  We introduce shortcut graphs and groups. Shortcut graphs are graphs in which
cycles cannot embed without metric distortion. Shortcut groups are groups which
act properly and cocompactly on shortcut graphs. These notions unify a
surprisingly broad family of graphs and groups of interest in geometric group
theory and metric graph theory, including: the 1-skeletons of systolic and
quadric complexes (in particular finitely presented C(6) and C(4)-T(4) small
cancellation groups), 1-skeletons of finite dimensional CAT(0) cube complexes,
hyperbolic graphs, standard Cayley graphs of finitely generated Coxeter groups
and the standard Cayley graph of the Baumslag-Solitar group BS(1,2). Most of
these examples satisfy a strong form of the shortcut property.
  The shortcut properties also have important geometric group theoretic
consequences. We show that shortcut groups are finitely presented and have
exponential isoperimetric and isodiametric functions. We show that groups
satisfying the strong form of the shortcut property have polynomial
isoperimetric and isodiametric functions.
