On the Nielsen method in free products with amalgamated subgroups
Using methods first developed by Nielsen, it can be shown that every subgroup of a free group has a generating set that satisfies two significant properties, namely: (1) the length of any (freely reduced) product of the generators exceeds or equals the maximum of the lengths of the generators involved; (2) every factor in a (freely reduced) product of the generators contains a letter that is not cancelled during reduction of the product to its normal form. A similar result can be established for free products, with due allowance being made for the fact that distinct letters from the same group may ''coalesce'' during reduction to normal form. On the other hand it is well known that no such result can be established in general in a free product with a non-trivial amalgamated subgroup. The aim of the present paper is to show that for a subgroup of an amalgamated free product either one can obtain a generating set for which both (1) and (2) hold or certain ''bad'' configurations occur. Our approach also represents a refinement of the traditional method of Nielsen, where a generating set is transformed into a minimal set by a sequence of operations, the typical one being the replacement of the pair \(\{\) x,y\(\}\) by the pair \(\{\) xy,y\(\}\). In the context of free products this may result in an element of finite order being replaced by an element of infinite order which may be undesirable for the problem to hand. We therefore modify the permitted operations so that in some cases we may replace \(\{\) x,y\(\}\) by \(\{y^{-1}xy,y\}\) but not by \(\{\) xy,y\(\}\). It turns out that this restriction does not impose any real limitation on the power of the Nielsen method.
scientific article; zbMATH DE number 4000201