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62 Publications visible to you, out of a total of 62
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

2π-Grafting and complex projective structures with generic holonomy

Abstract (Expand)

Let S be an oriented closed surface of genus at least two. We show that, given a generic representation }}{\backslashrho: \backslashpi_1(S) \backslashto {\backslashrm PSL} (2, … \backslashmathbb{C})}}}ρ:\pi1(S)\textrightarrowPSL(2,C)in the character variety, (}}{2\backslashpi}}}2\pi-)grafting produces all projective structures on S with holonomy }}{\backslashrho}}}ρ.

Author: Shinpei Baba

Date Published: 1st Oct 2017

Publication Type: Journal

Perturbations of the Spence-Abel equation and deformations of the dilogarithm function

Abstract (Expand)

We analyze existence, uniqueness and regularity of solutions for perturbations of the Spence-Abel equation for the Rogers’ dilogarithm. As an application we deduce a version of Hyers-Ulam stability for … the Spence-Abel equation.

Authors: Tobias Hartnick, Andreas Ott

Date Published: 1st Aug 2017

Publication Type: Journal

On weakly maximal representations of surface groups

Abstract (Expand)

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. We prove that weakly maximal representations are discrete … and injective and we describe the structure of the Zariski closure of their image. Furthermore, we prove that the set of weakly maximal representations is a closed subset of the representation variety and describe its relation to other geometrically significant subsets of the representations variety.

Authors: Gabi Ben Simon, Marc Burger, Tobias Hartnick, Alessandra Iozzi, Anna Wienhard

Date Published: 1st Mar 2017

Publication Type: Journal

Domains of discontinuity for almost-Fuchsian groups

Abstract (Expand)

An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1) . We show … that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual boundary of hyperbolic 3-space. This yields a necessary condition for a quasi-Fuchsian group to be almost-Fuchsian which involves only conformal geometry. As an application, we prove that there are no doubly-degenerate geometric limits of almost-Fuchsian groups.

Author: Andrew Sanders

Date Published: 1st Feb 2017

Publication Type: Journal

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