Building hyperbolic metrics suited to closed curves and applications to lifting simply

TL;DR

This paper constructs hyperbolic metrics on surfaces that control the length and self-intersection properties of closed curves, providing bounds on covers where these curves lift simply, with applications to hyperbolic geometry and surface topology.

Contribution

It introduces new hyperbolic metrics tailored to curves with self-intersections, establishing bounds on their lengths and lifting degrees, advancing understanding of curve behavior on hyperbolic surfaces.

Findings

Constructed hyperbolic metrics with length bounds proportional to square root of self-intersections

Established linear bounds on cover degrees for lifting curves simply

Provided bounds relating curve length on cusped surfaces to cover degrees

Abstract

Let $\gamma$ be an essential closed curve with at most $k$ self-intersections on a surface $\mathcal{S}$ with negative Euler characteristic. In this paper, we construct a hyperbolic metric $\rho$ for which $\gamma$ has length at most $M \cdot \sqrt{k}$, where $M$ is a constant depending only on the topology of $\mathcal{S}$. Moreover, the injectivity radius of $\rho$ is at least $1/(2\sqrt{k})$. This yields linear upper bounds in terms of self-intersection number on the minimum degree of a cover to which $\gamma$ lifts as a simple closed curve (i.e. lifts simply). We also show that if $\gamma$ is a closed curve with length at most $L$ on a cusped hyperbolic surface $\mathcal{S}$, then there exists a cover of $\mathcal{S}$ of degree at most $N \cdot L \cdot e^{L/2}$ to which $\gamma$ lifts simply, for $N$ depending only on the topology of $\mathcal{S}$.