  We give a combinatorial criterion that implies both the non-strong relative
hyperbolicity and the one-endedness of a finitely generated group. We use this
to show that many important classes of groups do not admit a strong relatively
hyperbolic group structure and have one end. Applications include surface
mapping class groups, the Torelli group, (special) automorphism and outer
automorphism groups of most free groups, and the three-dimensional Heisenberg
group. Our final application is to Thompson's group F.
