Title: Roots, Centralizers and the Conjugacy Problem in Subgroups of Homeomorphism Groups of R and S^1, in preparation, with Collin Bleak and Martin Kassabov. Abstract:  We describe centralizers of subgroups of the groups PL_+(S^1) of all orientation-preserving piecewise-linear homemorphisms of the circle. By using this description and extending the techniques introduced in this paper we are able to prove that the simultaneous conjugacy problem is solvable in all the instances we analyze. A preliminary version of this paper appears as Chapter 7 of my Dissertation Thesis. Back to top.