Generating the Mobius group with involution conjugacy classes

TL;DR

This paper establishes bounds on the word length of Möbius groups generated by involution conjugacy classes, revealing linear growth with dimension and asymptotic behavior of involution classes as dimension increases.

Contribution

It provides effective bounds for the word length of Möbius groups generated by involutions, and analyzes their growth and distribution as the dimension increases.

Findings

Word length bounds grow linearly with dimension.

Percentage of involution classes with length two approaches zero as dimension increases.

Growth behavior differs for even and odd involution codimensions.

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.