Shortest k-Geodesics on Hyperbolic Surfaces

TL;DR

The paper addresses the link between the length of the shortest non-simple closed geodesics on a hyperbolic surface and their self-intersection number, introducing a constructive approach rooted in the pair of pants determined by a shortest figure-eight curve. By explicitly building words in $\pi_1(X)$ (via the words $w(m,n,j)$ and $w'(m,n,j)$) that realize prescribed self-intersection counts and analyzing their self-intersection via a combinatorial framework, the authors derive new sharp bounds: $s_k(X) \le \left( (k+1/4)^{1/2} + (3/\sqrt{2})(k+1/4)^{1/4} - 1/2 \right) L_8(X)$, with a simpler form $s_k(X) < (\sqrt{k} + \tfrac{3}{\sqrt{2}} k^{1/4}) L_8(X)$. These estimates yield a tighter upper bound for $I_k(X)$, reducing the asymptotic constant from 512 to 128, thereby strengthening the quantitative connection between geometric length and combinatorial complexity of non-simple geodesics. The results advance understanding of how geodesic length interacts with self-intersection and offer a concrete method to translate combinatorial data into geometric bounds on hyperbolic surfaces.

Abstract

We study the relationship between the lengths of closed geodesics on hyperbolic surfaces and their topological complexity, measured by the self-intersection number. In particular, we provide explicit upper bounds for the length $s_k(X)$ of a shortest closed geodesic with exactly $k$ self-intersections in terms of the length $L_\textswab{8}(X)$ of a shortest figure eight curve, improving Basmajian's estimate. We analyze the geometry of a shortest figure eight curve and explicitly build families of words in $π_1(X)$ whose geodesic representatives realize prescribed self-intersection numbers. As a consequence, we improve existing estimates on the maximal self-intersection number $I_k(X)$ of shortest geodesics with at least $k$ self-intersections, reducing the asymptotic upper bound from 512 to 128. This provides a sharper quantitative connection between the geometry and combinatorial complexity of non-simple closed geodesics on hyperbolic surfaces.

Figures (3)

• Figure 1: A shortest figure eight curve $\gamma_\textswab{8}$ (in red) determines an embedded pair of pants $P$ (in blue); The loops $a$ and $b$ are the subloops of $\gamma_\textswab{8}$ generating $\pi_1(P)$ and the word $ab$ represents $\gamma_\textswab{8}$.
• Figure 2: A curve represented by the word $w(3,3,3)$, drawn so that it realizes its self-intersection number, as described in Lemma \ref{['self intersection number of w']}
• Figure 3: For $1\le k\le m$ and $1\le l\le n$, the infinite words $x_kx_kx_k\cdots$ and $x_{m+l}x_{m+l}\cdots$, as well as their inverses, initially agree along the black arrows before diverging. Hence, $\alpha_k \cap \alpha_{m+l}\neq0$.

Theorems & Definitions (20)

• Theorem 1.1: =Theorem\ref{['s_k L_8']}
• Theorem 1.2: =Theorem \ref{['I_k']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Definition 3.1
• Definition 3.2
• Theorem 3.3: diop2022self