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

Dumas, David; Sanders, Andrew

doi:10.2140/gt.2020.24.1615

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

David Dumas and Andrew Sanders

Geometry & Topology 24 (2020) 1615–1693

DOI: 10.2140/gt.2020.24.1615

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 $G$ . We compute the homology of the manifolds obtained from $G$ –Fuchsian representations and their Anosov deformations, where $G$ is simple. We show that in sufficiently high rank, these quotient complex manifolds are not Kähler. We also obtain results about their Picard groups and existence of meromorphic functions.

In a final section, we apply our topological results to some explicit families of domains and derive closed formulas for certain topological invariants. We also show that the manifolds associated to Anosov deformations of ${PSL}_{3} ℂ$ –Fuchsian representations are topological fiber bundles over a surface, and we conjecture this holds for all simple $G$ .

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Keywords

Anosov representations, complex manifolds, flag varieties

Mathematical Subject Classification 2010

Primary: 32Q30, 57M50

References

Publication

Received: 9 June 2017

Revised: 25 June 2019

Accepted: 14 October 2019

Published: 10 November 2020

Proposed: Anna Wienhard

Seconded: Jean-Pierre Otal, Benson Farb

Authors

David Dumas
Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Chicago, IL United States

Andrew Sanders
Mathematisches Institut Universität Heidelberg Heidelberg Germany

Volume 24, issue 4 (2020)