The continuous part of the axial distance spectrum for Kleinian groups

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 $Γ$ of $Isom^+(\IH^3)$ has an initial discrete spectrum $0 =δ_0< δ_1 < δ_2 < \ldots <δ_\infty,$ each value taken with finite multiplicity, and above $δ_\infty$ this spectrum of possible distances is continuous. The value $δ_\infty$ is the smallest number with the property that for each $λ<1$ there are only finitely many discrete groups generated by elements of order $p$ and $q$ whose axes are no more than $λδ_\infty(p,q)$ apart. Geometrically $δ_\infty$ places a bound on embedded tubular neighbourhoods of components of the singular set in the orbifold quotients $\IH^3/Γ$ and provides other geometric information about this set. The value $δ_1(p,q)$ is known and tends to $\infty$ with $\min\{p,q\}$. Here we seek to determine - actually find asymptotically sharp upper-bounds for - $δ_\infty(p,q)$. We also show that the gap $δ_\infty(p,q)-δ_1(p,q)$ 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 $q$.

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