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

Abstract

Riley "defined" the Heckoid groups for 2-bridge links as Kleinian groups, with nontrivial torsion, generated by two parabolic transformations, and he constructed an infinite family of epimorphisms from 2-bridge link groups onto Heckoid groups. In this paper, we make Riley's definition explicit, and give a systematic construction of epimorphisms from 2-bridge link groups onto Heckoid groups, generalizing Riley's construction.In honour of J. Hyam Rubinstein and his contribution to mathematics

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.…”

“…We note that (1) a loop in the orbifold S(r; n) is null-homotopic in S(r; n) if and only if it determines the trivial conjugacy class of the Heckoid group G(r; n), and (2) two loops in S(r; n) are homotopic in S(r; n) if and only if they determine the same conjugacy class in G(r; n) (see [1,2] for the concept of homotopy in orbifolds). We say that a loop in S(r; n) is peripheral if and only if it is homotopic to a loop in the paring annulus naturally associated with S(r; n) (see [8,Section 6]), i.e., it represents the conjugacy class of a power of a meridian of G(r; n). We also say that a loop in S(r; n) is torsion if it represents the conjugacy class of a non-trivial torsion element of G(r; n).…”

Homotopically equivalent simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links (I)

“…Let K(r) be the 2-bridge link of slope r ∈ Q and let n be an integer or a half-integer greater than 1. In [9], following Riley's work [19], we introduced the Heckoid group G(r; n) of index n for K(r) as the orbifold fundamental group of the Heckoid orbifold S(r; n) of index n for K(r). The classical Hecke groups introduced in [4] are essentially the simplest Heckoid groups.…”

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

“…Let K(r) be the 2-bridge link of slope r ∈ Q and let n be an integer or a halfinteger greater than 1. In [8], following Riley's work [12], we introduced the Heckoid group G(r; n) of index n for K(r) as the orbifold fundamental group of the Heckoid orbifold S(r; n) of index n for K(r). According to whether n is an integer or a non-integral half-integer, the Heckoid group G(r; n) and the Heckoid orbifold S(r; n) are said to be even or odd.…”