Automatic presentations for finitely generated groups

Abstract

A structure is said to be computable if its domain can be represented by a set which is accepted by a Turing machine and if its relations can then be checked using Turing machines. Restricting the Turing machines in this definition to finite automata gives us a class of structures with a particularly simple computational structure; these structures are said to have automatic presentations. Given their nice algorithmic properties, these have been of interest in a wide variety of areas. An area of particular interest has been the classification of automatic structures. One of the prime examples under consideration has been the class of groups. We give a complete characterization in the case of finitely generated groups and show that such a group has an automatic presentation if and only if it is virtually abelian. © Springer-Verlag Berlin Heidelberg 2005.