ABSTRACT:

Thompson's Group F is a finitely presented infinite group which has proved a rich source of counterexamples for group theorists. We re-consider the group as represented by polygons in the Farey triangulation of the
hyperbolic plane. Through this lens, the group is linked to the associahedra, an infinite family of convex polytopes. By analogy with the type B associahedra, known as the cyclohedra, we define a type B version
of Thompson's Group F, show that this group is contained in Thompson's Group T, and give finite and infinite presentations and normal forms of F_B by utilizing a quotient of the group which is isomorphic to T.