On groups that have normal forms computable in logspace

TL;DR

This paper studies groups with normal forms computable in logspace, showing their closure properties, including certain free products, and demonstrating that all groups of polynomial growth are logspace embeddable.

Contribution

It establishes closure properties of groups with logspace computable normal forms and introduces the concept of logspace embeddability, including all groups of polynomial growth.

Findings

Closure under finite extensions, subgroups, direct and wreath products.

Includes solvable Baumslag-Solitar groups and non-residually finite examples.

All groups of polynomial growth are logspace embeddable.

Abstract

We consider the class of finitely generated groups which have a normal form computable in logspace. We prove that the class of such groups is closed under finite extensions, finite index subgroups, direct products, wreath products, and also certain free products, and includes the solvable Baumslag-Solitar groups, as well as non-residually finite (and hence non-linear) examples. We define a group to be logspace embeddable if it embeds in a group with normal forms computable in logspace. We prove that finitely generated nilpotent groups are logspace embeddable. It follows that all groups of polynomial growth are logspace embeddable.