Geodesic edge paths in nonpositively curved complexes (Q1971134)

Recall that a group admitting a presentation satisfying the small cancellation condition \(C''(p)-T(q)\) is hyperbolic in the sense of Gromov except possibly if the couple \((p,q)\) is one of \((3,6)\), \((4,4)\) or \((6,3)\). Therefore, the powerful methods of the theory of hyperbolic groups do not apply to these three cases, which correspond nonetheless to non-positively curved Cayley complexes. The aim of the paper under review is to remedy this situation by providing a careful geometric analysis of these complexes, which correspond in the above order to Cayley complexes with triangular, square and hexagonal \(2\)-cells, respectively. The approach chosen by the author is not restricted to Cayley complexes attached to group presentations, but includes naturally kindred \(2\)-complexes. A typical result among those presented by the author is an algorithm for the straightening of edge paths. Some of the results pin down more rigorously certain ``folklore'' facts previously known to the experts.