On the Algebra of Semigroup Diagrams

Abstract

In this work we enrich the geometric method of semigroup diagrams to study semigroup presentations. We introduce a process of reduction on semigroup diagrams which leads to a natural way of multiplying semigroup diagrams associated with a given semigroup presentation. With respect to this multiplication the set of reduced semigroup diagrams is a groupoid. The main result is that the groupoid [Formula: see text] of reduced semigroup diagrams over the presentation S = <X:R> may be identified with the fundamental groupoid γ (KS) of a certain 2-dimensional complex KS. Consequently, the vertex groups of the groupoid [Formula: see text] are isomorphic to the fundamental groups of the complex KS. The complex we discovered was first considered in the paper of Craig Squier, published only recently. Steven Pride has also independently defined a 2-dimensional complex isomorphic to KS in relation to his work on low-dimensional homotopy theory for monoids. Some structural information about the fundamental groups of the complex KS are presented. The class of these groups contains all finitely generated free groups and is closed under finite direct and finite free products. Many additional results on the structure of these groups may be found in the paper of Victor Guba and Mark Sapir.

Hyperbolic diagram groups are free

“…A diagram is called reduced if it does not contain dipoles. By reducing dipoles, a diagram can be transformed into a reduced diagram, and a result of Kilibarda [Kil94] proves that this reduced form is unique. If ∆ 1 , ∆ 2 are two diagrams for which ∆ 1 • ∆ 2 is well defined, let us denote by ∆ 1 · ∆ 2 the reduced form of ∆ 1 • ∆ 2 .…”

“…The definition of diagram groups was first given by Meakin and Sapir, with the first results found by their student Kilibarda in her thesis [Kil94]. Although it was proved that diagram groups define a large class of groups with strong properties [GS97,GS99,GS06a,GS06b], very little is known on their geometric properties.…”

On subgroups of R. Thompson's group $ F$ and other diagram groups

“…It was implicitely defined by Squier in [31]. The same complex was independently constructed by Kilibarda [17,18] and Pride [26]. The important role of this complex is justified by the fact that equal diagrams over P correspond to homotopic paths in K(P).…”

“…The important role of this complex is justified by the fact that equal diagrams over P correspond to homotopic paths in K(P). The following Kilibarda's theorem [17,18] plays an important role in this paper: The diagram group D(P, w) is isomorphic to the fundamental group π 1 (K, w) of the Squier complex K = K(P).…”

Low dimensional homotopy for monoids II: groups

“…A typical element of this monoid is a congruence class W& (WPF), and we have an isomorphism from this monoid to G, given by W& U 3 W N W P F X We will often identify W& and [W]N (if no confusion can arise) and will denote this element by " W. Now in [12] (see also [11]) we associated with any monoid presentation a 2-complex h( ) (``the 2-complex of monoid pictures'') and we showed that the ®rst homology group H 1 (h( )) has considerable signi®cance. The fundamental groups of h( ) are also of considerable interest and have been investigated by Guba and Sapir [7], and Kilibarda [8].…”