SIMPLE LOOPS ON 2-BRIDGE SPHERES IN HECKOID ORBIFOLDS FOR 2-BRIDGE LINKS

Following Riley's work, for each $2$-bridge link $K(r)$ of slope $r∈\mathbb{R}$ and an integer or a half-integer $n$ greater than $1$, we introduce the Heckoid orbifold $S(r;n)$ and the Heckoid group $G(r;n)=\pi_1(S(r;n))$ of index $n$ for $K(r)$ . When $n$ is an integer, $S(r;n)$ is called an even Heckoid orbifold; in this case, the underlying space is the exterior of $K(r)$, and the singular set is the lower tunnel of $K(r)$ with index $n$. The main purpose of this note is to announce answers to the following questions for even Heckoid orbifolds. (1) For an essential simple loop on a $4$-punctured sphere $S$ in $S(r;n)$ determined by the $2$-bridge sphere of $K(r)$, when is it null-homotopic in $S(r;n)$? (2) For two distinct essential simple loops on $S$, when are they homotopic in $S(r;n)$? We also announce applications of these results to character varieties, McShane's identity, and epimorphisms from $2$-bridge link groups onto Heckoid groups.