Polycyclic-by-finite Groups are C*-unique

Given a reduced *-algebra A admitting an enveloping C*-algebra, there are in general many non-isomorphic C*-completions of A. However, if the enveloping C*-algebra is the unique C*-completion of A up to isomorphism, we say A is C*-unique. The study of C*-uniqueness for L^1-algebras of locally compact groups began in the late 70s, but little progress has been made since the late 80s. In this talk we give an introduction to the question of C*-uniqueness for L^1-algebras of groups and groupoids and show how one can use groupoid techniques to prove that L^1(G) has a unique C*-norm whenever G is a polycyclic-by-finite group.