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Limit cones of multi-Fuchsian representations

Jeffrey Danciger, François Guéritaud, Fanny Kassel

TL;DR

The paper investigates the limit cone Λρ for multi-Fuchsian representations ρ: π1(S) → (PSL2)^d, focusing on how its projectivization can be polygonal or strictly convex in the d=3 case. It develops a framework using azimuthal supporting hyperplanes and the simple hull Λρ^s to certify boundary points, connecting to Thurston’s asymmetric metric in lower rank. It constructs explicit finite-sided limit cones for higher genus and a strictly convex cone in the one-holed-torus setting, employing geodesic currents, Markoff-type combinatorics, and triangle slack estimates. The results illuminate the diverse geometric shapes Λρ can assume, revealing a rich interplay between simple laminations, currents, and the asymptotic growth of multi-length data in higher Teichmüller theory.

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.

Limit cones of multi-Fuchsian representations

TL;DR

The paper investigates the limit cone Λρ for multi-Fuchsian representations ρ: π1(S) → (PSL2)^d, focusing on how its projectivization can be polygonal or strictly convex in the d=3 case. It develops a framework using azimuthal supporting hyperplanes and the simple hull Λρ^s to certify boundary points, connecting to Thurston’s asymmetric metric in lower rank. It constructs explicit finite-sided limit cones for higher genus and a strictly convex cone in the one-holed-torus setting, employing geodesic currents, Markoff-type combinatorics, and triangle slack estimates. The results illuminate the diverse geometric shapes Λρ can assume, revealing a rich interplay between simple laminations, currents, and the asymptotic growth of multi-length data in higher Teichmüller theory.

Abstract

We study the set of normalized multi-lengths for representations of closed surface groups and free groups into whose projections to are all convex cocompact. These multi-lengths define a convex cone in , called the limit cone. When , 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.

Paper Structure

This paper contains 24 sections, 13 theorems, 49 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The limit cone
  3. Main results
  4. Finding supporting hyperplanes to the limit cone
  5. Preliminaries
  6. Reminders: geodesic currents
  7. Basic facts about multi-Fuchsian representations
  8. Asymptotic slack
  9. A simpleness criterion
  10. Azimuthality and slacks
  11. Surgeries on long curves and slacks
  12. Self-intersecting currents and long crossing curves
  13. Proof of Theorem \ref{['thm:azimut-limit-cone-switched']}
  14. Proof of Theorem \ref{['thm:azimut-limit-cone']}
  15. Finite-sided limit cones
  16. ...and 9 more sections

Key Result

Theorem 1.2

For any $g\geq 2$, let $S_g$ be a closed orientable surface of genus $g$. Then there exist multi-Fuchsian representations $\boldsymbol{\rho} : \pi_1(S_g) \to (\mathrm{PSL}_2\mathbf{R})^3$ whose projectivized limit cone $\mathbf{P} \Lambda_{\boldsymbol{\rho}}$ is a finite polygon with at least $4g-1$

Figures (8)

  • Figure 1: Some projectivized co-oriented hyperplanes in $\mathbf{P}(\mathbf{R}^3_{\geq 0})$: we shade off their negative sides. Green: azimuthal. Red: non-azimuthal.
  • Figure 2: Resolutions $[\gamma']$ and $[\gamma"] \cup [\gamma"']$ of a self-crossing of $[\gamma]$ in an immersed pair of pants in $\rho(\pi_1(S)) \backslash \mathbb{H}^2 \simeq S$
  • Figure 3: A thin right-angled convex hyperbolic $(2g+2)$-gon $P(x_0, \dots, x_{2g-2})$, with a chord $c$ (dotted)
  • Figure 4: The projectivized limit cone $\mathbf{P} \Lambda_{\boldsymbol{\rho}} = \mathbf{P} \Lambda_{\boldsymbol{\rho}}^s$ of a multi-Fuchsian representation $\boldsymbol{\rho} = (\rho_{P_1}, \rho_{P_2}, \rho_{P_3}) : \pi_1(S_P)\to (\mathrm{PSL}_2\mathbf{R})^3$ as in Section \ref{['subsec:g-gon']}. It is a polygon with $\geq 4g-1$ sides. Here $g=3$.
  • Figure 5: The Markoff tree (black) superimposed on the Farey triangulation (white) in the hyperbolic plane (gray)
  • ...and 3 more figures

Theorems & Definitions (30)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 20 more