Last year, Monod proved the vanishing of the bounded cohomology of many lamplighters, and via coamenability he obtained the same result for many groups acting on the line, such as Thompson's group F. We will see that the techniques involved translate well to asymptotic cohomology, and allow to prove Ulam stability of such groups. Some of the steps of the proof establish more general facts about asymptotic cohomology and stability, most notably we prove a relative version of Kazhdan's Theorem on stability of amenable groups.