Mathematical Research Letters

Volume 24 (2017)

Number 3

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

Pages: 593 – 617



Tarik Aougab (Department of Mathematics, Brown University, Providence, Rhode Island, U.S.A.)

Jonah Gaster (Department of Mathematics, Boston College, Boston, Massachusetts, U.S.A.)

Priyam Patel (Department of Mathematics, University of California at Santa Barbara)

Jenya Sapir (Department of Mathematics, University of Illinois, Urbana, Il., U.S.A.)


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 $\mathcal{S}$.

Received 27 May 2016

Accepted 24 October 2016

Published 1 September 2017