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All Outputs (22)

Quantum Unique Ergodicity for Cayley graphs of quasirandom groups (2023)
Journal Article
Magee, M., Thomas, J., & Zhao, Y. (2023). Quantum Unique Ergodicity for Cayley graphs of quasirandom groups. Communications in Mathematical Physics, https://doi.org/10.1007/s00220-023-04801-x

A finite group G is called C-quasirandom (by Gowers) if all non-trivial irreducible complex representations of G have dimension at least C. For any unit ℓ2 function on a finite group we associate the quantum probability measure on the group given by... Read More about Quantum Unique Ergodicity for Cayley graphs of quasirandom groups.

Near optimal spectral gaps for hyperbolic surfaces (2023)
Journal Article
Hide, W., & Magee, M. (2023). Near optimal spectral gaps for hyperbolic surfaces. Annals of Mathematics, 198(2), 791-824. https://doi.org/10.4007/annals.2023.198.2.6

We prove that if X is a finite area non-compact hyperbolic surface, then for any ϵ > 0, with probability tending to one as n → ∞, a uniformly random degree n Riemannian cover of X has no eigenvalues of the Laplacian in [0, 1 4 − ϵ) other than those o... Read More about Near optimal spectral gaps for hyperbolic surfaces.

The Asymptotic Statistics of Random Covering Surfaces (2023)
Journal Article
Magee, M., & Puder, D. (2023). The Asymptotic Statistics of Random Covering Surfaces. Forum of mathematics. Pi, 11, Article e15. https://doi.org/10.1017/fmp.2023.13

Let Γg be the fundamental group of a closed connected orientable surface of genus g ≥ 2. We develop a new method for integrating over the representation space Xg,n = Hom(Γg, Sn) where Sn is the symmetric group of permutations of {1, . . . , n}. Equiv... Read More about The Asymptotic Statistics of Random Covering Surfaces.

Random Unitary Representations of Surface Groups II: The large n limit (2023)
Journal Article
Magee, M. (in press). Random Unitary Representations of Surface Groups II: The large n limit. Geometry & Topology,

Let Σg be a closed surface of genus g ≥ 2 and Γg denote the fundamental group of Σg. We establish a generalization of Voiculescu’s theorem on the asymptotic ∗-freeness of Haar unitary matrices from free groups to Γg. We prove that for a random repres... Read More about Random Unitary Representations of Surface Groups II: The large n limit.

Core surfaces (2022)
Journal Article
Magee, M., & Puder, D. (2022). Core surfaces. Geometriae Dedicata, 216(4), Article 46. https://doi.org/10.1007/s10711-022-00706-6

Let Γg be the fundamental group of a closed connected orientable surface of genus g≥2. We introduce a combinatorial structure of core surfaces, that represent subgroups of Γg. These structures are (usually) 2-dimensional complexes, made up of vertice... Read More about Core surfaces.

A random cover of a compact hyperbolic surface has relative spectral gap 3/16 - ϵ (2022)
Journal Article
Magee, M., Naud, F., & Puder, D. (2022). A random cover of a compact hyperbolic surface has relative spectral gap 3/16 - ϵ. Geometric And Functional Analysis, 32(3), 595-661. https://doi.org/10.1007/s00039-022-00602-x

Let X be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature −1. For each n ∈ N, let Xn be a random degree-n cover of X sampled uniformly from all degree-n Riema... Read More about A random cover of a compact hyperbolic surface has relative spectral gap 3/16 - ϵ.

Random Unitary Representations of Surface Groups I: Asymptotic expansions (2021)
Journal Article
Magee, M. (2022). Random Unitary Representations of Surface Groups I: Asymptotic expansions. Communications in Mathematical Physics, 391(1), 119-171. https://doi.org/10.1007/s00220-021-04295-5

In this paper, we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott and Goldman. Let Σg denote a topological surface o... Read More about Random Unitary Representations of Surface Groups I: Asymptotic expansions.

Surface Words are Determined by Word Measures on Groups (2021)
Journal Article
Magee, M., & Puder, D. (2021). Surface Words are Determined by Word Measures on Groups. Israel Journal of Mathematics, 241, 749-774. https://doi.org/10.1007/s11856-021-2113-5

Every word w in a free group naturally induces a probability measure on every compact group G. For example, if w = [x, y] is the commutator word, a random element sampled by the w-measure is given by the commutator [g, h] of two independent, Haar-ran... Read More about Surface Words are Determined by Word Measures on Groups.

Kesten-McKay law for the Markoff surface mod p (2021)
Journal Article
Courcy-Ireland, M. D., & Magee, M. (2021). Kesten-McKay law for the Markoff surface mod p. Annales Henri Lebesgue, 4, 227-250. https://doi.org/10.5802/ahl.71

For each prime p, we study the eigenvalues of a 3-regular graph on roughly vertices constructed from the Markoff surface. We show they asymptotically follow the Kesten–McKay law, which also describes the eigenvalues of a random regular graph. The pro... Read More about Kesten-McKay law for the Markoff surface mod p.

Explicit spectral gaps for random covers of Riemann surfaces (2020)
Journal Article
Magee, M., & Naud, F. (2020). Explicit spectral gaps for random covers of Riemann surfaces. Publications mathématiques de l'IHÉS, 132(1), 137-179. https://doi.org/10.1007/s10240-020-00118-w

We introduce a permutation model for random degree n covers Xn of a non-elementary convex-cocompact hyperbolic surface X = \H. Let δ be the Hausdorff dimension of the limit set of . We say that a resonance of Xn is new if it is not a resonance of X,... Read More about Explicit spectral gaps for random covers of Riemann surfaces.

On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials (2019)
Journal Article
Magee, M. (2019). On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials. Compositio Mathematica, 155(12), 2354-2398. https://doi.org/10.1112/s0010437x1900767x

J.-C. Yoccoz proposed a natural extension of Selberg’s eigenvalue conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg’s theorem to moduli spaces of abe... Read More about On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials.

An asymptotic formula for integer points on Markoff-Hurwitz varieties (2019)
Journal Article
Gamburd, A., Magee, M., & Ronan, R. (2019). An asymptotic formula for integer points on Markoff-Hurwitz varieties. Annals of Mathematics, 190(3), 751-809. https://doi.org/10.4007/annals.2019.190.3.2

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation x21+x22+⋯+x2n=ax1x2⋯xn+k. When n≥4, the previous best result is by Baragar (1998) that gives an exponential rate of growth with exponent β that is... Read More about An asymptotic formula for integer points on Markoff-Hurwitz varieties.

The cycle structure of a Markoff automorphism over finite fields (2019)
Journal Article
Cerbu, A., Gunther, E., Magee, M., & Peilen, L. (2020). The cycle structure of a Markoff automorphism over finite fields. Journal of Number Theory, 211, 1-27. https://doi.org/10.1016/j.jnt.2019.09.022

We begin an investigation of the action of pseudo-Anosov elements of Out(F2) on the Marko-type varieties X : x2 + y2 + z2 = xyz + 2 + over nite elds Fp with p prime. We rst make a precise conjecture about the permutation group generated by Out(F2) on... Read More about The cycle structure of a Markoff automorphism over finite fields.

Matrix group integrals, surfaces, and mapping class groups I: U(n) (2019)
Journal Article
Magee, M., & Puder, D. (2019). Matrix group integrals, surfaces, and mapping class groups I: U(n). Inventiones Mathematicae, 218(2), 341-411. https://doi.org/10.1007/s00222-019-00891-4

Since the 1970’s, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces.We establish a new aspect of this t... Read More about Matrix group integrals, surfaces, and mapping class groups I: U(n).

Counting One-Sided Simple Closed Geodesics on Fuchsian Thrice Punctured Projective Planes (2018)
Journal Article
Magee, M. (2020). Counting One-Sided Simple Closed Geodesics on Fuchsian Thrice Punctured Projective Planes. International Mathematics Research Notices, 2020(13), 3886-3901. https://doi.org/10.1093/imrn/rny112

We prove that there is a true asymptotic formula for the number of one-sided simple closed curves of length ≤ L on any Fuchsian real projective plane with three points removed. The exponent of growth is independent of the hyperbolic metric, and it is... Read More about Counting One-Sided Simple Closed Geodesics on Fuchsian Thrice Punctured Projective Planes.

Uniform congruence counting for Schottky semigroups in SL2(𝐙) (2017)
Journal Article
Magee, M., Oh, H., & Winter, D. (2019). Uniform congruence counting for Schottky semigroups in SL2(𝐙). Journal für die reine und angewandte Mathematik, 2019(753), 89-135. https://doi.org/10.1515/crelle-2016-0072

Let Γ be a Schottky semigroup in SL2(Z), and for q∈N, let Γ(q):={γ∈Γ:γ=e(modq)} be its congruence subsemigroup of level q. Let δ denote the Hausdorff dimension of the limit set of Γ. We prove the following uniform congruence counting theorem with res... Read More about Uniform congruence counting for Schottky semigroups in SL2(𝐙).