The continuous part of the axial distance spectrum for Kleinian groups

TL;DR

This paper investigates the spectrum of possible hyperbolic distances between axes of finite order elements in Kleinian groups, establishing bounds and continuity properties that inform geometric structures of associated orbifolds.

Contribution

It determines asymptotically sharp upper bounds for the continuous part of the distance spectrum and reveals the surprisingly small gap between initial and limiting spectra.

Findings

Established bounds for elta_(p,q)

Proved the gap elta_(p,q) - elta_1(p,q) is less than 1.4059

Showed elta_(p,q) tends to infinity with p,q

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 $q$.