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We give a criterion for the nonpositive sectional curvature of ( p , q , r ) (p,q,r) -complexes. As a consequence, we show that certain 2 2 -complexes have locally indicable, coherent and even locally quasiconvex fundamental groups.

This paper answers a question of Burns, Karrass and Solitar by giving examples of knot and link groups which are not subgroup-separable. For instance, it is shown that the fundamental group of the square knot complement is not subgroup separable. We characterise the Graph Manifolds with subgroup separable fundamental group as precisely the geometric ones, i.e. the Seifert Fibered 3-manifolds and the Sol manifolds, and show that there is a specific non-subgroup separable group which is a subgroup in all other cases.

In part 1, a construction of Rips is modified so that it produces a CAT ( − 1 ) (-1) group instead of a small-cancellation group. Thus, many of the applications of Rips’ construction to small-cancellation groups may be applied to CAT ( − 1 ) (-1) groups as well. Part 2 offers a simple way of producing incoherent groups.