Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Entropy rigidity for cusped Hitchin representations

Richard Canary, Tengren Zhang, Andrew Zimmer

TL;DR

The work proves entropy rigidity for Hitchin representations of geometrically finite Fuchsian groups, extending previous closed-surface results to cusped settings. It develops a broad transverse-geometry framework, introducing (1,1,2)-hypertransverse and projectively visible transverse representations, and shows that for these families the conical-limit-sets’ Hausdorff dimension matches the first simple-root entropy, with h^φ(ρ)=δ^φ(ρ) and sharp 1-bounds for simple-root entropies. The paper unifies Kleinian and higher Teichmüller phenomena by connecting shadows, singular-value data, and convex-domain dynamics to entropy functionals, and it proves that simple-root entropies of geometrically finite Hitchin representations are ≤1, attaining equality only for lattices. It also develops the theory of transverse representations, derives upper and lower bounds for shadows and Hausdorff dimensions, and introduces Quint’s indicator set to study the analytic structure of entropy in the Hitchin setting, offering a deep toolkit for entropy rigidity in higher-rank geometries.

Abstract

We establish an entropy rigidity theorem for Hitchin representations of all geometrically finite Fuchsian groups which generalizes a theorem of Potrie and Sambarino for Hitchin representations of closed surface groups. In the process, we introduce the class of (1,1,2)-hypertransverse groups and show for such a group that the Hausdorff dimension of its conical limit set agrees with its (first) simple root entropy, providing a common generalization of results of Bishop and Jones, for Kleinian groups, and Pozzetti, Sambarino and Wienhard, for Anosov groups. We also introduce the theory of transverse representations of projectively visible groups as a tool for studying discrete subgroups of linear groups which are not necessarily Anosov or relatively Anosov.

Entropy rigidity for cusped Hitchin representations

TL;DR

The work proves entropy rigidity for Hitchin representations of geometrically finite Fuchsian groups, extending previous closed-surface results to cusped settings. It develops a broad transverse-geometry framework, introducing (1,1,2)-hypertransverse and projectively visible transverse representations, and shows that for these families the conical-limit-sets’ Hausdorff dimension matches the first simple-root entropy, with h^φ(ρ)=δ^φ(ρ) and sharp 1-bounds for simple-root entropies. The paper unifies Kleinian and higher Teichmüller phenomena by connecting shadows, singular-value data, and convex-domain dynamics to entropy functionals, and it proves that simple-root entropies of geometrically finite Hitchin representations are ≤1, attaining equality only for lattices. It also develops the theory of transverse representations, derives upper and lower bounds for shadows and Hausdorff dimensions, and introduces Quint’s indicator set to study the analytic structure of entropy in the Hitchin setting, offering a deep toolkit for entropy rigidity in higher-rank geometries.

Abstract

We establish an entropy rigidity theorem for Hitchin representations of all geometrically finite Fuchsian groups which generalizes a theorem of Potrie and Sambarino for Hitchin representations of closed surface groups. In the process, we introduce the class of (1,1,2)-hypertransverse groups and show for such a group that the Hausdorff dimension of its conical limit set agrees with its (first) simple root entropy, providing a common generalization of results of Bishop and Jones, for Kleinian groups, and Pozzetti, Sambarino and Wienhard, for Anosov groups. We also introduce the theory of transverse representations of projectively visible groups as a tool for studying discrete subgroups of linear groups which are not necessarily Anosov or relatively Anosov.
Paper Structure (30 sections, 72 theorems, 383 equations, 5 figures)

This paper contains 30 sections, 72 theorems, 383 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Linear Algebra
  4. Cusped Anosov representations of geometrically finite Fuchsian groups
  5. Cusped Hitchin representations of Fuchsian groups
  6. Properly convex domains
  7. Special representations
  8. Skew-symmetric tensors
  9. Hermitian symmetric tensors
  10. Divergent and transverse subgroups
  11. Properties of the limit set
  12. Examples of transverse subgroups
  13. Conical limit points
  14. Transverse representations
  15. Upper bounds on shadows and Hausdorff dimension
  16. ...and 15 more sections

Key Result

Theorem 1.1

If $\Gamma \subset \mathsf{PSL}(2,\mathop{\mathrm{\mathbb{R}}}\nolimits)$ is geometrically finite, $\rho : \Gamma \rightarrow \mathsf{PSL}(d, \mathop{\mathrm{\mathbb{R}}}\nolimits)$ is Hitchin and $\phi = \sum c_j \alpha_j \in \left(\mathfrak{a}^*\right)^+$, then Moreover, equality occurs if and only if $\Gamma$ is a lattice and either

Figures (5)

  • Figure 1: Upper bound on the size of shadow.
  • Figure 2: Lower bound on the size of shadow.
  • Figure 3: $x\in\mathcal{O}_{r_0}(z,b_0)$.
  • Figure 4: $x\in \mathcal{O}_{2r_0+1}(b_0,w)$.
  • Figure 5: $D$, $H^-$, and $H^+$.

Theorems & Definitions (124)

  • Theorem 1.1: see Theorem \ref{['thm:main in body']}
  • Theorem 1.2: see Theorem \ref{['transverse image of visible1']}
  • Theorem 1.3: see Corollary \ref{['prop:upper_bound']}
  • Theorem 1.4: see Theorem \ref{['thm:KMO general in body']}
  • Theorem 1.5: see Theorem \ref{['thm:lower_bd']}
  • Proposition 1.6: see Proposition \ref{['pairwise transverse']}
  • Corollary 1.7: see Corollary \ref{['PSgen in body']} and Theorem \ref{['thm:main in body']}
  • Proposition 1.8: see Proposition \ref{['entropy and critical exponent']}
  • Proposition 1.9: see Proposition \ref{['prop: entropy drop']}
  • Corollary 1.10
  • ...and 114 more