Generating the Möbius group with involution conjugacy classes

Abstract:

A $k$-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 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 for $\text {M\"ob}(n)$ 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.