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

Dumas, David¹ ; Sanders, Andrew²

¹ Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, United States
² Mathematisches Institut, Universität Heidelberg, Heidelberg, Germany

Geometry & topology, Tome 24 (2020) no. 4, pp. 1615-1693.

Résumé

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 ${PSL}_3ℂ$–Fuchsian representations are topological fiber bundles over a surface, and we conjecture this holds for all simple $G$.

Détail

DOI : 10.2140/gt.2020.24.1615

Classification : 32Q30, 57M50
Keywords: Anosov representations, complex manifolds, flag varieties

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