Cayley Polynomial-Time Computable Groups

TL;DR

This paper introduces a broad class of groups called Cayley polynomial-time computable groups, extending automatic groups by allowing polynomial-time multiplication and exploring their properties and inclusions.

Contribution

It generalizes Cayley automatic groups by relaxing time constraints, demonstrating that many properties are preserved and including significant groups like nilpotent and Thompson's groups.

Findings

Includes all virtually polycyclic groups

Contains all finitely generated nilpotent groups

Encompasses Thompson's group F

Abstract

We propose a new generalisation 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 $\mathcal C$ 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 (quadratic time word problem), prove various closure properties, and show that the class is quite large; for example it includes all virtually polycyclic groups. We then generalise to groups which have normal form language in the class $\mathcal C$ and multiplication by generators computable in polynomial time on a (standard) Turing machine. Of particular interest is when $\mathcal C= \mathrm{REG}$ (the class of regular languages). We prove that $\mathrm{REG}$-Cayley polynomial-time computable groups includes all finitely generated nilpotent groups, the wreath product $\mathbb Z_2 \wr \mathbb Z^2$, and Thompson's group $F$.