Volume 24, issue 4 (2020)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
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 ${PSL}_{3}ℂ$–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
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