A closed ball compactification of a maximal component via cores of trees

TL;DR

This paper introduces a new compactification of the maximal component in the character variety of surface group representations into a product of Lie groups, showing it forms a closed ball with a well-understood boundary and group action.

Contribution

It constructs a closed ball compactification of the maximal component and describes its boundary points as geometric mixed structures, analyzing the mapping class group action.

Findings

The compactification is a closed ball.

The boundary points are described as mixed structures.

The mapping class group acts on the compactification.

Abstract

We show that, in the character variety of surface group representations into the Lie group $\mathrm{PSL}(2,\mathbb{R}) \times \mathrm{PSL}(2,\mathbb{R})$, the compactification of the maximal component introduced by the second author is a closed ball upon which the mapping class group acts. We study the dynamics of this action. Finally, we describe the boundary points geometrically as $(\overline{A_{1} \times A_{1}},2)$-valued mixed structures.