标题： 曲面群的均匀随机置换表示的强收敛性 显示英文标题

标题： Strong convergence of uniformly random permutation representations of surface groups

摘要： 设$\Gamma$为至少具有二 genus 的闭定向曲面的基本群。 考虑将$\mathrm{Hom}(\Gamma,S_n)$中的一个均匀随机元素与$S_n$的$(n-1)$维不可约表示的合成。 我们证明当$n\to\infty$趋于无穷时，这一随机表示序列以强收敛概率收敛到$\Gamma$的正则表示。 因此，对于任意闭合双曲曲面$X$，随着$n\to\infty$趋向于某个值时，具有概率接近一的均匀随机度-$n$覆盖空间的$X$几乎具有最优的相对谱隙——忽略由基底曲面$X$引起的特征值。为了实现这一点，我们证明了多项式方法证明强收敛可以扩展到超越有理设置的情况。为了满足这种扩展的要求，我们证明了两种新的结果。首先，我们展示了在随机同态映射到$S_n$的情况下，存在有效的多项式逼近$\Gamma$元素迹的期望值。 其次，我们估计在给定步数后，定义在$\Gamma$上的有限支撑随机游走成为真幂的概率的增长率。 显示英文摘要

摘要： Let $\Gamma$ be the fundamental group of a closed orientable surface of genus at least two. Consider the composition of a uniformly random element of $\mathrm{Hom}(\Gamma,S_n)$ with the $(n-1)$-dimensional irreducible representation of $S_n$. We prove the strong convergence in probability as $n\to\infty$ of this sequence of random representations to the regular representation of $\Gamma$. As a consequence, for any closed hyperbolic surface $X$, with probability tending to one as $n\to\infty$, a uniformly random degree-$n$ covering space of $X$ has near optimal relative spectral gap -- ignoring the eigenvalues that arise from the base surface $X$. To do so, we show that the polynomial method of proving strong convergence can be extended beyond rational settings. To meet the requirements of this extension we prove two new kinds of results. First, we show there are effective polynomial approximations of expected values of traces of elements of $\Gamma$ under random homomorphisms to $S_n$. Secondly, we estimate the growth rates of probabilities that a finitely supported random walk on $\Gamma$ is a proper power after a given number of steps.