Test Words, Generic Elements and Almost Primitivity

A test element in a group G is an element g with the property that if f(g) = g for an endomorphism f of G to G then f must be an automorphism. A test element in a free group is called a test word. Nielsen gave the first example of a test word by showing that in the free group on x, y the commutator [x, y] satisfies this property. T. Turner recently characterized test words as those elements of a free group contained in no proper retract. Since free factors are retracts, test words are therefore very strong forms of non-primitive elements. In this paper we give some new examples of test words and examine the relationship between test elements and several other concepts, in particular generic elements and almost-primitive elements (APE’s). In particular we show that an almost primitive element which lies in a certain type of verbal subgroup must be a test word. Further using a theorem of Rosenberger on equations in free products we prove a result on APE’s, generic elements and test words in certain free products of free groups. Finally we examine test elements in non-free groups and introduce the concept of the test rank of a group.