Limit cones of multi-Fuchsian representations

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 ${\mathbf{R}}_{\ge 0}^d$, 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.

28 pages, 8 figures. Added Section 6 on discontinuity of limit cones