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7 Publications visible to you, out of a total of 7
Surface groups acting on CAT (-1) spaces

Abstract (Expand)

Harmonic map theory is used to show that a convex cocompact surface group action on a \mathrmCAT(-1) metric space fixes a convex copy of the hyperbolic plane (i.e. the action is Fuchsian) if and only … if the Hausdorff dimension of the limit set of the action is equal to 1. This provides another proof of a result of Bonk and Kleiner. More generally, we show that the limit set of every convex cocompact surface group action on a \mathrmCAT(-1) space has Hausdorff dimension ≥1, where the inequality is strict unless the action is Fuchsian.

Authors: GEORGIOS DASKALOPOULOS, CHIKAKO MESE, ANDREW SANDERS, ALINA VDOVINA

Date Published: 1st Jul 2019

Publication Type: Journal

(G,P)-opers and global Slodowy slices

Abstract (Expand)

In this paper, we introduce a generalization of G-opers for arbitrary parabolic subgroups P<G. For parabolic subgroups associated to even nilpotents, we parameterize (G,P)-opers by an object … generalizing the base of the Hitchin fibration. In particular, we describe families of opers associated to higher Teichmuller spaces

Authors: Brian Collier, Andrew Sanders

Date Published: 2019

Publication Type: Misc

Entropy, minimal surfaces and negatively curved manifolds

Abstract (Expand)

Taubes [Minimal surfaces in germs of hyperbolic 3-manifolds. Proceedings of the Casson Fest, Geom. Topol. Monogr. 7 (2004), 69–100 (electronic)] introduced the space of minimal hyperbolic germs with … elements consisting of the first and second fundamental form of an equivariant immersed minimal disk in hyperbolic 3-space. Herein, we initiate a further study of this space by studying the behavior of a dynamically defined function which records the entropy of the geodesic flow on the associated Riemannian surface. We provide a useful estimate on this function which, in particular, yields a new proof of Bowen’s theorem on the rigidity of the Hausdorff dimension of the limit set of quasi-Fuchsian groups. These follow from new lower bounds on the Hausdorff dimension of the limit set which allow us to give a quantitative version of Bowen’s rigidity theorem. To demonstrate the strength of the techniques, these results are generalized to convex-cocompact surface groups acting on n-dimensional \mathrmCAT(-1) Riemannian manifolds.

Author: ANDREW SANDERS

Date Published: 1st Feb 2018

Publication Type: Journal

The pre-symplectic geometry of opers and the holonomy map

Abstract (Expand)

Given a connected complex semi-simple Lie group G and a Riemann surface X, a G-oper on X is a higher rank generalization of a complex projective structure on X. These objects play an important role in …in integrable systems and geometric representation theory, a status that was cemented by the seminal work of Beilinson-Drinfeld \citeBD91. For G a connected complex simple Lie group of adjoint type, we study the global deformation theory of G-opers on a connected, closed, oriented smooth surface Σof genus at least two. We exhibit the deformation space of G-opers on Σas a holomorphic fiber bundle over Teichmüller space, and elucidate the relationship with the deformation space of complex projective structures. Then, we show that there is a family of identifications of the deformation space of G-opers with a holomorphic vector bundle \mathcalB_G(Σ) over Teichmüller space whose typical fiber over a Riemann surface X is a sum of spaces of pluri-canonical sections. Finally, we show that the holonomy map from the deformation space of G-opers to the deformation space of flat G-bundles on Σis a holomorphic immersion. As a consequence of this result, we show that the deformation space of G-opers carries a (pre-symplectic) closed holomorphic differential 2-form of constant rank, and we prove that a sub-family of the identifications of \mathcalB_G(Σ) with the deformation space of G-opers is a holomorphic pre-symplectic map for a natural holomorphic pre-symplectic form on \mathcalB_G(Σ). These results generalize the fundamental features of the deformation space of complex projective structures on Σto the setting of G-opers.

Author: Andrew Sanders

Date Published: 2018

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

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

Bi-Lagrangian structures and Teichmüller theory

Abstract (Expand)

This paper has two purposes: the first is to study several structures on manifolds in the general setting of real and complex differential geometry; the second is to apply this study to Teichmüller … theory. We primarily focus on bi-Lagrangian structures, which are the data of a symplectic structure and a pair of transverse Lagrangian foliations, and are equivalent to para-Kähler structures. First we carefully study real and complex bi-Lagrangian structures and discuss other closely related structures and their interrelationships. Next we prove the existence of a canonical complex bi-Lagrangian structure in the complexification of any real-analytic Kähler manifold and showcase its properties. We later use this bi-Lagrangian structure to construct a natural almost hyper-Hermitian structure. We then specialize our study to moduli spaces of geometric structures on closed surfaces, which tend to have a rich symplectic structure. We show that some of the recognized geometric features of these moduli spaces are formal consequences of the general theory, while revealing other new geometric features. We also gain clarity on several well-known results of Teichmüller theory by deriving them from pure differential geometric machinery.

Authors: Brice Loustau, Andrew Sanders

Date Published: 2017

Publication Type: Misc

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