Simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links

TL;DR

This paper investigates the homotopy properties of simple loops on 2-bridge spheres within Heckoid orbifolds associated with 2-bridge links, providing criteria for null-homotopy and homotopy, and exploring applications to character varieties and group epimorphisms.

Contribution

It introduces new criteria for when simple loops are null-homotopic or homotopic in Heckoid orbifolds of 2-bridge links, extending Riley's work and connecting to various geometric and algebraic applications.

Findings

Criteria for null-homotopic loops in Heckoid orbifolds.

Conditions for homotopy between simple loops.

Applications to character varieties and group epimorphisms.

Abstract

Following Riley's work, for each 2-bridge link $K(r)$ of slope $r\in\QQ$ and an integer or a half-integer $n$ greater than 1, we introduce the {\it Heckoid orbifold $\orbs(r;n)$} and the {\it Heckoid group $\Hecke(r;n)=\pi_1(\orbs(r;n))$ of index $n$ for $K(r)$}. When $n$ is an integer, $\orbs(r;n)$ is called an {\it 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 $\PConway$ in $\orbs(r;n)$ determined by the 2-bridge sphere of $K(r)$, when is it null-homotopic in $\orbs(r;n)$? (2) For two distinct essential simple loops on $\PConway$, when are they homotopic in $\orbs(r;n)$? We also announce applications of these results to character varieties, McShane's identity, and epimorphisms from 2-bridge link groups onto Heckoid groups.