The continuous part of the axial distance spectrum for Kleinian groups

Abstract

Elements $f$ of finite order in the isometry group of hyperbolic three-space $\IH^3$ have a hyperbolic line as a fixed point set, this line is the axis of $f$. The possible hyperbolic distances between axes of elements of order $p$ and $q$, not both two, among {\em all} discrete subgroups $\Gamma$ of $Isom^+(\IH^3)$ has an initial discrete spectrum \[ 0 =\delta_0< \delta_1 < \delta_2 < \ldots <\delta_\infty,\] each value taken with finite multiplicity, and above $\delta_\infty$ this spectrum of possible distances is continuous. The value $\delta_\infty$ is the smallest number with the property that for each $\lambda<1$ there are only finitely many discrete groups generated by elements of order $p$ and $q$ whose axes are no more than $\lambda \delta_\infty(p,q)$ apart. Geometrically $\delta_\infty$ places a bound on embedded tubular neighbourhoods of components of the singular set in the orbifold quotients $\IH^3/\Gamma$ and provides other geometric information about this set. The value $\delta_1(p,q)$ is known and tends to $\infty$ with $\min\{p,q\}$. Here we seek to determine - actually find asymptotically sharp upper-bounds for - $\delta_\infty(p,q)$. We also show that the gap $\delta_\infty(p,q)-\delta_1(p,q)$ is surprisingly small, less than $1.4059\ldots$, the sharp value for the Fuchsian case, independent of $p$ and $q$. This is despite both of these numbers tending to $\infty$ with either $p$ or $q$.