Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem

As application of these characterizations we show that the conjugacy problem in fundamental groups of finite graphs of groups with finitely generated free abelian vertex groups can be solved in polynomial time on a strongly generic set. Furthermore, the conjugacy problem in groups with more than one end can be solved with a strongly generic algorithm which has essentially the same time complexity as the word problem. These are rather striking results as the word problem might be easy, but the conjugacy problem might be even undecidable. Finally, our results yield a new proof that the set where the conjugacy problem of the Baumslag group G1,2 is decidable in polynomial time is also strongly generic.