Martin R. Bridson and Henry Wilton
We prove that there is no algorithm that can determine whether or not a finitely
presented group has a non-trivial finite quotient; indeed, it remains undecidable
among the fundamental groups of compact, non-positively curved square complexes.
We deduce that many other properties of groups are undecidable. For hyperbolic
groups, there cannot exist algorithms to determine largeness, the existence of
a linear representation with infinite image (over any infinite field), or the
rank of the profinite completion.
35 pages, no figures.