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

Abstract

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

The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces.With the appendix by Khalid Bou–Rabee

“…Part (3) directly implies part (2). Since the simplicial splittings that appear in Definition 2·12 are very small, part (2) also directly implies part (1).…”

Bounds on the Number of Non-Simple Closed Geodesics on a Surface

“…Note that despite the choice we must make, this function is still well-defined. In [AGPS16], we show that for each orbit…”

“…Basmajian shows the lower bound, and proves that it is tight, in [Bas13]. The fact that l(γ short ) ≤ c 2 K is shown in [AGPS16]. Another, simpler, method to show this upper bound was communicated to us by Malestein [Mal].…”

Short closed geodesics with self-intersections

“…Note that if X is closed, setting ε := min{ 1 2 , sys(X) 2 } where sys(X) is the systole length of X, then X = X T . In particular γ is entirely contained in the thick part of X and we have a lower bound on its length that grows like the root of its intersection.…”