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

## TL;DR

This paper investigates the topology and geometry of compact complex manifolds derived from Anosov representations of surface groups in complex semisimple Lie groups, revealing their homological properties, non-Kähler nature, and complex structure features.

## Contribution

It provides explicit computations of homology, Picard groups, and cohomology for these manifolds, and introduces new topological invariants and conjectures about their fiber bundle structures.

## Key findings

- Computed homology of domains and quotients for Fuchsian representations
- Showed quotient manifolds are not Kähler for certain groups
- Identified conditions for existence of nonconstant meromorphic functions

## 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 K\"{a}hler. 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 $\mathrm{PSL}_3(\mathbb{C})$ is a fiber bundle over a surface, and we conjecture that this holds for all simple $G$.