REVIEW 5 cited by
Strong convergence of uniformly random permutation representations of surface groups
Not yet reviewed by Pith; the record is open.
This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.
SPECIMEN: schema-true, not a live event
T0 review · schema-true
One-sentence machine reading of the paper's core claim.
pith:XXXXXXXX · record.json · timestamp
abstract
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.
Forward citations
Cited by 5 Pith papers
-
Nearly optimal spectral gaps for random Belyi surfaces
Random Brooks–Makover surfaces have first Laplacian eigenvalue > 1/4 − ε with probability → 1 for every ε > 0.
-
Word maps and surface relations in symmetric groups
For a shortest representative word w of a non-trivial surface group element, the stable-character-weighted integral with the surface relator is O(n^{-k}); with a unique shortest representative it improves to O(n^{-k-1}).
-
Spectral gaps for noncompact hyperbolic surfaces with linearly many cusps
Random and planted graphs with degree-3 interior and degree-1 boundary vertices yield expander families, and their pants decompositions give hyperbolic surfaces with n comparable to g cusps and a uniform spectral gap.
-
On the spectral stability of finite coverings
Random finite covers of manifolds with Ricci curvature bounded below have no new Laplacian eigenvalues in [0,Λ] when the fundamental group satisfies strong convergence of permutation representations.
-
Bass notes of random hyperbolic surfaces of large genus
A survey of recent results proving that random hyperbolic surfaces of large genus have near-optimal spectral gaps, after Hide–Magee, Anantharaman–Monk, and Hide–Macera–Thomas.
Discussion (0). Continue with ORCID to comment.