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Quasi-isometric embeddings for shrinking maps from surfaces into the moduli space

Yibo Zhang

TL;DR

The paper develops a comprehensive framework to compare shrinking maps from a cusped hyperbolic surface $B$ into the moduli space $\mathcal{M}_h$ with their lifts to Teichmüller space $\mathcal{T}_h$, focusing on three quasi-isometric notions: $\text{q.i.i.}(F)$, $\widetilde{\text{q.i.i.}}(F)$, and $\widetilde{\text{q.i.e.}}(F)$. It shows that for holomorphic $F$, all three properties are completely governed by peripheral monodromies: $F$ satisfies $\text{q.i.i.}(F)$ (and its lifted versions) if and only if these peripheral monodromies have infinite order, with the monodromy criterion extending to the quasi-isometric embedding of $F_*$ into $\mathrm{Mod}_h$ acting on $\mathcal{T}_h$. The work extends these characterisations to $\lambda_0$-shrinking differentiable maps, establishing precise cusp-region and compact-region estimates, and proving that shrinking maps are quasi-isometric immersions with quasi-isometric immersed or embedded lifts under natural monodromy assumptions. By linking monodromy, cusp geometry, and Teichmüller dynamics, the results provide a complete characterization of when shrinking maps and their lifts preserve coarse geometric information, with Teichmüller curves illustrating when isometric lifts arise. The findings contribute a rigorous bridge between hyperbolic geometry, mapping class group actions, and moduli-space geometry, and have potential implications for understanding holomorphic curves in moduli spaces and their geometric embeddings.

Abstract

We investigate shrinking maps from a cusped hyperbolic surface into the moduli space of closed Riemann surfaces. For such a map and its lift to the Teichmüller space, we consider whether they are quasi-isometric embeddings with respect to natural metrics like the Teichmüller distance and the intrinsic distance. Under a mild condition, we prove that these properties are characterised solely by the map's monodromy. These characterisations apply, in particular, to holomorphic maps.

Quasi-isometric embeddings for shrinking maps from surfaces into the moduli space

TL;DR

The paper develops a comprehensive framework to compare shrinking maps from a cusped hyperbolic surface into the moduli space with their lifts to Teichmüller space , focusing on three quasi-isometric notions: , , and . It shows that for holomorphic , all three properties are completely governed by peripheral monodromies: satisfies (and its lifted versions) if and only if these peripheral monodromies have infinite order, with the monodromy criterion extending to the quasi-isometric embedding of into acting on . The work extends these characterisations to -shrinking differentiable maps, establishing precise cusp-region and compact-region estimates, and proving that shrinking maps are quasi-isometric immersions with quasi-isometric immersed or embedded lifts under natural monodromy assumptions. By linking monodromy, cusp geometry, and Teichmüller dynamics, the results provide a complete characterization of when shrinking maps and their lifts preserve coarse geometric information, with Teichmüller curves illustrating when isometric lifts arise. The findings contribute a rigorous bridge between hyperbolic geometry, mapping class group actions, and moduli-space geometry, and have potential implications for understanding holomorphic curves in moduli spaces and their geometric embeddings.

Abstract

We investigate shrinking maps from a cusped hyperbolic surface into the moduli space of closed Riemann surfaces. For such a map and its lift to the Teichmüller space, we consider whether they are quasi-isometric embeddings with respect to natural metrics like the Teichmüller distance and the intrinsic distance. Under a mild condition, we prove that these properties are characterised solely by the map's monodromy. These characterisations apply, in particular, to holomorphic maps.

Paper Structure

This paper contains 19 sections, 17 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quasi-isometric embeddings for a holomorphic map
  3. Quasi-isometric embeddings for a shrinking map
  4. Preliminary results
  5. Hyperbolic geodesics
  6. Teichmüller distance
  7. Relations between quasi-isometric properties
  8. Monodromy representations
  9. Peripheral monodromy
  10. Peripheral monodromies of finite order
  11. Peripheral monodromies of infinite order
  12. Quasi-isometric embedding of the cusp region
  13. Quasi-isometric lengths along the path
  14. Bi-Lipschitz lengths along the path joining boundary points
  15. Conclusions
  16. ...and 4 more sections

Key Result

Theorem A

Let $F:B\rightarrow \mathcal{M}_h$ be a holomorphic map, where $B$ is a hyperbolic surface of type $(g,n)$ with $g\ge 2$, $n\ge 0$ and $h\ge 2$. Then $F$ satisfies $\text{q.i.i.}(F)$ if and only if all peripheral monodromies of $F$ are of infinite order.

Figures (1)

  • Figure 1: Different selections of the geodesic segment $\kappa$ joining the boundaries of two given cusp regions $U_1$ and $U_2$, each providing a pair of peripheral monodromies $(\phi_1,\phi_2)$.

Theorems & Definitions (47)

  • Remark 1.1
  • Remark 1.2
  • Theorem A: Theorem A - 2 in Zhang2024holomorphic
  • Theorem B
  • Theorem C
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 1.7
  • ...and 37 more