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62 Publications visible to you, out of a total of 62
Mapping class group orbits of curves with self-intersections

Abstract

Not specified

Authors: Patricia Cahn, Federica Fanoni, Bram Petri

Date Published: 1st Feb 2018

Publication Type: Journal

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

Graphs of curves on infinite-type surfaces with mapping class group actions

Abstract (Expand)

We study when the mapping class group of an infinite-type surface S admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on S. We introduce a topological … invariant for infinite-type surfaces that determines in many cases whether there is such an action. This allows us to conclude that, as non-locally compact topological groups, many big mapping class groups have nontrivial coarse geometry in the sense of Rosendal.

Authors: Matthew Gentry Durham, Federica Fanoni, Nicholas G. Vlamis

Date Published: 2018

Publication Type: Book

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

An invitation to higher Teichmüller theory

Abstract

The goal of this article is to invite the reader to get to know and to get involved into higher Teichmüller theory by describing some of its many facets.

Author: Anna Wienhard

Date Published: 2018

Publication Type: Journal

Boundary maps and maximal representations on infinite dimensional Hermitian symmetric spaces

Abstract (Expand)

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the … target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite dimensional totally geodesic subspace on which the action is maximal. In the opposite direction we construct examples of geometrically dense maximal representation in the infinite dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, that we are able to construct in low ranks or under some suitable Zariski-density assumption, circumventing the lack of local compactness in the infinite dimensional setting.

Authors: Bruno Duchesne, Jean Lécureux, Maria Beatrice Pozzetti

Date Published: 2018

Publication Type: Journal

Hitchin components for orbifolds

Abstract (Expand)

We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichmüller spaces. We show that the Hitchin component of an orbifold … group is homeomorphic to an open ball and we compute its dimension explicitly. We then give applications to the study of the pressure metric, cyclic Higgs bundles, and the deformation theory of real projective structures on 3-manifolds.

Authors: Daniele Alessandrini, Gye-Seon Lee, Florent Schaffhauser

Date Published: 2018

Publication Type: Journal

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