Free Subgroups in Certain Generalized Triangle Groups of Type (2, m, 2)

Abstract

A generalized triangle group is a group that can be presented in the form G = x, y | x p = y q = w(x, y) r = 1 where p, q, r ≥ 2 and w(x, y) is a cyclically reduced word of length at least 2 in the free product Z p * Z q = x, y | x p = y q = 1 . Rosenberger has conjectured that every generalized triangle group G satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple (p, q, r) is one of (3, 3, 2), (3, 4, 2), (3, 5, 2), or (2, m, 2) where m = 3, 4,5,6,10,12,15, 20, 30, 60. In this paper we show that the Tits alternative holds in the cases (p, q, r) = (2, m, 2) where m = 6, 10, 12, 15, 20, 30, 60.

Generalised triangle groups of type (3,5,2)

“…In the present note we prove it in the case (p, q, r) = (3, 5, 2). In conjunction with previously known results [8,3,11,15,4,5,1,2,6,13,14,17] (see for example the survey [9] or the Introduction to [12] for details), this reduces the conjecture to the cases (p, q, r) = (2, q, 2) for q ∈ {3, 4, 5}.…”

Generalised triangle groups of type (2,3,2) with no cyclic essential representations

“…Moreover, τ W (λ) is a polynomial of degree k in λ, and the combinatorics of the word W determine the coefficients of τ W in a well-understood way. Details can be found, for example, in [6].…”

“…Now suppose that c + e > 0. We analyse the coefficients of λ −2 and λ −4 in τ W (λ), using the formulae in [6]. Write W = xy α(1) • • • xy α( ) with α(j) ∈ {1, 2} for each j.…”

Pseudo-elementary generalized triangle groups

“…(See [9] or [13] for a survey of this problem, and [3], [2], [15] for more recent results.) Related to this conjecture is the question of which generalized triangle groups contain Ree-Mendelsohn pairs.…”