  Tsirelson's problem asks whether the commuting operator model for two-party
quantum correlations is equivalent to the tensor-product model. We give a
negative answer to this question by showing that there are non-local games
which have perfect commuting-operator strategies, but do not have perfect
tensor-product strategies. The weak Tsirelson problem, which is known to be
equivalent to Connes embedding problem, remains open.
  The examples we construct are instances of (binary) linear system games. For
such games, previous results state that the existence of perfect strategies is
controlled by the solution group of the linear system. Our main result is that
every finitely-presented group embeds in some solution group. As an additional
consequence, we show that the problem of determining whether a linear system
game has a perfect commuting-operator strategy is undecidable.
