Limits of Cubic Differentials and Buildings

TL;DR

This paper studies the asymptotic behavior of surface group representations into SL(3,R) using cubic differentials, revealing convergence properties of harmonic maps and geometric structures related to Hitchin components.

Contribution

It establishes an asymptotic formula for holonomy, describes the limit of harmonic maps into asymptotic cones, and introduces a new compactification for the Hitchin component for triangle groups.

Findings

Holonomy along rays can be expressed asymptotically via local invariants of cubic differentials.

Harmonic maps converge to maps into the asymptotic cone of the symmetric space.

The geometric image is a weakly convex, one-third translation surface.

Abstract

In the Labourie-Loftin parametrization of the Hitchin component of surface group representations into SL(3,R), we prove an asymptotic formula for holonomy along rays in terms of local invariants of the holomorphic differential defining that ray. Globally, we show that the corresponding family of equivariant harmonic maps to a symmetric space converge to a harmonic map into the asymptotic cone of that space. The geometry of the image may also be described by that differential: it is weakly convex and a (one-third) translation surface. We define a compactification of the Hitchin component in this setting for triangle groups that respects the parametrization by Hitchin differentials.