Metrics on Hitchin Components from Hölder Distortion

Abstract

We observe Thurston's asymmetric metric on Teichmüller space may be expressed in terms of the Hölder regularity of boundary maps. We then associate $2$-dimensional stratified loci in $\mathbb{RP}^{n-1}$ to $\text{PSL}_n(\mathbb{R})$ Hitchin representations with $n > 3$. We prove that measuring the relative Hölder distortion of these loci gives asymmetric metrics on the Hitchin component $\text{Hit}_n(S)$ with complete and geometrically meaningful symmetrizations. These are the first known geometrically significant complete metrics on $\text{Hit}_n(S)$ for $n > 3$.

Figures (1)

• Figure 1: Left: the part of $\mathcal{B}_\rho$ for a $\text{PSL}_4(\mathbb R)$-Hitchin representation in a coordinate box in an affine chart for $\mathbb R \mathbb P^3$. This is the "Fuchsian" example, i.e. is preserved by an irreducible $\text{PSL}_2(\mathbb R)$-subgroup of $\text{PSL}_4(\mathbb R)$. The singular curve is the image of $\xi^1_\rho$. Right: sketch of the intersection of $\mathcal{B}_\rho$ with a $3$-dimensional subspace of the form $\xi^4_\rho(x)$ for a $\text{PSL}_8(\mathbb R)$-Hitchin representation $\rho$. The point where the drawn curves intersect is $\xi^1_\rho(x)$. The term "bouquet" is motivated by how the tori $T_\rho^k$ come together near the image of $\xi^1_\rho.$

Theorems & Definitions (65)

• Definition 1.1
• Corollary 1.2: sorvali1973dilatation, wolpert1979length
• proof
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Theorem 2.2: Coupling is Stretch
• proof
• Claim 2.3: Proximality Refinement
• proof