Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\Gamma$ 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 $q$.

35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
16
The categories of this paper are: stat
19
The categories of this paper are: eess
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc