Series: Penn State Logic Seminar Date: Tuesday, March 22, 2005 Time: 2:30 - 3:45 PM Place: 103 Pond Laboratory Speaker: Stephen G. Simpson, Penn State, Mathematics Title: Residual Finiteness and the Word Problem for Groups Abstract: Novikov 1955 and Boone 1959 have constructed finitely presented groups with unsolvable word problem. Therefore, it is of interest to find sufficient conditions for a finitely presented group to have solvable word problem. One such condition is residual finiteness: the intersection of all subgroups of finite index is trivial. We prove that every finitely generated nilpotent group is finitely presented and residually finite (Hirsch 1946), hence has solvable word problem. On the other hand, Kharlampovich 1981 has constructed a finitely presented solvable group with unsolvable word problem. We review some unpublished constructions of S. Aanderaa 1994 and H. Gravir 1995 which yield a Trakhtenbrot-style inseparability theorem for finitely presented groups.