# Spectra/Moduli seminar

### We meet at 15:00 Monday in MCS2068

13/03/2023 Kevin Boucher Spherical representations of hyperbolic groups
[Abstract]
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.
10/03/2023 Antoine Dahlqvist The Planar Master Field and the Yang-Mills measure on surfaces
[Abstract]
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.
06/02/2023 Timothée Bénard The local limit theorem on nilpotent Lie groups
[Abstract]
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.
26/01/2023 Petr Kravchuk Automorphic Spectra and the Conformal Bootstrap
[Abstract]
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.
09/01/2023 Thi Dang Equidistribution of flat periodic tori
[Abstract]
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.

## 2022

28/11/2022 Lars Louder What is a one-relator group?
[Abstract]
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.
21/11/2022 Laura Monk Friedman-Ramanujan functions in random hyperbolic geometry
[Abstract]
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.
14/11/2022 Bram Petri How do you efficiently cut a hyperbolic surface in two?
[Abstract]
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).
31/10/2022 Mark Pollicott Estimates on Hausdorff dimension and Lyapunov exponents and their applications
[Abstract]
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.
24/10/2022 Sam Edwards Temperedness and the growth indicator function
[Abstract]
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.
10/10/2022 Joe Thomas Poisson statistics, short geodesics and small eigenvalues on hyperbolic punctured spheres
[Abstract]
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.
16/05/2022 Michael Magee Near optimal spectral gaps for hyperbolic surfaces
25/04/2022 Sebastian Hurtado Height Gaps, the Margulis Lemma and Almost Laws
[Abstract]
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
14/03/2022 Thibaut Lemoine Large N limit of Yang-Mills partition function
[Abstract]
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.
07/03/2022 Natalia Jurga Cover times in dynamical systems
[Abstract]
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.
28/02/2022 Yunhui Wu Degenerating hyperbolic surfaces and spectral gaps for large genus
[Abstract]
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.
07/02/2022 Joe Thomas Quantum Unique Ergodicity for Cayley graphs of quasirandom groups

## 2021

06/12/2021 Sean Eberhard Spectral gap for random Schreier graphs of $$\mathrm{SL}_n(2)$$
[Abstract]
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.)
29/11/2021 Jungwon Lee Another view of Ferrero-Washington Theorem
[Abstract]
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)
22/11/2021 Doron Puder Measures induced by words on $$GL_N(\mathbb{F}_q)$$ and free group algebras
[Abstract]
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.
15/11/2021 Wenyu Pan Exponential mixing of flows for geometrically finite hyperbolic manifolds with cusps
[Abstract]
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.
08/11/2021 Amitay Kamber Optimal Lifting and Spectral gap
[Abstract]
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.
01/11/2021 Tuomas Sahlsten Spectral gap, sum-product estimates and Fourier decay
[Abstract]
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.
25/10/2021 Will Hide Spectral theory of cusped hyperbolic surfaces
18/10/2021 Irving Calderón Effective $$\mathbf{Z}_S$$-equivalence of integral quadratic forms and applications
11/10/2021 Joe Thomas Delocalization of eigenfunctions on random surfaces