 Then (see  and Theorem 13.3 in ), under the above assumptions on (∗), "Y" is a classifying space for "G", that is "G" = "π"1("Y") and the universal cover of "Y" is contractible.In particular, this implies that "G" is torsion-free and has cohomological dimension two.More generally, it is possible to define various sorts of local "curvature" on any van Kampen diagram to be - very roughly - the average excess of vertices + faces - edges (which, by Euler's formula, must total 2) and, by showing, in a particular group, that this is always non-positive (or - even better - negative) internally, show that the curvature must all be on or near the boundary and thereby try to obtain a solution of the word problem.