In this paper we use the terminology and notation of [12] and [51]. One finds there the concepts and results concerning combinatorial descriptions of groups which are used (but not always further explained) below.

THE NIELSEN REDUCTION METHOD IN AMALGAMATED FREE PRODUCTS AND HNN GROUPS

One of the most important methods in the theory of free groups and some similar groups, is the Nielsen reduction method. If F is the free group with free generators a, b, … one can define a notion of free length L in F (relative to the generators a, b, …) and a certain lexicographical ordering. The Nielsen reduction method in F concerns Nielsen transformation from systems {gj}jϵJ to systems which are shorter with respect to the length L and ordering (cf. [21], [22], [54]).

If we apply this method to a finite system {x1, …, xm}, we arrive after a finite number of steps at a system in which no element and no inverse of an element can shorten another by more than half, and no two can shorten another to nothing - i. e. a system which possesses the Nielsen property with respect to L. Nielsen ([21], [22]) used the property for the proof that subgroups of free groups are free. To be sure, one cannot define a meaningful length function in every group (cf. [4], [8], [13]). A length satisfying certain ‘natural’ axioms exists essentially only in free products.