On the difficulty of presenting finitely presentable groups

TL;DR

This paper explores the computational complexity of presenting finitely presentable groups, showing that certain classes have solvable word problems but lack algorithms for finite presentations of subgroups, revealing fundamental limitations in group theory computations.

Contribution

It demonstrates the existence of groups with solvable word problems but no algorithm for finite subgroup presentations, and constructs a finitely presented group with a polynomial Dehn function but no Betti number algorithm.

Findings

Classes of groups with solvable word problems but no finite subgroup presentation algorithms

Existence of a finitely presented group with polynomial Dehn function and no Betti number algorithm

Discussion of groups where finite presentations of subgroups are computable

Abstract

We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there does exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of the finitely presentable subgroups.