# Geometric structures and discrete groups

### May 2-4, @ The University of Texas-Austin

This workshop will focus on locally homogeneous geometric structures (aka (G,X) structures) on manifolds and the corresponding spaces of representations that parameterize these structures.

Support for early career mathematicians will be available through the NSF Geometry and Topology RTG at UT Austin, and through the GEAR network. We encourage graduate students and postdocs working in and around this subject to apply for funding.

**Mini-course:**

#### Bill Goldman (Maryland)

*Affine Structures on Manifolds.* An affine structure on a manifold is a coordinate atlas where the coordinate changes are local affine. Equivalently an affine structure is an affine connection with vanishing curvature and torsion. In these lectures I will explore the known examples and techniques for classifying such structures. I will begin with Benzecri’s theorem (1955) that the only closed surfaces which admit affine structures have Euler characteristic zero, and how this led to the Milnor-Wood inequality bounding the Euler class of flat bundles. Then I will describe the closely related theory of projective structures on surfaces where one has a complete classification in terms of the deformation space. I will describe the Auslander and Markus conjectures, which indicate the state of our ignorance concerning examples. In particular I hope to describe the result of Smillie (1978) giving, for example, nonexistence of affine structures on closed manifolds whose holonomy is a free product of finite groups

**Invited speakers:**

#### Sam Ballas (UCSB)

*The Structure of Properly Convex Manifolds.* Convex projective geometry is a flexible generalization of hyperbolic geometry that retains many of the nice features of its hyperbolic counterpart. It comes in two distinct flavors: properly convex geometry and strictly convex geometry. In the strictly convex setting work of Benoist and Cooper, Long, and Tillmann have shown that strictly convex manifolds are geometrically and structurally similar to hyperbolic manifolds. On the other hand, general properly convex manifolds can exhibit extremely non-hyperbolic behavior. However in recent work with Darren Long we show that if the properly convex manifold is “almost” strictly convex then it will remain structurally similar to a hyperbolic manifold. I will discuss this last phenomenon by way of a concrete example.

#### Todd Drumm (Howard)

*Bisectors of the Bidisk.*The bidisk, $\mathbf{H}^2 \times \mathbf{H}^2$, is a rank 2 symmetric space. We begin the study of the bidisk by looking at the bisector, surfaces which are equidistant between two points. We show that for certain cyclic groups, there exist Dirichlet domains bounded by two disjoint bisectors, and that there are other Dirichlet domains bounded by (two or more) intersecting bisectors. Also, we will indicate why bisectors are unique.

#### Subhojoy Gupta (Caltech)

*Dynamics of grafting deformations.* The operation of grafting yields paths in the space P(S) of complex projective structures on a surface S. In this talk I shall describe what is known of the dynamics of their projection to moduli space, and discuss a recent application (joint work with Shinpei Baba) concerning fibers of the holonomy map from P(S) to the PSL(2,C) character variety.

#### Sean Lawton (Texas – Pan American)

*Character Varieties and their Topology.* In this talk, we will begin by introducing character varieties with concrete examples, followed by a brief overview of their properties. We will then discuss recent advances in the study of their topology.

#### Sara Maloni (Brown)

*Polyhedra inscribed in the hyperboloid and anti-de Sitter geometry.*Let G be a 3-connected graph embedded in the sphere. In this talk we will show that G is the 1-skeleton of a Euclidean polyhedron inscribed in a hyperboloid if and only if it is the 1-skeleton of a polyhedron inscribed in a sphere and it has a Hamiltonian cycle. That result originates in statements on the geometry of ideal AdS polyhedra. Any hyperbolic metric on the sphere with n labelled cusps, and a distinguished “equator” and “top” and “bottom” polygon, can be uniquely realised as the induced metric on a convex ideal polyhedron in the anti-de Sitter space. Moreover we characterise the possible dihedral angles of those ideal AdS polyhedra, and show that each ideal polyhedron is characterised by its angles. (This is a joint work with J Danciger and J-M Schlenker.)

**Organizers:**

#### Jeffrey Danciger (Austin)

#### Alan Reid (Austin)