Brown University

Bowdoin College, September 23-25, 2016
AMS Special Section on Convex Cocompactness


Biringer, Ian -- Laminations on the boundary of a handlebody.

If H is a handlebody with boundary S, let Mod(H) be the subgroup of Mod(S) consisting of surface homeomorphisms that extend to H. While Mod(H) isn’t convex cocompact (it contains all Dehn twists around meridians), there is a rich theory surrounding its action on PML(S), and to a lesser extent, Teichmuller space. We’ll briefly survey what’s known, and talk a bit about the related problem of characterizing mapping classes that extend via their Nielsen-Thurston invariant laminations.

Burelle, Jean-Philippe -- Schottky Presentations of Maximal Representations.

Maximal Representations form a class of geometrically interesting representations of a surface group into a Lie group of Hermitian type. They are defined by requiring the Toledo invariant, a topological invariant of the representation, to be maximal. 

In the special case of a surface with non-empty boundary, we explain how to construct explicit presentations for any such representation using the theory of partial cyclic orders. In the case of the group Sp(2n,R), we define hypersurfaces in RP2n−1 that are pairwise identified by the generators of the group and bound a fundamental domain for a properly discontinuous action on an open dense subset of projective space.

Charette, Virginie -- Bisectors in the Bidisk.

Bisectors are equidistant hypersurfaces between two points and are basic objects in a metric geometry. They play an important part in understanding the action of subgroups of isometries on a metric space. In many metric geometries (spherical, Euclidean, hyperbolic, complex hyperbolic, to name a few) bisectors do not uniquely determine a pair of points, in the following sense : completely different sets of points share a common bisector. However, in the bidisk H2xH2, generic bisectors are such for a unique pair of points.

Collier, Brian -- Connected components of the SO(p,q) character varieties.

We will discuss the connected components of the moduli space of representations of the fundamental group of a closed surface of genus at least 2 into the Lie group SO(p,q). In particular, we will show there are more components than expected and discuss which components consist of discrete and faithful representations.

Durham, Matthew -- Characterizing subgroup stability in finitely generated groups. 

Stability for subgroups of finitely generated groups is a strong quasiconvexity condition which generalizes the Morse property, quasiconvexity in hyperbolic groups, and convex cocompactness in mapping class groups. We give a new characterization of stability which allows us to pull back stability under proper actions on proper metric spaces. This result has several applications to mapping class groups, outer automorphism groups of free groups, and relatively hyperbolic groups.

Hironaka, Eriko -- Coxeter mapping classes.

From a Coxeter graph it is possible to define a mapping class on a compact oriented surface whose homological action is conjugate to the Coxeter element of the Coxeter system. This holds not only for classical(simply-laced) Coxeter graphs, but also for Coxeter graphs with sign-labeled vertices. In this talk, we survey some ways that Coxeter mapping classes have been used to produce interesting examples of periodic and ”nearly-periodic” pseudo-Anosov mapping classes.

Koberda, Thomas -- The geometry of purely loxodromic subgroups of right-angled Artin groups.

We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group A(Γ) fulfill equivalent con- ditions that parallel characterizations of convex cocompactness in mapping class groups Mod(S). In particular, such subgroups are quasiconvex in A(Γ). In addition, we identify a milder condition for a finitely generated subgroup of A(Γ) that guarantees it is free, undistorted, and retains finite generation when intersected with A(Λ) for subgraphs Λ of Γ. These results have applications to both the study of convex cocompactness in Mod(S) and the way in which certain groups can embed in right-angled Artin groups.

Lawton, Sean -- Varieties of Characters.

Let G be a connected reductive affine algebraic group. We define the variety of G-characters of a finitely generated group Γ and show that the quotient of the G-character variety of Γ by the action of the trace preserving outer automorphisms of G normalizes the variety of G-characters when Γ is a free group, free abelian group, or a surface group. This is joint work with Adam Sikora.

Mangahas, Johanna -- Contexts for convex cocompactness.

I will introduce both the classical notion of convex cocompactness for Kleinian groups and its adaptation for subgroups of mapping class groups by Farb and Mosher. I will also talk about joint work with Sam Taylor relating mapping class group convex cocompactness to quasiconvexity within right-angled Artin groups.

Martone, Giuseppe -- Positively ratioed representations and length functions in higher rank. 

For a compact surface S with negative Euler characteristic, Hitchin and maximal representations are well studied higher rank analogues of Teichmuller space. In this setting, one can define many length functions generalizing the hyperbolic length on S. For a representation ρ and a length function l, we will define what it means to be a (\rho, l)-positively ratioed representation. This is a sufficient condition for the existence of a shortest non-peripheral simple closed curve on S. We relate the length of such curve to the topological entropy of the representation. The main tool is a construction of a measure on the space of geodesics of S associated to a (ρ,l)-positively ratioed representation. This is joint work with Tengren Zhang.

Ruane, Kim -- Splittings of CAT(0) Groups with Isolated Flats.

In joint work with C. Hruska, we prove that if a one-ended G is a group acting geometrically on a CAT(0) space with Isolated Flats Property, then the well-defined boundary ∂G is locally connected if and only if G does not have a “geometric” splitting in the sense of Mihalik-Ruane-Tschantz. To prove this result, we must first recognize the boundary as a tree of spaces in the sense of Swiatkowski and then prove a general topology result which says that a tree of connected and locally connected compacta is again locally connected.

Vlamis, Nicholas -- Basmajian’s identity in higher Teichmuller theory. 

Basmajian’s original identity gives the area of the boundary of a compact hyperbolic manifold as a summation over the orthospectrum. We will demonstrate an extension of this identity to the setting of Hitchin representations of surface groups. We will see that for 3-Hitchin representations the identity has a natural geometric interpretation analogous to the hyperbolic setting.

Zhang, Tengren (I) -- Anosov representations via examples.

An important notion in Higher Teichmuller theory is that of an Anosov representation. This was first introduced by Francois Labourie to study Hitchin representations. Later, the combined work of many others showed that this is a useful generalization of convex cocompactness to subgroups of higher rank reductive Lie groups. In this talk, I will define the notion of an Anosov representation and explain why this is a useful notion via a variety of examples.

Zhang, Tengren (II) -- Collar lemma for Hitchin representations.

The classical collar lemma is an important result and a useful tool used to study hyperbolic surfaces. In particular, it implies that if two geodesics intersect on a surface, then there is an explicit lower bound of the length of one in terms of the length of the other. We will explain an analog of this statement for Hitchin representations, which are a natural generalization of the holonomy representations of hyperbolic surfaces. This is joint work with Gye-Seon Lee.