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62 Publications visible to you, out of a total of 62
Bounded cohomology via partial differential equations, I

Abstract (Expand)

We present a new technique that employs partial differential equations in order to explicitly construct primitives in the continuous bounded cohomology of Lie groups. As anapplication, we prove a … a vanishing theorem for the continuous bounded cohomology of SL(2,R) in degree four, establishing a special case of a conjecture of Monod.

Authors: Tobias Hartnick, Andreas Ott

Date Published: 2015

Publication Type: Journal

Compactification of certain Clifford-Klein forms of reductive homogeneous spaces

Abstract (Expand)

We describe smooth compactifications of certain families of reductive homogeneous spaces such as group manifolds for classical Lie groups, or pseudo-Riemannian analogues of real hyperbolic spaces and … their complex and quaternionic counterparts. We deduce compactifications of Clifford-Klein forms of these homogeneous spaces, namely quotients by discrete groups Gamma acting properly discontinuously, in the case that Gamma is word hyperbolic and acts via an Anosov representation. In particular, these Clifford-Klein forms are topologically tame.

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

Date Published: 2015

Publication Type: Misc

ADS 3-manifolds and Higgs bundles

Abstract (Expand)

In this paper we investigate the relationships between closed \mathrmAdS 3-manifolds and Higgs bundles. We have a new way to construct AdS structures that allows us to see many of their properties … explicitly, for example we can recover the very recent formula by Tholozan for the volumes. We also find applications to the theory of minimal immersions into quadrics with their natural pseudo-Riemannian structure: using the geometry of the \mathrmAdS manifolds we can characterize the representations admitting equivariant minimal immersions of the Poincaré disc into the Klein quadric, the Grassmannian \mathrmGr(2,4), and understand the geometry of these minimal immersions.

Authors: Daniele Alessandrini, Qiongling Li

Date Published: 2015

Publication Type: Misc

Projective deformations of weakly orderable hyperbolic Coxeter orbifolds

Abstract (Expand)

A Coxeter n–orbifold is an n–dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order m, whose neighborhood is locally modeled … on \mathbbR^n modulo the dihedral group of order 2m generated by two reflections. For n ≥3, we study the deformation space of real projective structures on a compact Coxeter n–orbifold Q admitting a hyperbolic structure. Let e_+(Q) be the number of ridges of order greater than or equal to 3. A neighborhood of the hyperbolic structure in the deformation space is a cell of dimension e_+(Q)-n if n=3 and Q is weakly orderable, ie the faces of Q can be ordered so that each face contains at most 3 edges of order 2 in faces of higher indices, or Q is based on a truncation polytope.

Authors: Suhyoung Choi, Gye-Seon Lee

Date Published: 2015

Publication Type: Journal

2π–grafting and complex projective structures, I

Abstract (Expand)

Let S be a closed oriented surface of genus at least two. Gallo, Kapovich and Marden asked whether 2\pi–grafting produces all projective structures on S with arbitrarily fixed holonomy (the Grafting … conjecture). In this paper, we show that the conjecture holds true “locally” in the space \mathcalGL of geodesic laminations on S via a natural projection of projective structures on S into \mathcalGL in Thurston coordinates. In a sequel paper, using this local solution, we prove the conjecture for generic holonomy.

Author: Shinpei Baba

Date Published: 2015

Publication Type: Journal

Milnor-Wood type inequalities for Higgs bundles

Abstract (Expand)

We explain how the generalized Milnor–Wood inequality of Burger and Iozzi for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates under the non-abelian … Hodge correspondence into an inequality for topological invariants of the corresponding Higgs bundles. In this manner, we obtain in a uniform way a universal Milnor–Wood inequality for Higgs bundles over complex-hyperbolic manifolds of arbitrary dimensions and with arbitrary Hermitian structure group. This complements results of Biquard, Bradlow, García-Prada, Gothen, Mundet, Rubio and Chaput, Koziarz, Maubon.

Authors: Tobias Hartnick, Andreas Ott

Date Published: 2011

Publication Type: Misc

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