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 $G(r;n)=\pi_1(S(r;n))$ ofindex $n$ 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 tothe 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 loopson $S$, when are they homotopic in $S(r;n)$?We also announce applications of these results tocharacter varieties, McShane's identity, andepimorphisms from $2$-bridge link groups onto Heckoid groups.
Highlights
In the late 70’s, Robert Riley made a pioneering exploration of groups generated by two parabolic transformations. (His personal account of background history of the exploration can be found in [37].) The computer-drawn Figure 1 has been circulated among the experts and has inspired many researchers in the fields of Kleinian groups and knot theory
In [36], in which he mentions to the computer experiments and describes the algebra behind the experiments, Riley introduced an infinite collection of Laurent polynomials, which he calls the Heckoid polynomials associated with a 2-bridge link K, and observed that these Heckoid polynomials define the affine representation varieties of certain groups, the Heckoid groups for K
The classification of non-free 2-parabolic generator Kleinian groups announced by Agol [2], which generalizes the characterization of 2-bridge links obtained by [1, 7], justifies Riley’s anticipation
Summary
In the late 70’s, Robert Riley made a pioneering exploration of groups generated by two parabolic transformations. (His personal account of background history of the exploration can be found in [37].) The computer-drawn Figure 1 has been circulated among the experts and has inspired many researchers in the fields of Kleinian groups and knot theory. The classification of non-free 2-parabolic generator Kleinian groups announced by Agol [2], which generalizes the characterization of 2-bridge links obtained by [1, 7], justifies Riley’s anticipation. As noted in [2], the proof is analogous to the arguments in [14, Proof of Theorem 9] By this theorem and the topological description of odd Heckoid orbifolds ([24, Proposition 5.3]), we obtain the following proposition, which shows a significant difference between odd and even Heckoid groups (see [24, Section 6]).
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