Alon和Friedman理论在肖特基表面上的推广
Extension of Alon's and Friedman's conjectures to Schottky surfaces
摘要：
设$X=Λ\backslash\mathbb{H}$是一个Schottky曲面，即一个面积为无穷大的共形紧致双曲曲面.令$δ$表示$Λ$的极限集的豪斯多夫维数。我们证明了，对于任何紧子集$\mathcal{K} \subset\{\,s\,:\,\Re(s)>\fracδ{2}\,\}$，如果一个随机度$n$覆盖$X$的$X_{n}$均匀随机，然后与概率趋于1 $n\to\infty$，有没有共振的$X_{n}$在$\mathcal{K}$以外的其他已经属于$X$（并具有相同的多重性）。这一结果被证明是最佳的一个有界频率共振，是类似于阿隆和弗里德曼的定理随机图，这是现在的定理，由于弗里德曼和Bordenave-Collins，分别。
英文摘要：
Let $X=Λ\backslash\mathbb{H}$ be a Schottky surface, that is, a conformally compact hyperbolic surface of infinite area. Let $δ$ denote the Hausdorff dimension of the limit set of $Λ$. We prove that for any compact subset $\mathcal{K} \subset\{\,s\,:\,\Re(s)>\fracδ{2}\,\}$, if one picks a random degree $n$ cover $X_{n}$ of $X$ uniformly at random, then with probability tending to one as $n\to\infty$, there are no resonances of $X_{n}$ in $\mathcal{K}$ other than those already belonging to $X$ (and with the same multiplicity). This result is conjectured to be the optimal one for bounded frequency resonances and is analogous to both Alon's and Friedman's conjectures for random graphs, which are now theorems due to Friedman and Bordenave-Collins, respectively.
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