Montreal Geometric & Combinatorial Group Theory Seminar

Speaker: Ted Turner (SUNY Albany)

Title: “Test elements”

Date: 3:30PM,
Wednesday, March 10, 2004

Place: Room 920, Burnside Hall, McGill
University

Abstract:

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).