The Hilbert metric is classically studied on convex subsets of the real projective space. In this talk I will present a generalisation of that notion to certain non-convex subsets of complex projective space, and give a few examples.

This is joint work with Elisha Falbel and Antonin Guilloux.

Title: *Rigidity and geometricity for surface group actions on the circle*

Abstract: We consider representations of surface groups in the group of orientation-preserving homeomorphisms of the circle. Kathryn Mann has proved that the geometric representations (ie, those obtained by lifting to a finite cover the faithful and discrete representations in PSL(2,R)) are rigid, ie, all their deformations are semi-conjugate. In a joint work with her, we prove the converse: all rigid representations are semi-conjugate to a geometric one.

Title: *Hitchin representations and proper actions*

Abstract: We prove that SO(n,n)-Hitchin representations, when viewed as representations into SL(2n,R), are never Anosov with respect to the stabilizer of the n-dimensional subspace in SL(2n,R). Using this, we prove the following two results. First, the actions of surface groups on H^{n,n-1} induced by SO(n,n)-Hitchin representations are never proper. Second, SO(n,n-1)-Hitchin representations cannot be the linear part of an affine action of a surface group on the affine space E^{n,n-1}.

This is joint work with Jeff Danciger.