Groups, Languages and Automata

Abstract

Fascinating connections exist between group theory and automata theory, and a wide variety of them are discussed in this text. Automata can be used in group theory to encode complexity, to represent aspects of underlying geometry on a space on which a group acts, and to provide efficient algorithms for practical computation. There are also many applications in geometric group theory. The authors provide background material in each of these related areas, as well as exploring the connections along a number of strands that lead to the forefront of current research in geometric group theory. Examples studied in detail include hyperbolic groups, Euclidean groups, braid groups, Coxeter groups, Artin groups, and automata groups such as the Grigorchuk group. This book will be a convenient reference point for established mathematicians who need to understand background material for applications, and can serve as a textbook for research students in (geometric) group theory.

Polynomial-time proofs that groups are hyperbolic

“…Proof. It is shown in the proof of [14,Theorem 6.5.3] that G is hyperbolic and that all geodesic triangles in its Cayley graph of G are δ-slim with δ ≤ 96λr 2 + 4. In fact the result of [14, Lemma 6.5.1] can easily be improved from area(∆) ≥ mn/l 2 to area(∆) ≥ 4mn/l(l − 1), which results in the improved bound δ ≤ 24λr(r − 1) + 4.…”

“…In fact the result of [14, Lemma 6.5.1] can easily be improved from area(∆) ≥ mn/l 2 to area(∆) ≥ 4mn/l(l − 1), which results in the improved bound δ ≤ 24λr(r − 1) + 4. It is proved in [14,Theorem 6.1.3] that all geodesic triangles in the Cayley graph are 4δ-thin, and then [14, Theorem 6.4.1] implies that any word w with w = G 1 must contain a non-geodesic word of length at most 16δ, which is at most 384λr(r − 1) + 64.…”

“…In the cases (m, n) = (7,12), (8,7), (9,6), (10,5) and (14,4), the group G is automatic and infinite. In each of these examples, by using straightforward searches through the elements of G of bounded length, we were able to find a pair g, h of commuting elements that project onto a free abelian group of rank 2 in an abelian quotient of a suitably chosen subgroup of finite index in G. So these group all contain free abelian subgroups of rank 2, and hence they are not hyperbolic.…”

“…There are several good sources, such as [1], for an introduction to and development of the basic properties of hyperbolic groups. Another useful reference for the specific properties that we need in this paper is [14,Chapter 6]. In particular, hyperbolic groups are finitely presentable, and they admit a Dehn algorithm.…”

Word problems of groups: Formal languages, characterizations and decidability

“…When examining groups based on their word problem as a formal language it is quite common to try to classify groups based on what family of languages their word problem lies in. Whilst we will concentrate on the results most pertinent to the current paper here, the reader is referred to [4,9,15,21,33] (for example) for more information.…”