On discrete generalised triangle groups

Abstract

A generalised triangle group has a presentation of the form where R is a cyclically reduced word involving both x and y. When R=xy, these classical triangle groups have representations as discrete groups of isometries of S 2 , R 2 , H 2 depending on In this paper, for other words R, faithful discrete representations of these groups in Isom + H 3 = PSL(2,C) are considered with particular emphasis on the case /? = [x, y] and also on the relationship between the Euler characteristic x and finite covolume representations.1991 Mathematics subject classification: 20H15.

2-generator arithmetic Kleinian groups

“…When m = n = 2 and R(x, y) = (xy)3(x-1y)2, these groups are commensurable with the Fibonacci groups F2ℓ and are arithmetic Kleinian groups if and only if ℓ = 4, 5, 6, 8,12 [16], [24]. (3, 6; 2), (6, 6; 2), (6, 6; 3) can be shown to be nearly arithmetic, but neither arithmetic nor finite extensions of Fuchsian groups [14], [15]. These are arithmetic for (ℓ, m; n) = (3, 3; 3), (4, 4; 2), (3,4; 2) [15].…”

“…Thus (15) $ where $ is the norm map. where the τi here are the Galois embeddings of L such that τi|K = Id.…”

“…For example certain generalized triangle groups [3], [15], [24] and groups obtained from Dehn surgeries on various hyperbolic 2-bridge knot complements [33]. There are infinitely many finite covolume Kleinian groups generated by two elements of finite order.…”

One-Relator Products Induced from Generalized Triangle Groups

“…The arguments in [12] involve detailed analysis of one-relator products 'of type E(2, 3, 4) + ', which are induced from generalized triangle groups of the form x, y | x 2 = y 3 = ((xyxy 2 ) t xy) 4 , t ≥ 1. The 'type E' condition also occurs in [10] as an obstruction to the straightforward calculation of Euler characteristics of generalized triangle groups.…”

Some Special Kleinian Groups and Their Orbifolds

“…For it, triangle and Coxeter groups have been of great interest (see Coxeter-Moser [1]). Another examples are given by generalized triangle groups [2], [3], [4], [5]. In this paper, we consider certain type of generalized Coxeter groups (see next section for definitions).…”