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{{Short description|1992 book on abstract algebra}}
{{italic title}}
'''''Word Processing in Groups''''' is a monograph in mathematics on the theory of [[Automatic group|automatic group]]s, a type of [[Abstract algebra|abstract algebra]] whose operations are defined by the behavior of [[Deterministic finite automaton|finite automata]]. The book's authors are [[Biography:David B. A. Epstein|David B. A. Epstein]], [[Biography:James W. Cannon|James W. Cannon]], Derek F. Holt, Silvio V. F. Levy, [[Biography:Mike Paterson|Mike Paterson]], and [[Biography:William Thurston|William 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 ({{ISBN|0-86720-244-0}}).{{r|apanasov|baumslag|cohen}}

==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|topology]], and other related topics.{{r|apanasov}}

The first part has eight chapters. They cover automata theory and [[Regular language|regular language]]s, 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 graph]]s of automatic groups; [[Abelian group|abelian group]]s and the automaticity of [[Euclidean group]]s; the theory of determining whether a group is automatic, and its practical implementation by Epstein, Holt, and Sarah Rees; extensions to asynchronous automata; and [[Nilpotent group|nilpotent group]]s.{{r|apanasov|baumslag|thomas}}

The second part has four chapters, on [[Braid group|braid group]]s, [[Isoperimetric inequality|isoperimetric inequalities]], [[Geometric finiteness|geometric finiteness]], and the [[Fundamental group|fundamental group]]s of three-dimensional [[Manifold|manifold]]s.{{r|apanasov|thomas}}

==Audience and reception==
Although not primarily a textbook, the first part of the book could be used as the basis for a graduate course.{{r|apanasov|thomas}} More generally, reviewer [[Biography:Gilbert Baumslag|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 problem|undecidable]]. Despite tracing the origins of automatic groups to early 20th-century mathematician [[Biography:Max Dehn|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".{{r|baumslag}}

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.{{r|cohen}}

In 2009, mathematician Mark V. Lawson wrote that despite its "odd title," the book made [[Automata theory|automata theory]] more respectable among mathematicians stating that it became part of "a quiet revolution in the diplomatic relations between mathematics and computer science".{{r|lawson}}

==References==
{{reflist|refs=

<ref name=apanasov>{{citation|first=B. N.|last=Apanasov|journal=zbMATH|title=Review of ''Word Processing in Groups''|zbl=0764.20017}}</ref>

<ref name=baumslag>{{citation|last=Baumslag|first=Gilbert|doi=10.1090/S0273-0979-1994-00481-1|issue=1|journal=[[Organization:Bulletin of the American Mathematical Society|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}}</ref>

<ref name=cohen>{{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}}</ref>

<ref name=thomas>{{citation|title=Review of ''Word Processing in Groups''|journal=[[Mathematical Reviews]]|year=1993|first=Richard M.|last=Thomas|mr=1161694}}</ref>

<ref name=lawson>{{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}}</ref>

}}

[[Category:Mathematics books]]

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