Limit cones of multi-Fuchsian representations

TL;DR

This paper investigates the geometric structure of limit cones associated with multi-Fuchsian representations into $( ext{PSL}_2 extbf{R})^d$, revealing diverse regimes and discontinuities in their shape.

Contribution

It characterizes the shape and complexity of limit cones for multi-Fuchsian representations, including finite-sided cones, dense extremal rays, and discontinuous variations.

Findings

Limit cones can have finitely many sides, with the number growing with genus or free rank.

Extremal rays can be dense in the boundary of the limit cone.

Examples show the limit cone can vary discontinuously with the representation.

Abstract

We study the set of normalized multi-lengths for representations of closed surface groups and free groups into $(\mathrm{PSL}_2\mathbf{R})^d$ whose projections to $\mathrm{PSL}_2\mathbf{R}$ are all convex cocompact. These multi-lengths define a convex cone in $\mathbf{R}^d_{\geq 0}$, called the limit cone. When $d=3$, we show the coexistence of different regimes: for some representations the limit cone has only a finite number of sides, which we can force to grow like the genus (or free rank); for other representations, extremal rays are dense in the boundary of the limit cone. We also give examples where the limit cone varies discontinuously with the representation.