论文标题
克莱恩基团的轴向距离光谱的连续部分
The continuous part of the axial distance spectrum for Kleinian groups
论文作者
论文摘要
在双曲线三个空间$ \ ih^3 $的等轴测订单组中的元素$ f $具有双曲线线作为固定点集，该行是$ f $的轴。 {\ em all}离散子组中的$ p $和$ q $元素轴轴的轴轴的轴轴轴的轴轴的轴轴的轴轴的轴$^$^+（\ ih^3）$之间可能的双曲线距离（\ ih^3）$具有初始离散频谱\ [0 =Δ_0<δ_1<δ_2<δ_2<δ_2<δ____ $δ_\ infty $此可能的距离频谱是连续的。 $δ_\ infty $是最小的属性数字，每个属性对于每个$λ<1 $，只有订单$ p $和$ p $和$ q $的轴轴不超过$λΔ__\ infty（p，q）$的元素生成的许多离散组。几何$δ_\ infty $将一个嵌入式的管状邻域绑定到Orbifold商中的单数组件的嵌入式管状社区$ \ ih^3/γ$，并提供了有关此集合的其他几何信息。 $δ_1（p，q）$是已知的，并倾向于$ \ infty $，带有$ \ min \ {p，q \} $。在这里，我们试图确定 - 实际上发现 - $δ_\ infty（p，q）$的渐近上限。我们还表明，差距$δ_\ infty（p，q）-Δ_1（p，q）$很小，小于$ 1.4059 \ ldots $，fuchsian案件的尖锐价值，独立于$ p $和$ q $。尽管这两个数字都倾向于$ \ infty $，其中包括$ p $或$ q $。
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 $q$.
