Spectra/Moduli seminar

On a finite-area hyperbolic surface with genus g and n cusps, the eigenvalues of the Laplacian below 1/4 play an important role in understanding the geometry of the surface and dynamics of its geodesic flow. A natural question is to understand how many of these small eigenvalues a surface can have. Work of Otal and Rosas proves that there can be at most 2g+n-2 small eigenvalues and simple examples can be constructed to show that this is sharp. In this talk I will discuss forthcoming work with Will Hide (Oxford) proving that if the genus is not too large compared to the number of cusps, any surface has at least const*(2g+n-2)/log(2g+n-2) small eigenvalues. I will also discuss how the log(2g+n-2) factor can be removed under some mild assumptions on the surface geometry.

In this talk, we consider two models of random geometry of surfaces in high genus: combinatorial maps and hyperbolic surfaces. The geometric properties of random hyperbolic surfaces under the Weil–Petersson measure as the genus tends to infinity have been studied for around 15 years now, building notably on enumerative results of Mirzakhani. On the other hand, random combinatorial maps were first studied in the planar/finite genus case, and then in the high genus regime starting 10 years ago with unicellular (i.e. one-faced) maps. In a joint work with Svante Janson, we noticed some numerical coincidence regarding the counting of short, closed curves in unicellular maps/hyperbolic surfaces in high genus (comparing it to results of Mirzakhani and Petri). This leads us to conjecture some similarities between the two models in the limit, and raises several other open questions.

We are interested in studying the behavior of geometric invariants of hyperbolic 3-manifolds, such as the length of their geodesics. A way to do so is by using probabilistic methods. That is, we consider a set of hyperbolic manifolds, put a probability measure on it, and ask what is the probability that a random manifold has a certain property. There are several models of construction of random manifolds. In this talk, I will explain one of the principal probabilistic models for 3 dimensions and I will present a result concerning the length spectrum -the set of lengths of all closed geodesics- of a 3-manifold constructed under this model.

The notion of generic/mean points goes back to the seminal work of Krylov and Bogolyubov. The first to investigate the question of what happens when all points of a dynamical system are generic for some invariant measure seem to be Dowker and Lederer in 1964. As it turns out, combining this property with other topological regularity criteria yields measure-theoretic rigidity results of the dynamical system. For example, minimality of the system implies its unique ergodicity in this setting. Another natural topological criterion in place of minimality is to assume that the map, which assigns each point its invariant measure to which it is generic, is continuous. By several recent works by different authors, the following picture emerges for abelian group actions in this setting: each point is generic for some ergodic measure and even stronger, each orbit closure is uniquely ergodic. In my talk, I will show that this is no longer the case for general actions by topological amenable groups, providing concrete counter-examples involving the group of all orientation preserving homeomorphisms on the unit interval as well as the Lamplighter group. This is joint work with G. Fuhrmann and T. Hauser.

Small eigenvalues of a surface are eigenvalues of the Laplacian on the surface that are below the bottom of the spectrum of the Laplacian of the universal cover of the surface. In this talk I will give a brief history of the problems related to these eigenvalues and known results.

We will introduce and discuss a 'moduli space' of metric structures on hyperbolic groups that can be seen as a coarse geometric extension of Teichmüller Space or Outer Space. This space will provide us with a setting in which we can compare various distances coming from different parts of geometry. After equipping our space with a metric, we will look at its geometric properties and see how, by studying these properties, we can learn about metrics on groups. This is based on joint work with Eduardo Reyes.

Bernoulli convolutions are examples of stationary measures to the iterated function system {lambda x -1,lambda x +1} in R for 1/2 < lambda < 1. Studying smoothness of Bernoulli convolutions (e.g. absolute continuity, dimension, rate of decay of Fourier transform) is notoriously difficult due to overlaps in the IFS, which has seen much recent progress especially on the dimension theory by Hochman, Shmerkin, Varjú and various others. The rate of decay of Fourier transform of the Bernoulli convolution is mostly unexplored territory, and only some cases of Bernoulli convolutions are known to have explicit rate of decay, such as Garsia numbers where the decay is polynomial or for random lambda by the Erdös-Kahane method. In this talk I will discuss a consequence of a recent joint work with S. Baker (Loughborough), where by introducing a small non-linear perturbation to the IFS {lambda x -1,lambda x +1} defining the Bernoulli convolution, we have that any stationary measure to the perturbed IFS must have polynomial Fourier decay rate, even with any type of overlaps. The proof is based on Bourgain’s discretised sum-product theorem combined with a new uniform spectral gap theorem for complex transfer operators with overlaps, where we introduce a co-cycle version of Dolgopyat’s method that allows us to deal with the overlap structure.

An overview of some of the issues involved in studying the space of all compact group automorphisms modulo measurable and topological equivalence.

The problem of finding the maximal possible multiplicity of the first Laplacian eigenvalues has been studied at least since the 1970's. I will present a recent work in collaboration with Simon Machado (ETH Zürich) in which we proved, for negatively curved surfaces, the first upper bound which is sublinear in the genus g. Our method also yields an upper bound on the number of eigenvalues in small spectral windows, and this upper bound is shown to be nearly sharp. We also obtain results for higher-dimensional manifolds. Our proof combines a trace argument for the heat kernel and a geometric idea introduced in the context of graphs of bounded degree in a paper by Jiang-Tidor-Yao-Zhang-Zhao (2021). Our work provides new insights on a conjecture by Colin de Verdière and a new way to transfer spectral results from graphs to surfaces.

Let $F$ be a family of finite coverings of a hyperbolic surface S. A spectral gap of F is an interval I = [0, a] such that the eigenvalues in I (counted with multiplicity) of the Laplacian \Delta_S of S and \Delta_X, any X in F, are the same. I will present a joint work with M. Magee where we give a spectral gap for congruence coverings when S is the surface associated to a Schottky subgroup of SL(2,Z) with thick enough limit set. The proof exploits the link between eigenvalues of the Laplacian and zeros of dynamical zeta functions attached to S via the thermodynamic formalism.

The aim of this talk will be to construct functions on a cyclic group of odd order whose ''convolution square'' is proportional to their square. For that, we will have to interpret the cyclic group as a subgroup of an abelian variety with complex multiplication, and to use the modularity properties of their theta functions.

After a brief introduction to subject of spherical representations of hyperbolic groups, I will present a new construction motivated by a spectral formulation of the so-called Shalom conjecture.This a joint work with Dr Jan Spakula.

In this talk I will discuss about models of random unitary matrices of large size. Considering r matrices of size N, this can be thought of as a random, N dimensional, unitary representation of the free group of rank r, and leads to a random state on the group algebra of the free group. Since the seminal work of Voiculescu in the 90’s, it is well established that independent, uniform unitary matrices are asymptotically, freely independent: the latter random state converges weakly, in probability towards the character of the regular representation of the free group, as N goes to infinity.
On Friday, I will consider models of unitary matrices which are not independent, but with conditions on their commutators. I will review recent results of M. Magee on the so called Atiyah-Bott-Goldman measure — a model of random unitary representation of surface groups — and of two recent joint works of myself with T. Lemoine about the Yang-Mills measure on surfaces, which can be thought of as a model interpolating between Voiculescu’s and Atiyah-Bott-Goldman’s settings. For the torus, the latter joint works lead to a new interpolation between classical and free independence.

We establish the local limit theorem for biased random walks on a simply connected nilpotent Lie group G. The result allows to approximate at scale 1 the n-step distribution of the walk by the time n of a smooth diffusion process for a new group structure on G. We also show this approximation is robust under deviation. The proof uses a Gaussian replacement scheme, combining Fourier analysis and a swapping argument inspired by the work of Diaconis-Hough. As a consequence, we obtain a probabilistic version of Ratner's theorem: Ad-unipotent random walks on finite-volume homogeneous spaces equidistribute toward algebraic measures. Joint work with Emmanuel Breuillard.

Conformal Bootstrap is an active area of research in theoretical physics, which aims to constrain a subclass of quantum field theories starting only from a set of basic self-consistency conditions. In this talk I will review the relation between this program and the spectral theory of hyperbolic manifolds. As I will show, hyperbolic manifolds (orbifolds, more generally) provide solutions to the self-consistency equations used in conformal bootstrap. Furthermore, a direct application of standard conformal bootstrap techniques leads to strong constraints on the spectral gap of hyperbolic 2-orbifolds, which is in many cases nearly saturated by known surfaces. I will also mention some new results from an ongoing work. Based on https://arxiv.org/abs/2111.12716 and work in progress with James Bonifacio, Dalimil Mazac and Sridip Pal.

Bowen and Margulis in the 70s proved that closed geodesics on compact hyperbolic surfaces equidistribute towards the measure of maximal entropy. From a homogeneous dynamics point of view, this measure is the quotient of the Haar measure.
In a joint work with Jialun Li, we study a higher rank generalization of this homogeneous dynamics problem.

One-relator groups \( G=\langle x_i | w \rangle \) as a class are something of an outlier in geometric group theory. On the one hand they have some good algorithmic properties, e.g. solvable word problem, but pathological examples abound, and they have therefore been resistant to most of the geometric tools we have available - for instance, small cancellation theory tells us nothing. I will relate the subgroup structure of a one-relator group \(G \) to the primitivity rank, a notion introduced by Puder, \( \pi(w) \) of \( w \), in his work on word maps in free groups. Our results seem to provide a conceptual explanation for some strong analogies between one-relator groups on the one hand and surface and three-manifold groups on the other. This is joint work with Henry Wilton.

The Weil-Petersson model is a very nice and natural way to sample random hyperbolic surfaces. Unfortunately, it is not easy to compute expectations and probabilities in this probabilistic setting. We are only able to compute expectations of quantities that depend on lengths of *simple* closed geodesics, i.e. geodesics with no self-intersections, thanks to breakthrough work by Mirzakhani. The aim of this talk is to present new ideas that allow to deal with non-simple geodesics, developed in an ongoing collaboration with Nalini Anantharaman. We show that certain averages can be expanded in powers of 1/g and provide information on the terms appearing in this expansion. I will discuss the implications of these results in spectral geometry, and the inspiration we found in Friedman's work on random regular graphs.

The Cheeger constant of a Riemannian manifold measures how hard it is to cut out a large part of the manifold. If the Cheeger constant of a manifold is large, then, through Cheeger's inequality, this implies that Laplacian of the manifold has a large spectral gap. In this talk, I will discuss how large Cheeger constants of hyperbolic surfaces can be. I will discuss recent joint work with Thomas Budzinski and Nicolas Curien in which we prove that the Cheeger constant of a closed hyperbolic surface of large genus cannot be much larger than 2/pi (approximately 0.6366). This in particular proves that there is a uniform gap between the maximal possible Cheeger constant of a hyperbolic surface of large enough genus and the Cheeger constant of the hyperbolic plane (which is equal to 1).

We will consider useful numerical invariants for simple dynamically defined sets (in this case Hausdorff dimension) and transformations (in this case Lyapunov exponents). We will consider the problem of getting rigorous estimates and the implications for some problems in number theory (in particular, the Lagrange spectrum in diophantine approximation) and Euclidean and Hyperbolic geometry (via barycentric subdvision of triangles and random walks in hyperbolic space) ... time permitting.

The shape of the spectrum of the Laplacian on a geometrically finite hyperbolic manifold \( M = \Gamma \backslash \mathbb{H}^{d+1} \) is closely connected to the size of the critical exponent \(\delta\) of \(\Gamma\). In particular, results of Patterson, Sullivan, and Lax-Phillips imply that when \(\delta > \frac{d}{2}\), the base eigenfunction is square integrable, and the corresponding eigenvalue is isolated in the \(L^2\) spectrum. These facts allow one to use tools from representation theory to study dynamics on \(M\). On the other hand, when \(\delta\) is at most \(\frac{d}{2}\), one knows that \( L^2 ( \mathrm{SO}^o(d+1,1) ) \) is a tempered representation of \(\mathrm{SO}^o(d+1,1) \). I will discuss joint work with Hee Oh in which we investigate to what extend these properties hold for higher-rank infinite volume locally symmetric spaces. In particular, I will introduce the growth indicator function, a higher-rank analogue of the critical exponent, and discuss the role it plays in dynamics on higher-rank infinite volume homogeneous spaces.

For hyperbolic surfaces, there is a deep connection between the geometry of closed geodesics and their spectral theoretic properties. In this talk, I will discuss recent work with Will Hide (Durham), where we study both sides of this relationship for hyperbolic punctured spheres. In particular, we consider Weil-Petersson random surfaces and demonstrate Poisson statistics for counting functions of closed geodesics with lengths on scales 1/sqrt(number of cusps), in the large cusp regime. Using similar ideas, we show that typical hyperbolic punctured spheres with many cusps have lots of arbitrarily small eigenvalues. Throughout, I will contrast these findings to the setting of closed hyperbolic surfaces in the large genus regime.

We discuss some open problems and some new results about the topology of arithmetic locally symmetric spaces. Among the new results is a proof of a conjecture of Gelander stating that the topology of these manifolds can be bounded just in terms of the volume. The tools in the proofs include an arithmetic version of the classical Margulis lemma about discrete subgroups of Lie groups and makes use of the existence of certain curious word maps called almost laws.

All notions will be explained. Based on joint projects with M. Fraczyk-J. Raimbault and Lvzhou Chen-Homin Lee

The 2-dimensional Yang-Mills theory, although initially depicted as a gauge theory, was more recently studied through the prism of probability, and more precisely, random matrices. In this talk, I will briefly explain why and how it was realised, then I will discuss one of its building blocks: the partition function. More precisely, I will define it formally and discuss its convergence on compact surfaces for several matrix groups of large rank.

Let \( f:I \to I\) be a map of an interval equipped with an ergodic measure \(\mu\). In this talk we will introduce and discuss the cover time of the system \((f,\mu)\). Roughly speaking this describes the asymptotic rate at which orbits become dense in the state space \(I\), that is, what is the expected amount of time one has to wait for an orbit to reach a given density in the state space. We will discuss how one can combine probabilistic tools and operator theoretic methods in order to estimate the expected cover time in terms of the local scaling properties of the measure \(\mu\). This is based on joint work with Mike Todd.

We study the differences of two consecutive eigenvalues \( \lambda_{i}-\lambda_{i-1} \) up to \( i=2g-2 \) for the Laplacian on hyperbolic surfaces of genus \( g \), and show that the supremum of such spectral gaps over the moduli space has infimum limit at least \( \frac{1}{4} \) as genus goes to infinity. A min-max principle for eigenvalues on degenerating hyperbolic surfaces is also established. This is a joint work with Haohao Zhang and Xuwen Zhu.

This talk will be about Schreier graphs of \( \mathrm{SL}_n(2) \) acting on \( F_2^n \) with respect to random generators. I will outline a proof that there is a uniform spectral gap with high probability as long as there are \( \geq C \) generators for some constant \( C \) (conjecturally \( C = 2 \) is enough). The proof is based on the trace method used by Broder and Shamir for the standard action of \( S_n \). Time permitting I will also describe an application to Babai's conjecture on diameters of finite simple groups.
(Joint work with Urban Jezernik.)

Ferrero--Washington observed the equidistribution of digits of p-adic integers, which describes the growth of class number in a certain tower of number fields. We introduce an ergodic proof of this equidistribution lemma based on the use of two-sided Bernoulli shift.
(joint with Bharathwaj Palvannan)

Fix a finite field \( \mathbb{K} \) and a word \( w \) in a free group \( F \). A w-random element in \( GL_N(\mathbb{K}) \) is obtained by substituting the letters of \( w \) with uniformly random elements of \( GL_N(\mathbb{K})\). For example, if \( w=abab^{-2} \), a \(w\)-random element is \( ghgh^{-2} \) with \( g,h \) independent and uniformly random in \( GL_N(\mathbb{K}) \). The moments of \( w \)-random elements reveal a surprising structure which relates to the free group algebra \( \mathbb{K}[F] \), and give rise to interesting analogies with \(w\)-random permutations.
I will describe what we know about this structure, and how it fits in the larger picture of word measures on groups.
This is joint work with Danielle Ernst-West and Matan Seidel.

Let \( \mathbb{H}^n \) be the hyperbolic \(n\)-space and \(\Gamma\) be a geometrically finite discrete subgroup in \(\mathrm{Isom}(\mathbb{H}^n)\) with parabolic elements. We investigate whether the geodesic flow (resp. the frame flow) over the unit tangent bundle \( T^1(\Gamma \backslash \mathbb{H}^n ) \) (resp. the frame bundle \( F(\Gamma \backslash \mathbb{H}^n \) ) mixes exponentially. This result has many applications, including spectral theory, orbit counting, equidistribution, prime geodesic theorems, etc.
In the joint work with Jialun LI, we show that the geodesic flow mixes exponentially. I will describe some ingredients in the proof. If there is time, I will also discuss the difficulty of obtaining exponential mixing of the frame flow.

Let \( \Gamma \) be a discrete group equipped with some length function. Given a finite index subgroup \(\Gamma'\), and a coset \(x\) in \(\Gamma/\Gamma'\), the lifting problem is to find a lift of \(x\) to \(\Gamma\) of small length.
It turns out that this problem is closely related to spectral problems, and in particular, an optimal spectral gap leads to an "optimal lifting" for almost all \(x\). When \(\Gamma\) is a free group, the problem translates into bounding the "almost diameter" of a certain Schreier graph, and this question was studied by Sardari and Lubetzky-Peres. When \(\Gamma\) is arithmetic and \(\Gamma'\) is a congruence subgroup, the spectral gap is related to the Generalized Ramanujan Conjecture from automorphic forms, and this question was studied by Gorodnik-Nevo.
In general, getting the optimal spectral gap is either hopelessly hard or simply false. I will describe how one may relax the strong spectral gap condition, following the work of Sarnak and Xue.

Rajchman measures (i.e. measures with decaying Fourier transform at infinity) have emerged in the historical study of sets of uniqueness and multiplicity for trigonometric series. Over the recent years, it has become an active topic to study Rajchman measures defined by groups or dynamical systems such as stationary measures or Gibbs equilibrium states. Roughly speaking stationarity allows one to study the measure using ”discretised word distributions” given by products of group elements or products of derivatives of the map defining the dynamical system. As such, given enough non-linearity of the dynamics (or ”non-concentration” in the group), using multiplicative energy this discretised word distribution must exhibit multiplicative structure at a positive proportion of scales. Then by the discretised sum-product phenomenon, such distributions cannot simultaneously have additive structure, which, using additive energy can be translated for them to have small Fourier coefficients. In this talk we will attempt to give a gentle and general introduction to this field and the methods, and outline some future challenges.