Affine laminations and coaffine representations

TL;DR

This paper investigates surface subgroups of SL(4,R) acting convex cocompactly on projective space, analyzing their boundary stratification and bending laminations, revealing a rich geometric structure related to convex projective structures and affine laminations.

Contribution

It introduces a detailed study of bending data and laminations for convex cocompact surface subgroups in SL(4,R), linking them to convex projective structures and holonomy representations.

Findings

The boundary of the convex core has a stratified structure with bending laminations.

The space of compatible bending data forms a sphere of dimension 6g-7.

Bending data include convex RP^2 structures and affine measured laminations.

Abstract

We study surface subgroups of $\mathrm{SL}(4,\mathbb R)$ acting convex cocompactly on $\mathbb R \textrm P^3$ with image in the coaffine group. The boundary of the convex core is stratified, and the one dimensional strata form a pair of bending laminations. We show that the bending data on each component consist of a convex $\mathbb R \textrm P^2$ structure and an affine measured lamination depending on the underlying convex projective structure on $S$ with (Hitchin) holonomy $\rho: \pi_1S \to \mathrm{SL}(3,\mathbb R)$. We study the space $\mathcal {ML}^\rho(S)$ of bending data compatible with $\rho$ and prove that its projectivization is a sphere of dimension $6g-7$.