Montreal Geometric & Combinatorial Group Theory Seminar
Speaker: Ted Turner (SUNY Albany)
Title: “Test elements”
Wednesday, March 10, 2004
Place: Room 920, Burnside Hall, McGill University
A 'test element' g in a group G is an element that tests whether an arbitrary endomorphism is an automorphism. In particular, g is a test element if g has the property that for any endomorphism f of G,
f(g) = g implies that f is an automorphism.
A natural first reaction is that such elements are quite rare and that even if they exist, their usefulness is very limited. We will show, on the contrary, that in many very interesting groups (in particular free groups), test elements exist in profusion and that furthermore, they can be used very effectively to identify automorphisms. Many examples will be given and effective methods of recognizing test elements will be described.
These ideas were first introduced in free groups (test elements in a free group are called 'test words'), but have been generalized to other groups (e.g., hyperbolic groups) and other categories (e.g., Lie algebras).