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Spectral gap with polynomial rate for random covering surfaces
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Spectral gap with polynomial rate for random covering surfaces
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In this note we show that the recent work of Magee, Puder and van Handel [MPvH25] can be applied to obtain an optimal spectral gap result with polynomial error rate for uniformly random covers of closed hyperbolic surfaces. Let $X$ be a closed hyperbolic surface. We show there exists $b,c>0$ such that a uniformly random degree-$n$ cover $X_{n}$ of $X$ has no new Laplacian eigenvalues below $\frac{1}{4}-cn^{-b}$ with probability tending to $1$ as $n\to\infty$.
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