On Dehn functions of infinite presentations of groups

TL;DR

This paper introduces two new Dehn functions suitable for infinite group presentations, establishing their connection to the word problem's solvability and providing bounds for specific groups.

Contribution

It defines new Dehn functions for infinite presentations and proves their equivalence to the word problem's solvability, unlike the standard Dehn function.

Findings

New Dehn functions are more suitable for infinite presentations.

Equivalence between solvability of word problem and function computability.

Upper bounds established for specific infinite groups.

Abstract

We introduce two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and prove the fundamental equivalence between the solvability of the word problem for a group presentation defined by a decidable set of defining words and the property of being computable for one of the newly introduced functions (this equivalence fails for the standard Dehn function). Elaborating on this equivalence and making use of this function, we obtain a characterization of finitely generated groups for which the word problem can be solved in nondeterministic polynomial time. We also give upper bounds for these functions, as well as for the standard Dehn function, for two well-known periodic groups one of which is an infinite 2-group of intermediate growth and the other is a free Burnside group of sufficiently large exponent.