Cayley polynomial–time computable groups

Abstract

We propose a new generalization of Cayley automatic groups, varying the time complexity of computing multiplication, and language complexity of the normal form representatives. We first consider groups which have normal form language in the class C and multiplication by generators computable in linear time on a certain restricted Turing machine model (position–faithful one–tape). We show that many of the algorithmic properties of automatic groups are preserved, prove various closure properties, and show that the class is quite large. We then generalize to groups which have normal form language in the class C and multiplication by generators computable in polynomial time on a (standard) Turing machine. Of particular interest is when C=REG (the class of regular languages). We prove that REG–Cayley polynomial–time computable groups include all finitely generated nilpotent groups, the wreath product Z2≀Z2, and Thompson's group F.