Abstract
We show that for any $\varepsilon>0$, $\alpha \in 0,\frac {1}{2})$, as $g\to \infty $ a generic finite-area genus $g$ hyperbolic surface with $n=O\left (g^{\alpha }\right )$ cusps, sampled ...with probability arising from the Weil–Petersson metric on moduli space, has no non-zero eigenvalue of the Laplacian below $\frac {1}{4}-\left (\frac {2\alpha +1}{4}\right )^{2}-\varepsilon $. For $\alpha =0$ this gives a spectral gap of size $\frac {3}{16}-\varepsilon $ and for any $\alpha <\frac {1}{2}$ gives a uniform spectral gap of explicit size.
Let \(X\) be a finite-area non-compact hyperbolic surface. We study the spectrum of the Laplacian on random covering surfaces of X and on random unitary bundles over X. We show that there is a ...constant \(c > 0\) such that, with probability tending to 1 as \(n \to \infty\), a uniformly random degree-\(n\) Riemannian covering surface \(X_n\) of \(X\) has no Laplacian eigenvalues below \(\frac{1}{4}-c\frac{(\log\log\log n)^2}{\log \log n}\) other than those of \(X\) and with the same multiplicities. We also show that with probability tending to 1 as \(n\to \infty\), a random unitary bundle \(E_{\phi}\) over \(X\) of rank \(n\) has no Laplacian eigenvalues below \(\frac{1}{4}-c\frac{(\log\log n)^2}{\log n}\).
We show that for any \(\epsilon>0\), \(\alpha\in0,\frac{1}{2})\), as \(g\to\infty\) a generic finite-area genus g hyperbolic surface with \(n=O\left(g^{\alpha}\right)\) cusps, sampled with ...probability arising from the Weil-Petersson metric on moduli space, has no non-zero eigenvalue of the Laplacian below \(\frac{1}{4}-\left(\frac{2\alpha+1}{4}\right)^{2}-\epsilon\). For \(\alpha=0\) this gives a spectral gap of size \(\frac{3}{16}-\epsilon\) and for any \(\alpha<\frac{1}{2}\) gives a uniform spectral gap of explicit size.
We study the geometry and spectral theory of Weil-Petersson random surfaces
with genus-$g$ and $n$ cusps in the large-$n$ limit. We show that for a random
hyperbolic surface in $\mathcal{M}_{g,n}$ ...with $n$ large, the number of small
Laplacian eigenvalues is linear in $n$ with high probability. By work of
Ballmann, Matthiesen and Mondal 7, this result is optimal up to a
multiplicative constant. We also study the relative frequency of simple and
non-simple closed geodesics, showing that on random surfaces with many cusps,
most closed geodesics with lengths up to $\log(n)$ scales are non-simple. Our
main technical contribution is a novel large-$n$ asymptotic formula for the
Weil-Petersson volume $V_{g,n}\left(\ell_{1},\dots,\ell_{k}\right)$ of the
moduli space $\mathcal{M}_{g,n}\left(\ell_{1},\dots,\ell_{k}\right)$ of
genus-$g$ hyperbolic surfaces with $k$ geodesic boundary components and $n-k$
cusps with $k$ fixed, building on work of Manin and Zograf 31.
We prove that if \(X\) is a finite area non-compact hyperbolic surface, then for any \(\epsilon>0\), with probability tending to one as \(n\to\infty\), a uniformly random degree \(n\) Riemannian ...cover of \(X\) has no eigenvalues of the Laplacian in \(0,\frac{1}{4}-\epsilon)\) other than those of \(X\), and with the same multiplicities. As a result, using a compactification procedure due to Buser, Burger, and Dodziuk, we settle in the affirmative the question of whether there exist a sequence of closed hyperbolic surfaces with genera tending to infinity and first non-zero eigenvalue of the Laplacian tending to \(\frac{1}{4}\).
We study the number of short geodesics and small eigenvalues on Weil-Petersson random genus zero hyperbolic surfaces with n cusps in the regime \(n\to\infty\). Inspired by work of Mirzakhani and ...Petri (2019), we show that the random multi-set of lengths of closed geodesics converges, after a suitable rescaling, to a Poisson point process with explicit intensity. As a consequence, we show that the Weil-Petersson probability that a hyperbolic punctured sphere with n cusps has at least \(k=o(n^{\frac{1}{3}})\) arbitrarily small eigenvalues tends to \(1\) as \(n\to\infty\).
We prove that all finitely generated fully residually free groups (limit groups) have a sequence of finite dimensional unitary representations that `strongly converge' to the regular representation ...of the group. The corresponding statement for finitely generated free groups was proved by Haagerup and Thorbjørnsen in 2005. In fact, we can take the unitary representations to arise from representations of the group by permutation matrices, as was proved for free groups by Bordenave and Collins. As for Haagerup and Thorbjørnsen, the existence of such representations implies that for any non-abelian limit group, the Ext-invariant of the reduced \(C^{*}\)-algebra is not a group (has non-invertible elements) An important special case of our main theorem is in application to the fundamental groups of closed orientable surfaces of genus at least two. In this case, our results can be used as an input to the methods previously developed by the authors of the appendix. The output is a variation of our previous proof of Buser's 1984 conjecture that there exist a sequence of closed hyperbolic surfaces with genera tending to infinity and first eigenvalue of the Laplacian tending to \(\frac{1}{4}\). In this variation of the proof, the systoles of the surfaces are bounded away from zero and the surfaces can be taken to be arithmetic.
We study the geometry and spectral theory of Weil-Petersson random surfaces with genus-\(g\) and \(n\) cusps in the large-\(n\) limit. We show that for a random hyperbolic surface in ...\(\mathcal{M}_{g,n}\) with \(n\) large, the number of small Laplacian eigenvalues is linear in \(n\) with high probability. By work of Ballmann, Matthiesen and Mondal 7, this result is optimal up to a multiplicative constant. We also study the relative frequency of simple and non-simple closed geodesics, showing that on random surfaces with many cusps, most closed geodesics with lengths up to \(\log(n)\) scales are non-simple. Our main technical contribution is a novel large-\(n\) asymptotic formula for the Weil-Petersson volume \(V_{g,n}\left(\ell_{1},\dots,\ell_{k}\right)\) of the moduli space \(\mathcal{M}_{g,n}\left(\ell_{1},\dots,\ell_{k}\right)\) of genus-\(g\) hyperbolic surfaces with \(k\) geodesic boundary components and \(n-k\) cusps with \(k\) fixed, building on work of Manin and Zograf 31.
