Test elements in groups and on their boundaries
by
Ted Turner
University at Albany

A test element g in a group G is an element with the property that any endomorphism of G that fixes g must be an automorphism. In groups for which automorphic equivalence is algorithmically decidable (like free groups for example) this provides a method for recognizing automorphisms.

Initially one might expect test elements to be quite rare, but in interesting classes of groups (e.g., free, hyperbolic, products of hyperbolic groups) they exist in profusion. I will describe results in this direction and progress on the following conjecture.

Conjecture All torsion free hyperbolic groups have test elements.

The notion of a test element on the boundary of a group makes sense as well and is interesting in certain cases. The infinite word [a, b]^\infty, for example, is a test element in the boundary of F(a, b). We will show that on the boundary, most elements are test elements.

Date received: February 29, 2000