  We prove that there is no algorithm that can determine whether or not a
finitely presented group has a non-trivial finite quotient; indeed, this
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.
