Word automatic groups of nilpotency class 2

Abstract

We consider word automaticity for groups that are nilpotent of class 2 and have exponent a prime p. We show that the infinitely generated free group in this variety is not word automatic. In contrast, the infinite extra-special p-group Ep is word automatic, as well as an intermediate group Hp which has an infinite centre. In the last section we introduce a method for showing automaticity of central extensions of abelian groups via co-cycles.