The talks for the academic year 1997-98 are held in Chauvenet 116 at 3:45 pm unless otherwise stated.
Time: Wednesday, October 8Speaker Tony Gaglione, USNA
Title: Lyndon's free exponential groups and applications
Abstract: Lyndon introduced a tool which he used to solve one variable equations in free groups. If F is a free group and x1,...,xr are independent integral parameters, then the object brought into play by Lyndon was the "freest" group, FZ[x1,...,xr], containing F and admitting exponents from the polynomial ring Z[x1,...,xr] (satisfying various natural axioms). Myasnikov and Remelennikov recently have simplified Lyndon's explicit construction. Benjamin Baumslag defined a group G to be fully residually free provided that for every finite nonempty set S contained in G-{1} there is a free group FS and an epimorphism phiS: G -> FS such that phiS(g) not= 1 for all g in S. Myasnikov and Remeslennikov had conjectured that any finitely generated group is fully residually free iff it is embeddable in FZ[x] where F is any non-Abelian free group and x is an integral parameter. (I have spoken in this colloquium before about this conjecture!) The great news is that O. Kharlampovich and Myasnikov have now announced a verification of this conjecture!