Embeddings into efficient groups

Embeddings into k-Efficient Groups

“…Moreover, one can check that the proof of the cases k = 1 and 2 of Theorem 1 also applies to non-finite G. We thus have the following result, which strengthens Corollary 4.3 in [5]. A graph of groups construction was used in [6] to provide an alternative proof of the fact that any finitely presented group G embeds into a finitely presented efficient group.…”

The efficiency of standard wreath product

“…Examples of efficient groups are finitely generated abelian groups, fundamental groups of closed 3-manifolds [12]; also, finite groups with balanced presentations (such finite groups have trivial Schur multiplier [13]). Finite metacyclic groups are efficient.…”

“…Infinite metacyclic groups, however, need not be efficient, a result due to Baik and Pride [5] (see also [3]). In [13], Harlander proved that a finitely presented group embeds into an efficient group. In [16], Johnson showed that all finite p-groups are efficient under direct products and standard wreath products (for p odd).…”

On defining groups efficiently without using inverses

“…For example, Conway [8], in the solution to his problem regarding the Fibonacci group F (2,5), defined by a, b, c, d, e | ab = c, bc = d, cd = e, de = a, ea = b , remarks that the semigroup defined by this presentation is also F (2,5). Adian [1] initiated the investigation of conditions under which a semigroup S can be embedded into a group G defined by the same presentation; see also [10]. Adian [1] initiated the investigation of conditions under which a semigroup S can be embedded into a group G defined by the same presentation; see also [10].…”

“…This led to a more detailed study of Fibonacci groups and semigroups in [6]. Adian [1] initiated the investigation of conditions under which a semigroup S can be embedded into a group G defined by the same presentation; see also [10]. A dual question as to when S contains G as a subsemigroup was considered in [7].…”