We give two flexible and degenerate constructions related to a theorem of Thurston. First, we produce geodesic segments for Thurston's asymmetric metric on Teichm\"uller space $\mathcal{ T}(S_g)$ that remain geodesics after adding arbitrary $\varepsilon$-Lipschitz noise to all but one Fenchel-Nielsen coordinate. Then, for all $2 < n \leq 3g-3$ we construct open sets in $\mathcal{ T}(S_g)^n$ for which the limit cones of the corresponding representations in $\mathrm{ PSL}_2(\mathbb{ R})^n$ are cones over explicit finite-sided polyhedra. Each construction is as degenerate as possible and has applications to the basic structure and local non-rigidity of the involved objects.
Categories: math.GT math.DG