Space functions of groups

TL;DR

This paper studies the space complexity of the word problem in finitely presented groups, introducing space functions as a geometric analog and providing criteria for polynomial space decidability.

Contribution

It defines space functions for groups, relates them to the word problem complexity, and establishes a criterion linking polynomial space solvability to subgroup embeddings.

Findings

Space functions measure the memory needed to reduce words to identity.

A group's word problem is polynomial space decidable iff it embeds into a group with polynomial space function.

Provides a geometric perspective on computational complexity in group theory.

Abstract

We consider space functions $s(n)$ of finitely presented groups $G =< A\mid R> .$ (These functions have a natural geometric analog.) To define $s(n)$ we start with a word $w$ over $A$ of length at most $n$ equal to 1 in $G$ and use relations from $R$ for elementary transformations to obtain the empty word; $s(n)$ bounds from above the tape space (or computer memory) one needs to transform any word of length at most $n$ vanishing in $G$ to the empty word. One of the main obtained results is the following criterion: A finitely generated group $H$ has decidable word problem of polynomial space complexity if and only if $H$ is a subgroup of a finitely presented group $G$ with a polynomial space function.