Commensurators offinitely generated nonfree Kleinian groups

Historically, the prominence of the commensurator was due in large part to its density in the arithmetic setting being closely related to the abundance of Hecke operators attached to arithmetic lattices. These operators are fundamental objects in the theory of automorphic forms associated to arithmetic lattices (see Shimura [38] for example). More recently, the commensurator of various classes of groups has come to the fore due to its growing role in geometry, topology and geometric group theory; for example in classifying lattices up to quasi-isometry, classifying graph manifolds up to quasiisometry, and understanding Riemannian metrics admitting many “hidden symmetries” (for more on these and other topics see Bartholdi and Bogopolski [2], Behrstock and Neumann [4], Farb and Weinberger [17; 18], Leininger and Margalit [25], Schwartz [34] and Shalom [37]).