Generating the Mobius group with involution conjugacy classes

Abstract

A {\it $k$-involution} is an involution with a fixed point set of codimension $k$. The conjugacy class of such an involution, denoted $S_k$, generates $\text{M\"ob}(n)$-the 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 $S_k$ word length of $\text{M\"ob}(n)$ if $k$ is odd, and the $S_k$ word length of $\text{M\"ob}^+(n)$, if $k$ is even. As a consequence, for a fixed codimension $k$ the length of $\text{M\"ob}^{+}(n)$ with respect to $S_k$, $k$ even, grows linearly with $n$ with the same statement holding in the odd case. Moreover, the percentage of involution conjugacy classes for which $\text{M\"ob}^{+}(n)$ has length two approaches zero, as $n$ approaches infinity.