USNA Pure Mathematics Seminar

Title:

Abstract: A group G is termed discriminating iff given any finite set, S, of nontrivial elements in GxG (its direct square) there exists a homomorphism f_S:GxG --->G such that f_S does not kill off any element of S. Clearly if GxG embeds into G, then G is discriminating. In this case, we say that G is trivially discriminating. It is not hard to show that any torsion free abelian group is discriminating. Thus there are nontrivially discriminating groups. For some non-abelian examples, let G₀ be your favorite non-abelian group and let G be the direct power (G₀)^I, where I is any infinite index set. Then GxG is isomorphic to G so G is discriminating. An example of a finitely presented non-abelian discriminating group is R. Thompson's group F of orientation-preserving piecewise linear homeomorphisms from the unit interval [0,1] to itself that are differentiable except at finitely many dyadic rational numbers and such that on intervals of differentiability the derivatives are powers of 2. It can be shown that FxF embeds into F. Thus F is also trivially discriminating. Thus comes the question in the title of this talk. To find out the answer, come to the talk!