Resonance of Basis-conjugating Automorphism Groups

1. Resonance of PΣn Let Fn be the free group generated by x1, . . . , xn. The basis-conjugating automorphism group, or pure symmetric automorphism group, is the group PΣn of all automorphisms of Fn which send each generator xi to a conjugate of itself. Results of Dahm [4] and Goldsmith [6] imply that this group may also be realized as the “group of loops,” the group of motions of a collection of n unknotted, unlinked oriented circles in 3-space, where each circle returns to its original position. McCool [8] found the following presentation for the basis-conjugating automorphism group: