Volume 24, issue 4 (2020)

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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
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 PSL3–Fuchsian representations are topological fiber bundles over a surface, and we conjecture this holds for all simple G.

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