Groups, Geometry, and Dynamics

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Volume 8, Issue 1, 2014, pp. 157–198
DOI: 10.4171/GGD/221

Published online: 2014-05-13

From automatic structures to automatic groups

Olga Kharlampovich[1], Bakhadyr Khoussainov[2] and Jan Cannizzo[3]

(1) McGill University, Montreal, Canada
(2) University of Auckland, New Zealand
(3) Stevens Institute of Technology, Hoboken, USA

In this paper we introduce the concept of a Cayley graph automatic group (CGA group or graph automatic group, for short) which generalizes the standard notion of an automatic group. Like the usual automatic groups graph automatic ones enjoy many nice properties: these groups are invariant under the change of generators, they are closed under direct and free products, certain types of amalgamated products, and finite extensions. Furthermore, the word problem in graph automatic groups is decidable in quadratic time. However, the class of graph automatic groups is much wider then the class of automatic groups. For example, we prove that all finitely generated 2-nilpotent groups and Baumslag–Solitar groups BS(1,n) are graph automatic, as well as many other metabelian groups.

Keywords: Automatic groups, Cayley graphs, word problem, nilpotent and metabelian groups

Kharlampovich Olga, Khoussainov Bakhadyr, Cannizzo Jan: From automatic structures to automatic groups. Groups Geom. Dyn. 8 (2014), 157-198. doi: 10.4171/GGD/221