Abstract

We show that the finitely presented groups with unsolvable word problem given by the Boone-Brit ton construction have cohomological dimension 2. More precisely we show these groups can be obtained from a free group by successively forming HNN-extensions where the associated subgroups are finitely generated free groups. Also the presentations obtained for these groups are aspherical. Using this we show there is no algorithm to determine whether a presentation is aspherical. There is no algorithm to determine whether a finite 2-complex is aspherical.

Introduction

Fundamental algorithms in combinatorial group theory (the original due to Nielsen [6]) enable one to decide membership in a finitely generated subgroup of a free group. It follows that the free product of two free groups with finitely generated amalgamation has a solvable word problem. Similarly, an HNN-extension of a free group with finitely generated associated subgroups has a solvable word problem. (However, such groups can have unsolvable conjugacy problem and the problem of deciding whether an arbitrary pair of them are isomorphic is unsolvable – see [7].) More generally, the fundamental group of a finite graph of groups whose edge and vertex groups are all finitely generated and free is finitely presented and has a solvable word problem.