arXiv subjects

Michael Magee

Publications and source records attributed to Michael Magee.

36 records · Page 2Linked to original sources

arXiv2021

Extension of Alon's and Friedman's conjectures to Schottky surfaces

Let $X=Λ\backslash\mathbb{H}$ be a Schottky surface, that is, a conformally compact hyperbolic surface of infinite area. Let $δ$ denote the Hausdorff dimension of the limit set of $Λ$. We prove that for any compact subset $\mathcal{K} \subset\{\,s\,:\,\Re(s)>\fracδ{2}\,\}$, if one picks a random degree $n$ cover $X_{n}$ of $X$ uniformly at random, then with probability tending to one as $n\to\infty$, there are no resonances of $X_{n}$ in $\mathcal{K}$ other than those already belonging to $X$ (and with the same multiplicity). This result is conjectured to be the optimal one for bounded frequency resonances and is analogous to both Alon's and Friedman's conjectures for random graphs, which are now theorems due to Friedman and Bordenave-Collins, respectively.

math.SP↗

arXiv2021

Random Unitary Representations of Surface Groups I: Asymptotic expansions

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 $Σ_{g}$ denote a topological surface of genus $g\geq2$. We establish the existence of a large $n$ asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of $π_{1}(Σ_{g})$ under a random representation of $π_{1}(Σ_{g})$ into $\mathsf{SU}(n)$. Each such expected value involves a contribution from all irreducible representations of $\mathsf{SU}(n)$. The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.

math.RT↗

arXiv2020

Explicit spectral gaps for random covers of Riemann surfaces

We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X=Γ\backslash\mathbb{H}$. Let $δ$ be the Hausdorff dimension of the limit set of $Γ$. We say that a resonance of $X_{n}$ is new if it is not a resonance of $X$, and similarly define new eigenvalues of the Laplacian. We prove that for any $ε>0$ and $H>0$, with probability tending to $1$ as $n\to\infty$, there are no new resonances $s=σ+it$ of $X_{n}$ with $σ\in[\frac{3}{4}δ+ε,δ]$ and $t\in[-H,H]$. This implies in the case of $δ>\frac{1}{2}$ that there is an explicit interval where there are no new eigenvalues of the Laplacian on $X_{n}$. By combining these results with a deterministic `high frequency' resonance-free strip result, we obtain the corollary that there is an $η=η(X)$ such that with probability $\to1$ as $n\to\infty$, there are no new resonances of $X_{n}$ in the region $\{\,s\,:\,\mathrm{Re}(s)>δ-η\,\}$.

math.SP↗

arXiv2019

Automorphism-invariant positive definite functions on free groups

In this article we raise some new questions about positive definite functions on free groups, and explain how these are related to more well-known questions. The article is intended as a survey of known results that also offers some new perspectives and interesting observations; therefore the style is expository.

math.OA↗

arXiv2019

Surface Words are Determined by Word Measures on Groups

Every word $w$ in a free group naturally induces a probability measure on every compact group $G$. For example, if $w=\left[x,y\right]$ is the commutator word, a random element sampled by the $w$-measure is given by the commutator $\left[g,h\right]$ of two independent, Haar-random elements of $G$. Back in 1896, Frobenius showed that if $G$ is a finite group and $ψ$ an irreducible character, then the expected value of $ψ\left(\left[g,h\right]\right)$ is $\frac{1}{ψ\left(e\right)}$. This is true for any compact group, and completely determines the $\left[x,y\right]$-measure on these groups. An analogous result holds with the commutator word replaced by any surface word. We prove a converse to this theorem: if $w$ induces the same measure as $\left[x,y\right]$ on every compact group, then, up to an automorphism of the free group, $w$ is equal to $\left[x,y\right]$. The same holds when $\left[x,y\right]$ is replaced by any surface word. The proof relies on the analysis of word measures on unitary groups and on orthogonal groups, which appears in separate papers, and on new analysis of word measures on generalized symmetric groups that we develop here.

math.GR↗

arXiv2019

Matrix Group Integrals, Surfaces, and Mapping Class Groups I: $U(n)$

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 theory: for random matrices sampled from the group $\mathcal{U}\left(n\right)$ of unitary matrices. More concretely, we study measures induced by free words on $\mathcal{U}\left(n\right)$. Let $F_{r}$ be the free group on $r$ generators. To sample a random element from $\mathcal{U}\left(n\right)$ according to the measure induced by $w\in F_{r}$, one substitutes the $r$ letters in $w$ by $r$ independent, Haar-random elements from $\mathcal{U}\left(n\right)$. The main theme of this paper is that every moment of this measure is determined by families of pairs $\left(Σ,f\right)$, where $Σ$ is an orientable surface with boundary, and $f$ is a map from $Σ$ to the bouquet of $r$ circles, which sends the boundary components of $Σ$ to powers of $w$. A crucial role is then played by Euler characteristics of subgroups of the mapping class group of $Σ$. As corollaries, we obtain asymptotic bounds on the moments, we show that the measure on $\mathcal{U}\left(n\right)$ bears information about the number of solutions to the equation $\left[u_{1},v_{1}\right]\cdots\left[u_{g},v_{g}\right]=w$ in the free group, and deduce that one can ``hear'' the stable commutator length of a word through its unitary word measures.

math.GT↗

arXiv2018

Kesten-McKay law for the Markoff surface mod p

For each prime $p$, we study the eigenvalues of a 3-regular graph on roughly $p^2$ 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 proof is based on the method of moments and takes advantage of a natural group action on the Markoff surface.

math.NT↗

arXiv2018

On Selberg's Eigenvalue Conjecture for moduli spaces of abelian differentials

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 $\frac{3}{16}$ Theorem to moduli spaces of abelian differentials on surfaces of genus $\geq 2$.

math.DS↗

arXiv2018

Counting saddle connections in a homology class modulo $q$

We give effective estimates for the number of saddle connections on a translation surface that have length $\leq L$ and are in a prescribed homology class modulo $q$. Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur-Veech measure on the stratum.

math.DS↗

arXiv2018

An asymptotic formula for integer points on Markoff-Hurwitz varieties

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation \[ x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}=ax_{1}x_{2}\ldots x_{n}+k. \] When $n\geq4$ the previous best result is by Baragar (1998) that gives an exponential rate of growth with exponent $β$ that is not in general an integer when $n\geq 4$. We give a new interpretation of this exponent of growth in terms of the unique parameter for which there exists a certain conformal measure on projective space.

math.NT↗

arXiv2018

The cycle structure of a Markoff automorphism over finite fields

We begin an investigation of the action of pseudo-Anosov elements of $\mathrm{Out}(\mathbf{F}_{2})$ on the Markoff-type varieties \[ \mathbb{X}_κ:\:x^{2}+y^{2}+z^{2}=xyz+2+κ\] over finite fields $\mathbb{F}_{p}$ with $p$ prime. We first make a precise conjecture about the permutation group generated by $\mathrm{Out}(\mathbf{F}_{2})$ on $\mathbb{X}_{-2}(\mathbb{F}_{p})$ that shows there is no obstruction at the level of the permutation group to a pseudo-Anosov acting `generically'. We prove that this conjecture is sharp. We show that for a fixed pseudo-Anosov $g\in\mathrm{Out}(\mathbf{F}_{2})$, there is always an orbit of $g$ of length $\geq C\log p+O(1)$ on $\mathbb{X}_κ(\mathbb{F}_{p})$ where $C>0$ is given in terms of the eigenvalues of $g$ viewed as an element of $\mathrm{GL}_{2}(\mathbf{Z})$. This improves on a result of Silverman (2007) that applies to general morphisms of quasi-projective varieties. We have discovered that the asymptotic $(p\to\infty)$ behavior of the longest orbit of a fixed pseudo-Anosov $g$ acting on $\mathbb{X}_{-2}(\mathbb{F}_{p})$ is dictated by a dichotomy that we describe both in combinatorial terms and in algebraic terms related to Gauss's ambiguous binary quadratic forms, following Sarnak. This dichotomy is illustrated with numerics, based on which we formulate a precise conjecture.

math.NT↗

arXiv2017

Counting one sided simple closed geodesics on Fuchsian thrice punctured projective planes

We prove that there is a true asymptotic formula for the number of one sided simple closed curves of length $\leq L$ on any Fuchsian real projective plane with three points removed. The exponent of growth is independent of the hyperbolic structure, and it is noninteger, in contrast to counting results of Mirzakhani for simple closed curves on orientable Fuchsian surfaces.

math.GT↗

arXiv2017

Uniform congruence counting for Schottky semigroups in $\mathrm{SL}_2(\mathbf{Z})$

Let $Γ$ be a Schottky semigroup in $\mathrm{SL}_2(\mathbf{Z})$, and for $q\in \mathbf N$, let $Γ(q):=\{γ\in Γ: γ= e \text{ (mod $q$)}\}$ be its congruence subsemigroup of level $q$. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls $B_R$ in $M_2(\mathbf{R})$ of radius $R$: for all $q$ with no small prime factors, $ (Γ(q) \cap B_R )= c_Γ\frac{R^{2δ}}{ (\mathrm{SL}_2(\mathbf{Z}/q\mathbf{Z}))} +O(q^C R^{2δ-ε})$ as $R\to \infty$ for some $c_Γ>0, C>0, ε>0$ which are independent of $q$. Our technique also applies to give a similar counting result for the continued fractions semigroup of $\mathrm{SL}_2(\mathbf{Z})$, which arises in the study of Zaremba's conjecture on continued fractions.

math.NT↗

arXiv2016

Word Measures on Unitary Groups

We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More particularly, we study measures induced by free words on the unitary groups $U(n)$. Every word $w$ in the free group $F_r$ on $r$ generators determines a word map from $U(n)^r$ to $U(n)$, defined by substitutions. The $w$-measure on $U(n)$ is defined as the pushforward via this word map of the Haar measure on $U(n)^r$. Let $Tr_w(n)$ denote the expected trace of a random unitary matrix sampled from $U(n)$ according to the $w$-measure. It was shown by Voiculescu [Voic 91'] that for $w \ne 1$ this expected trace is $o(n)$ asymptotically in $n$. We relate the numbers $Tr_w(n)$ to the theory of commutator length of words and obtain a much stronger statement: $Tr_w(n)=O(n^{1-2g})$, where $g$ is the commutator length of $w$. Moreover, we analyze the number $\lim_{n\to\infty}n^{2g-1} \cdot Tr_w(n)$ and show it is an integer which, roughly, counts the number of (equivalence classes of) solutions to the equation $[u_1,v_1]...[u_g,v_g]=w$ with $u_i,v_i \in F_r$. Similar results are obtained for finite sets of words and their commutator length, and we deduce that one can 'hear' the stable commutator length of a word by 'listening' to its unitary measures.

math.GR↗

arXiv2015

Thermodynamic expansion to arbitrary moduli

We extend the thermodynamic expansion results in [BourgainGamburdSarnak2011, MageeOhWinter2015] from square-free to arbitrary moduli by developing a novel decoupling technique and applying [BourgainVarju2012].

math.NT↗

arXiv2015

Arithmetic, zeros, and nodal domains on the sphere

We obtain lower bounds for the number of nodal domains of Hecke eigenfunctions on the sphere. Assuming the generalized Lindelof hypothesis we prove that the number of nodal domains of any Hecke eigenfunction grows with the eigenvalue of the Laplacian. By a very different method, we show unconditionally that the average number of nodal domains of degree l Hecke eigenfunctions grows significantly faster than the uniform growth obtained under Lindelof.

math.NT↗

arXiv2015

Expanding maps and continued fractions

We obtain a power saving in the error term for a semigroup congruence lattice point count related to continued fractions. This is done by adapting arguments from recent work of Oh and Winter (2014) that give uniform bounds for certain transfer operators in the congruence aspect. Our arguments also build crucially on work of Naud (2005) and Bourgain, Gamburd and Sarnak (2011). The result we obtain, together with a certain conjecture about the multiplicative combinatorics of $\mathrm{SL}_2(\mathbf{Z})$ that we highlight in the sequel, can be used to obtain an improvement on the size of the exceptional set in Bourgain and Kontorovich's work (2014) on Zaremba's conjecture.

math.NT↗

arXiv2013

Quantitative spectral gap for thin groups of hyperbolic isometries

Let $Λ$ be a subgroup of an arithmetic lattice in SO(n+1,1). The quotient $\mathbb{H}^{n+1} / Λ$ has a natural family of congruence covers corresponding to primes in some ring of integers. We establish a super-strong approximation result for Zariski-dense $Λ$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).

math.SP↗