On the Margulis constant for Kleinian groups, I curvature

Abstract

The Margulis constant for Kleinian groups is the smallest constant $c$ such that for each discrete group $G$ and each point $x$ in the upper half space ${\mathbf{H}}^3$, the group generated by the elements in $G$ which move $x$ less than distance c is elementary. We take a first step towards determining this constant by proving that if $⟨f,g⟩$ is nonelementary and discrete with $f$ parabolic or elliptic of order $n\ge 3$, then every point $x$ in ${\mathbf{H}}^3$ is moved at least distance $c$ by $f$ or $g$ where $c=.1829\dots$. This bound is sharp.