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

Abstract:

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.

SEEK ID: https://publications.h-its.org/publications/956

Research Groups: Groups and Geometry

Publication type: Misc

Journal: (to be published in Geometry and Topology)

Citation: arXiv,math.GT,1704.01091

Date Published: 2017

URL: https://arxiv.org/abs/1704.01091

Registered Mode: imported from a bibtex file

Authors: David Dumas, Andrew Sanders

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Created: 30th Jan 2020 at 09:39

Last updated: 5th Mar 2024 at 21:24

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