On A Property And Two Conjectures In Infinite Groups

There are many properties and results on free groups and free products widely used in topology and geometry. Of course some more problems are worth while to discuss only from the algebraic aspect. For example, whether the properties of free groups can be generalized naturally to free products and so on. Lubotski [3] and Lue [2] showed that every normal automorphism of a non-cyclic free group is inner. Our studies mainly focus on the free groups with finite ranks . Free groups with finite ranks are the fundamental subjects of combinatorial group theory, and they have central importance for the studies of both topology and algebra.The major technique to get the information of structure of those free groups is to sudy the action of automorphisms on them. In this paper we show that every normal automorphism of a nontrivial free product of groups is inner as well. Traub[16] made a purely algebraic conjecture and showed that it implied Poincare's Conjecture. He also showed that Poincare's Conjecture implied this algebraic conjecture modulo another topological hypothesis. Afterward Waldhausen ([7]) showed that this topological hypothesis was true. Besides, Stallings ([11]) gave another conjecture that was equivalent to Poincare's Conjecture. Thus we know both conjectures made respec- tively by Traub and Stallings are equivalent to Poincare's Conjecture. Consequently the two conjectures are also equivalent. In this paper we mainly show that conjecture 2 is directly implied by conjecture 1.