GENERATING THE MÖBIUS GROUP WITH INVOLUTION CONJUGACY CLASSES.

A k-involution is an involution with a fixed point set of codimension k. The conjugacy class of such an involution, denoted Sk, generates Möb(n)(the group of isometries of hyperbolic n-space) if k is odd and its orientation-preserving subgroup if k is even. In this paper, we supply effective lower and upper bounds for the Sk word length of Möb(n) if k is odd and the Sk word length of Möb+(n) if k is even. As a consequence, for a fixed codimension k, the length of Möb+(n) with respect to Sk, k even, grows linearly with n, with the same statement holding for Möb(n) in the odd case. Moreover, the percentage of involution conjugacy classes for which Möb+(n) has length two approaches zero as n approaches infinity. [ABSTRACT FROM AUTHOR]