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81 Publications visible to you, out of a total of 81
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

Horofunction Compactifications of Symmetric Spaces

Abstract (Expand)

We consider horofunction compactifications of symmetric spaces with respect to invariant Finsler metrics. We show that any (generalized) Satake compactification can be realized as a horofunction … compactification with respect to a polyhedral Finsler metric.

Authors: Thomas Haettel, Anna-Sofie Schilling, Cormac Walsh, Anna Wienhard

Date Published: 2017

Publication Type: Misc

Anosov representations and proper actions

Abstract (Expand)

We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using … a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of reductive groups.

Authors: François Guéritaud, Olivier Guichard, Fanny Kassel, Anna Wienhard

Date Published: 2017

Publication Type: Journal

Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations

Abstract (Expand)

We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group G. These manifolds … are obtained as quotients of the domains of discontinuity in generalized flag varieties G/P constructed by Kapovich-Leeb-Porti (arXiv:1306.3837), and in some cases by Guichard-Wienhard (arXiv:1108.0733). For G-Fuchsian representations and their Anosov deformations, where G is simple, we compute the homology of the domains of discontinuity and of the quotient manifolds. For G-Fuchsian and G-quasi-Fuchsian representations in simple G of rank at least two, we show that the quotient manifolds are not Kähler. We also describe the Picard groups of these quotient manifolds, compute the cohomology of line bundles on them, and show that for G of sufficiently large rank these manifolds admit nonconstant meromorphic functions. In a final section, we apply our topological results to several explicit families of domains and derive closed formulas for topological invariants in some cases. We also show that the quotient manifold for a G-Fuchsian representation in \mathrmPSL_3(C) is a fiber bundle over a surface, and we conjecture that this holds for all simple G.

Authors: David Dumas, Andrew Sanders

Date Published: 2017

Publication Type: Misc

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