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

Donghi Lee; Makoto Sakuma

doi:10.3934/era.2012.19.97

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2012, Volume 19: 97-111. Doi: 10.3934/era.2012.19.97

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

Donghi Lee^1, and
Makoto Sakuma^2,

1.
Department of Mathematics, Pusan National University, San-30 Jangjeon-Dong, Geumjung-Gu, Pusan, 609-735
2.
Department of Mathematics,, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526

Received: May 31, 2012

Published: December 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 57M25; Secondary: 20F06.

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