Generalizing Specker's result [7] for a countable case without using the continuum hypothesis, NΓΆbeling [6] proved that the subgroup of a direct product I consisting of all finite valued functions is free for any index set I As a non-commutative version of this theorem in case I is countable, Zastrow [8] proved that the subgroup of Γ Γ consisting of elements which have certain restricted presentations is free, where Γ Γ is isomorphic to the fundamental group of the Hawaiian earring [3]. His proof is involved with complicated notions related to the inverse limit of free groups. Here, we use the notion "words of infinite length" from [3,4] and give a simplified proof. Since we do not depend on countability in the proof, we prove Zastrow's theorem for the free complete product Γ Γ I for any index set I On the other hand, Cannon and Conner [1] proved that another subgroup of Γ Γ is free. They used the notion "words of infinite length," but did not use a simple reduction for a finite concatenation of reduced words (see Lemma 1.7). We give a short proof of their theorem. We remark that the group Γ Γ I is called "a big free group" in [1].
Advantages of using words are the existence of reduced words and that of a simple reduction based on Proposition 1.3, which were used to investigate group theoretic properties in [3,4,2]. In Section 1, we state definitions and preliminary facts about words of infinite length, including the above facts about reduced words. In Section 2, we state the main theorem precisely and prove it. In the Appendix, we explain relationships between loops in the Hawaiian earring and the above two free subgroups of Γ Γ are free.
598