There is a long history of curvature conditions known as "[small cancellation conditions][1]" which apply to group presentations, or if you prefer to the Cayley 2-complex of the presentation. These are combinatorial conditions on the structure of the presentation.

Gromov introduced several different measurements of curvature. One of them, now known as "[Gromov hyperbolicity][2]", is an asymptotic measurement that applies to path metric spaces, in particular to finitely generated groups via their Cayley graph; this property generalizes the properties of large triangles in the hyperbolic plane. Another one, actually one for each real number $K$ and known as $CAT(K)$, is a local measurement that generalizes properties of simply connected complete Riemannian manifolds whose sectional curvature is bounded above by $K$; as in ordinary Riemannian geometry, the cases $K<0$, $K=0$, $K>0$ have interesting theoretical differences. 

Close to Ryan's comment and Joseph's answer, another curvature measurement introduced by Januskiewicz and Swiatkowski, and known simply as "simplicial nonpositive curvature" (the title of their paper), is a purely combinatorial condition that applies to simplicial complexes.


  [1]: http://en.wikipedia.org/wiki/Small_cancellation_theory
  [2]: http://en.wikipedia.org/wiki/Hyperbolic_group