Combinatorial $k$-systoles on a punctured tori and a pairs of pants

ElHadji Abdou Aziz Diop, Masseye Gaye, Abdoul Karim Sane

Published 2021-08-18Version 1

In this paper, $S$ denotes a hyperbolic surface homeomorphic to a punctured torus or a pairs of pants. Our interest is the study of \emph{\textbf{combinatorial $k$-systoles}} that is geodesics with self-intersection number greater than $k$ and with minimal combinatorial length. We show that the maximal intersection number $I_k$ of combinatorial $k$-systoles grows like $k$ and $\underset{k\rightarrow+\infty}{\limsup}(I_k(S)-k)=+\infty$. This answer -- in the case of a pairs of pants and a punctured torus -- a weak version of Erlandsson-Palier conjecture, originally stated for the geometric length.