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Convex projective structures on nonhyperbolic three-manifolds

Abstract (Expand)

Y Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many submanifolds … each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoist’s theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be convexly glued together whenever the geometry at the boundary matches up. In particular, we prove that many doubles of cusped hyperbolic three-manifolds admit convex projective structures.

Authors: Samuel Ballas, Jeffrey Danciger, Gye-Seon Lee

Date Published: 2018

Publication Type: Journal

Deformations of convex real projective manifolds and orbifolds

Abstract (Expand)

In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with … an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric structures on orbifolds, and Hilbert geometry. The main examples of finitely generated groups for us will be Fuchsian groups, 3-manifold groups and Coxeter groups.

Authors: Suhyoung Choi, Gye-Seon Lee, Ludovic Marquis

Date Published: 2018

Publication Type: InCollection

A complete description of the antipodal set of most symmetric spaces of compact type

Abstract (Expand)

It is known that the antipodal set of a Riemannian symmetric space of compact type G/K consists of a union of K-orbits. We determine the dimensions of these K-orbits of most irreducible symmetric spaces … of compact type. The symmetric spaces we are not going to deal with are those with restricted root system \mathfrak a_r and a non-trivial fundamental group, which is not isomorphic to \mathbb Z_2 or \mathbb Z_r+1. For example, we show that the antipodal sets of the Lie groups Spin(2r+1)r≥5, E_8 and G_2 consist only of one orbit which is of dimension 2r, 128 and 6, respectively; \SO(2r+1) has also an antipodal set of dimension 2r; and the Grassmannian Gr_r,r+q(R) has a rq-dimensional orbit as antipodal set if r≥5 and r≠q>0.

Author: Jonas Beyrer

Date Published: 2018

Publication Type: Journal

Cubulations and cross ratios on contracting boundaries

Abstract (Expand)

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations … and geodesic currents. We add to this picture by studying cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary \partial_∞G. A consequence of our results is that essential, hyperplane-essential cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work in arXiv:1903.02447. Along the way, we describe the relationship between the Roller boundary of a \mathrmCAT(0) cube complex, its Gromov boundary and - in the non-hyperbolic case - the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

Authors: Jonas Beyrer, Elia Fioravanti

Date Published: 2018

Publication Type: Journal

CAT(0) cube complexes are determined by their boundary cross ratio

Abstract (Expand)

We introduce a \Z–valued cross ratio on Roller boundaries of \mathrmCAT cube complexes. We motivate its relevance by showing that every cross-ratio preserving bijection of Roller boundaries uniquely … extends to a cubical isomorphism. Our results are strikingly general and even apply to infinite dimensional, locally infinite cube complexes with trivial automorphism group.

Authors: Jonas Beyrer, Elia Fioravanti, Merlin Incerti-Medici

Date Published: 2018

Publication Type: Misc

Schottky presentations of positive representations

Abstract (Expand)

We show that the notion of 3-hyperconvexity on oriented flag manifolds defines a partial cyclic order. Using the notion of interval given by this partial cyclic order, we construct Schottky groups and … show that they correspond to images of positive representations in the sense of Fock and Goncharov. We construct polyhedral fundamental domains for the domain of discontinuity that these groups admit in the projective space or the sphere, depending on the dimension.

Authors: Jean-Philippe Burelle, Nicolaus Treib

Date Published: 2018

Publication Type: Journal

Rigidity of the saddle connection complex

Abstract (Expand)

For a half-translation surface (S,q), the associated \emphsaddle connection complex \A(S,q) is the simplicial complex where vertices are the saddle connections on (S,q), with simplices spanned by sets …s of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. We prove that any simplicial isomorphism φ\colon \A(S,q) \to \A(S’,q’) between saddle connection complexes is induced by an affine diffeomorphism F \colon (S,q) \to (S’,q’). In particular, this shows that the saddle connection complex is a complete invariant of affine equivalence classes of half-translation surfaces. Throughout our proof, we develop several combinatorial criteria of independent interest for detecting various geometric objects on a half-translation surface.

Authors: Valentina Disarlo, Anja Randecker, Robert Tang

Date Published: 2018

Publication Type: Journal

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