  We initiate the study of torsion-free algebraically hyperbolic groups; these
groups generalise, and are intricately related to, groups with no
Baumslag-Solitar subgroups. Indeed, for groups of cohomological dimension 2 we
prove that algebraic hyperbolicity is equivalent to containing no
Baumslag-Solitar subgroups. This links algebraically hyperbolic groups to two
famous questions of Gromov; recent work has shown these questions to have
negative answers in general, but they remain open for groups of cohomological
dimension 2.
  We also prove that algebraically hyperbolic groups are CSA, and so have
canonical abelian JSJ-decompositions. In the two-generated case we give a
precise description of the form of these decompositions.
