Geometry of hyperconvex representations of surface groups

TL;DR

This paper explores the geometric properties of hyperconvex surface group representations in complex projective groups, extending classical maps and analyzing limit set dimensions to understand their deformation spaces.

Contribution

It introduces a holomorphic extension of the Ahlfors--Bers map for hyperconvex representations and characterizes when their limit sets have Hausdorff dimension 1.

Findings

Holomorphic extension of the Ahlfors--Bers map to Teichmüller space products

Limit set Hausdorff dimension equals 1 iff conjugate to real projective group

Provides insights into deformation spaces of hyperconvex representations

Abstract

We study the geometry of hyperconvex representations of surface groups in ${\rm PSL}(d,\mathbb{C})$ and their deformation spaces: We produce a natural holomorphic extension of the classical Ahlfors--Bers map to a product of Teichm\"uller spaces of a canonical Riemann surface lamination and prove that the limit set of a hyperconvex representation in the full flag space has Hausdorff dimension 1 if and only if the representation is conjugate in ${\rm PSL}(d,\mathbb{R})$.