The Tits Alternative for Generalized Triangle Groups

Int. J. Algebra Comput.

2012

Largeness, SQ-universality, and the existence of free subgroups of rank 2 are measures of the complexity of a finitely presented group. We obtain conditions under which a cyclically presented group possesses one or more of these properties. We apply our results to a class of groups introduced by Prishchepov which contain, amongst others, the various generalizations of Fibonacci groups introduced by Campbell and Robertson. Using the techniques developed we give a new, purely group-theoretic, proof of the (almost complete) classification of the finite Cavicchioli-HegenbarthRepovš groups.

Section: Proofmentioning

confidence: 98%

Largeness and Sq-Universality of Cyclically Presented Groups

Int. J. Algebra Comput.

2012

“…We first recall some definitions and well-known facts concerning generalized triangle groups; further details are available in (for example) [9]. Let G be as defined in the Main Theorem, but with m ≥ 3.…”

Section: Preliminariesmentioning

confidence: 99%

“…(See [9] for a survey of these results.) In recent work Benyash-Krivets [3,4] considers the case (2, m, 2).…”

Section: Introductionmentioning

confidence: 99%

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

Howie¹,

2006

Geom Dedicata

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.

“…(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.…”

Section: Introductionmentioning

confidence: 99%

“…We classify all such groups for k ¼ 1 (Section 4), for n d 3 (Section 5), and provide partial results in the case n ¼ 2 (Sections 3 and 6). For n d 3 the classification of the pseudo-elementary generalized triangle groups was already implicit in the papers of Fine, Rosenberger, et al [6], [8], [9], [10], [13], [14], and for n ¼ 2 the partial classification was already implicit in [16], but never formally stated. Here we codify the results and simplify the methods.…”

Section: Introductionmentioning

confidence: 99%

Pseudo-elementary generalized triangle groups

Journal of Group Theory

2007

Abstract.A generalized triangle group is a group that can be presented in the form G ¼ hx; y j x l ¼ y m ¼ wðx; yÞ n ¼ 1i where l; m; n A f2; 3; 4; . . .g U fyg and wðx; yÞ is an element of the free product hx; y j x l ¼ y m ¼ 1i involving both x and y. A homomorphism f : G ! G is said to be essential if fðxÞ, fð yÞ, fðwðx; yÞÞ have orders l, m, n respectively. Every generalized triangle group G admits an essential representation to PSLð2; CÞ. In most cases there will be such a representation with infinite or non-elementary image.Vinberg and Kaplinsky say that G is pseudo-finite if the image of any essential representation G ! PSLð2; CÞ is finite and they have obtained a partial classification of such groups. Extending this concept, we call G pseudo-elementary if the image of any essential representation G ! PSLð2; CÞ is elementary. In this paper we classify the pseudo-elementary generalized triangle groups G with n d 3 and obtain partial results in the case n ¼ 2.