Fibre products, non-positive curvature, and decision problems

We give a criterion for fibre products to be finitely presented and use it as the basis of a construction that encodes the pathologies of finite group presentations into pairs of groups P subset of G where G is a product of h yperbolic groups and P is a finitely presented subgroup. This enables us to prove that there is a finitely presented subgroup P in a biautomatic group G such that the generalized word problem for P subset of G is unsolvable a nd P has an unsolvable conjugacy problem. An additional construction shows that there exists a compact non-positively curved polyhedron X such that pi (1)X is biautomatic and there is no algorithm to decide isomorphism among t he finitely presented subgroups of pi(1)X.