On the SQ-universality of T(6)-groups

Abstract

It is shown, in almost all cases, that a group given by a presentation satisfying the small-cancellation conditions C(3) and T (6) is SQ-universal. Among the exceptions to this result are a class of presentations called "special", whose star graphs are isomorphic to the incidence graphs of projective planes. An "example machine" is described, which constructs an example of a special presentation, beginning from the Desarguesian plane over any finite field.

On the SQ-universality of groups with special presentations

“…Examples of ð3; 3Þ-special presentations can be found in [9] and, more generally, in a construction due to Howie [14] and also the so-called triangle presentations of Cartwright, Mantero, Steger and Zappa [6]. In fact as we shall see later it turns out that these two latter constructions produce the same presentations.…”

“…Here we apply Theorem 1 together with known results on buildings to answer negatively a question of J. Howie in [14]. In particular we show that the Cð3Þ-Tð6Þ groups from Question 6.11 of [14] turn out to be just infinite (infinite groups all of whose non-trivial normal subgroups have finite index) and so are not SQ-universal.…”

“…In particular we show that the Cð3Þ-Tð6Þ groups from Question 6.11 of [14] turn out to be just infinite (infinite groups all of whose non-trivial normal subgroups have finite index) and so are not SQ-universal. In fact, in the hope that it may be of interest to the reader, we generalize the notion of so-called special group presentations from [14] and, using known results, analyse (in Theorem 2) when groups from this more general class are SQ-universal.…”

“…The group G is defined to have a special presentation if G can be defined by an ðm; kÞ-special presentation for some m and k. (We remark that our definition is inspired by Howie [14] whose definition of special presentation coincides with being ð3; 3Þ-special. )…”

“…Recall that a countable group G is SQ-universal if every countable group can be embedded in a quotient of G. Examples of SQ-universal groups include many HNN-extensions and amalgamated free products [16], [17]; groups of deficiency 2, groups of deficiency 1 in which at least one relator is a proper power, and many groups having balanced presentations in which at least two of the relators are proper powers (see [8] and the references there); most Cð3Þ-Tð6Þ groups [14]; non-elementary hyperbolic groups [21]; and, more recently, non-elementary groups hyperbolic relative to a collection of proper subgroups (including groups with an infinite number of ends) [1] and many mapping tori of free group endomorphisms [5]. On the other hand it is shown in [12] that if K is an algebraic number field and R an order of K then SL n ðRÞ is SQ-universal if and only if n ¼ 2 and either K ¼ Q or K is an imaginary quadratic number field.…”