G is an element with the property that any endomorphism fixing g is an automorphism.
At first glance, one would imagine test elements to be rather rare, but quite the opposite seems to be the case. A test element in a non-alelian free group is characterized by the fact that it lies in no proper retract and such elements abound. I will discuss the techniques used to understand test elements and progress on the following conjecture. Every torsion free hyperbolic group has test elements.