ON FINITENESS OF THE GEODESICS JOINING A PAIR OF POINTS IN CURVE COMPLEX

Abstract

Let $S$ be a genus $g$ orientable surface with $c$ boundary components and $p$ punctures. Then $\mathcal{C}^0(S)$ denotes the 0 -skeleton of the curve complex of $S$. This paper presents the following results:
(1) If " $g=1, c+p \geq 3$ " or " $g \geq 2, c+p \geq 1$ ", then, there exist $a_0, a_2 \in \mathcal{C}^0(S)$ such that $d_S\left(a_0, a_2\right)=2$ and the number of the geodesics joining $a_0$ and $a_2$ is exactly 2 .
(2) If " $g=2, c+p \geq 1$ " or " $g \geq 3$ ", then, there exist $a_0, a_2 \in$ $\mathcal{C}^0(S)$ such that $d_S\left(a_0, a_2\right)=2$ and the number of the geodesics joining $a_0$ and $a_2$ is exactly 3 .

Received: March 13, 2024
Accepted: April 12, 2024