Epimorphisms from 2-bridge link groups onto Heckoid groups (II)

Abstract

In Part I of this series of papers, we made Riley's definition of Heckoid groups for 2-bridge links explicit, and gave a systematic construction of epimorphisms from 2-bridge link groups onto Heckoid groups, generalizing Riley's construction. In this paper, we give a complete characterization of upper-meridian-pair-preserving epimorphisms from 2-bridge link groups onto even Heckoid groups, by proving that they are exactly the epimorphisms obtained by the systematic construction.

Epimorphisms from 2-bridge link groups onto Heckoid groups (I)

“…The result (iii) is an analogy of the systematic construction of epimorphisms between 2-bridge link groups given in [14,Theorem 1.1]. In the sequel [10] of this paper, we prove, by using small cancellation theory, that the epimorphisms in Theorem 2.3 are the only upper-meridian-pair-preserving epimorphisms from 2-bridge link groups onto even Heckoid groups. This in turn forms an analogy of [9,Main Theorem 2.4], which gives a complete characterization of upper-meridian-pair-preserving epimorphisms between 2-bridge link groups.…”

Simple Loops on 2-Bridge Spheres in Heckoid Orbifolds for the Trivial Knot

“…For r a rational number and n an integer or a half-integer greater than 1, let C r (2n) be the group of automorphisms of D generated by the parabolic transformation, centered on the vertex r, by 2n units in the clockwise direction, and let Γ(r; n) be the group generated by Γ ∞ and C r (2n). The answer to Question 2.1 obtained in [8,9,10,11], for the general case when r is nonintegral, is given in terms of the action of Γ(r; n) on ∂H 2 =R. The answer to the remaining case when r is an integer is also given in a similar way.…”

“…This question originated from Minsky's question [3,Question 5.4], and was completely solved in the series of papers [8,9,10,11] for the generic case when r is non-integral, that is, K(r) is not a trivial knot. See [7] for an overview of these works.…”

“…However these results were obtained for the generic case when r is non-integral and we deferred the results for the special case when r is integral (cf. [9,Remark 2.5]). The purpose of this note is to obtain similar results for the remaining special case when r is integral, that is, K(r) is a trivial knot.…”

Parabolic generating pairs of genus-one 2-bridge knot groups

“…(2) This follows from the proof of [15,Lemma 3.3]. In the lemma, pieces of the symmetrized set of relators generated by a power u k r with k ≥ 2 of the relator u r is treated.…”