Word automatic groups of nilpotency class 2

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. & COPY; 2023 Elsevier B.V. All rights reserved.

Automated summary

We investigate word automaticity for nilpotent groups of class 2 with prime exponent p. It is proven that the infinitely generated free group in this category is not word automatic. However, the infinite extra-special p-group Ep and an intermediate group Hp with an infinite center are both word automatic. Additionally, a method for demonstrating automaticity of central extensions of abelian groups via co-cycles is introduced.