''Word Processing in Groups'' is a monograph in mathematics on the theory of

automatic group In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata represent the Cayley graph of the group. That is, they can tell if a given word representation of a group element is in a " ...

s, a type of abstract algebra whose operations are defined by the behavior of finite automata. The book's authors are

David B. A. Epstein David Bernard Alper Epstein Fellow of the Royal Society, FRS (born 1937) is a mathematician known for his work in hyperbolic geometry, 3-manifolds, and group theory, amongst other fields. He co-founded the University of Warwick mathematics depa ...

James W. Cannon James W. Cannon (born January 30, 1943) is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University. Biographical data Jame ...

, Derek F. Holt, Silvio V. F. Levy,

Mike Paterson Michael Stewart Paterson, is a British computer scientist, who was the director of the Centre for Discrete Mathematics and its Applications (DIMAP) at the University of Warwick until 2007, and chair of the department of computer science in 2005 ...

, and

William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurston ...

. Widely circulated in preprint form, it formed the foundation of the study of automatic groups even before its 1992 publication by Jones and Bartlett Publishers ().

Topics

The book is divided into two parts, one on the basic theory of these structures and another on recent research, connections to geometry and topology, and other related topics. The first part has eight chapters. They cover automata theory and regular languages, and the closure properties of regular languages under logical combinations; the definition of automatic groups and biautomatic groups; examples from topology and "combable" structure in the Cayley graphs of automatic groups; abelian groups and the automaticity of

Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). ...

s; the theory of determining whether a group is automatic, and its practical implementation by Epstein, Holt, and

Sarah Rees Sarah Elizabeth Rees (born 1957) is Professor of Pure Mathematics at Newcastle University. Her focus of research is on geometrical, combinatorial and computational aspects of group theory. Rees obtained her Ph.D. in 1983 from the University of O ...

; extensions to asynchronous automata; and nilpotent groups. The second part has four chapters, on braid groups, isoperimetric inequalities, geometric finiteness, and the

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

s of three-dimensional

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

Audience and reception

Although not primarily a textbook, the first part of the book could be used as the basis for a graduate course. More generally, reviewer Gilbert Baumslag recommends it "very strongly to everyone who is interested in either group theory or topology, as well as to computer scientists." Baumslag was an expert in a related but older area of study, groups defined by finite presentations, in which research was eventually stymied by the phenomenon that many basic problems are undecidable. Despite tracing the origins of automatic groups to early 20th-century mathematician Max Dehn, he writes that the book studies "a strikingly new class of groups" that "conjures up the fascinating possibility that some of the exploration of these automatic groups can be carried out by means of high-speed computers" and that the book is "very likely to have a great impact". Reviewer Daniel E. Cohen adds that two features of the book are unusual. First, that the mathematical results that it presents all have names, not just numbers, and second, that the cost of the book is low. In 2009, mathematician Mark V. Lawson wrote that despite its "odd title," the book made automata theory more respectable among mathematicians stating that it became part of "a quiet revolution in the diplomatic relations between mathematics and computer science".

References

{{reflist, refs= {{citation, first=B. N., last=Apanasov, journal= zbMATH, title=Review of ''Word Processing in Groups'', zbl=0764.20017 {{citation, last=Baumslag, first=Gilbert, authorlink=Gilbert Baumslag, doi=10.1090/S0273-0979-1994-00481-1, issue=1, journal= Bulletin of the American Mathematical Society, mr=1568123, pages=86–91, series=New Series, title=Review of ''Word Processing in Groups'', volume=31, year=1994, doi-access=free {{citation, last=Cohen, first=D. E., date=November 1993, doi=10.1112/blms/25.6.614, issue=6, journal= Bulletin of the London Mathematical Society, pages=614–616, title=Review of ''Word Processing in Groups'', volume=25 {{citation, title=Review of ''Word Processing in Groups'', journal=

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

, year=1993, first=Richard M., last=Thomas, mr=1161694 {{citation, last=Lawson, first=Mark V., date=December 2009, issue=4, journal= SIAM Review, jstor=25662348, pages=797–799, title=Review of ''A Second Course in Formal Languages and Automata Theory'' by Jeffrey Shallit, volume=51 Computational group theory Mathematics books 1992 non-fiction books