On the spectrum of the number of geodesics and tight geodesics in the curve complex

TL;DR

This paper investigates the range of the number of geodesics and tight geodesics in the curve complex of a surface, revealing their relationship and how they depend on surface parameters.

Contribution

It establishes the inclusion relationship between the spectra of geodesics and tight geodesics and fully characterizes the spectrum for length 2 in terms of surface genus and punctures.

Findings

The spectrum of geodesics is contained within the spectrum of tight geodesics.

For length 2, the spectra of geodesics and tight geodesics are equal.

The spectra for length 2 are explicitly determined by the surface's genus and number of punctures.

Abstract

Let $S$ be an oriented surface of type $(g, n)$. We are interested in geodesics in the curve complex $\mathcal C(S)$ of $S$. In general, two $0$-simplexes in $\mathcal C(S)$ have infinitely many geodesics connecting the two simplexes while another geodesics called tight geodesics are always finitely many. On the other hand, we may find two $0$-simplexes in $\mathcal C(S)$ so that they have only finitely many geodesics between them. In this paper, we consider the spectrum of the number of geodesics with length $d (\geq 2)$ in $\mathcal C(S)$ and tight geodesics, which is denoted by $\mathfrak{Sp}_d(S)$ and $\mathfrak{Sp}_d^T(S)$, respectively. In our main theorem, it is shown that $\mathfrak{Sp}_d(S) \subset \mathfrak{Sp}_d^T(S)$ in general, but $\mathfrak{Sp}_2(S)= \mathfrak{Sp}_2^T(S)$. Moreover, we show that $\mathfrak{Sp}_2(S)$ and $\mathfrak{Sp}_2^T(g, n)$ are completely determined in terms of $(g, n)$.