UC San Diego Group Actions Seminar

Thursdays 10:00 - 10:50 AM

In-person meetings are held in AP&M 7218 and Zoom meetings are held here. Please send an email to one of the organizers to get the the password.

If you would like to be included or removed from our email announcements, please email Brandon Seward.

If you would like to give a talk, please send the title, abstract and related papers (if available) of your proposed talk to one of the organizers by email.

Organizers: Amir Mohammadi, Anthony Sanchez, Brandon Seward

Spring 2023 Speakers      Past Speakers

Winter 2023
  • January 12: David Aulicino (Brooklyn College and the CUNY Graduate Center)

    Title: Siegel-Veech Constants of Cyclic Covers of Generic Translation Surfaces

    Abstract: We consider generic translation surfaces of genus g>0 with n>1 marked points and take covers branched over the marked points such that the monodromy of every element in the fundamental group lies in a cyclic group of order d. Given a translation surface, the number of cylinders with waist curve of length at most L grows like L^2. By work of Veech and Eskin-Masur, when normalizing the number of cylinders by L^2, the limit as L goes to infinity exists and the resulting number is called a Siegel-Veech constant. The same holds true if we weight the cylinders by their area. Remarkably, the Siegel-Veech constant resulting from counting cylinders weighted by area is independent of the number of branch points n. All necessary background will be given and a connection to combinatorics will be presented. This is joint work with Aaron Calderon, Carlos Matheus, Nick Salter, and Martin Schmoll.

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  • January 19: Karl Winsor (Fields Institute)

    Title: Uniqueness of the Veech 14-gon

    Abstract: Teichmüller curves are algebraic curves in the moduli space of genus g Riemann surfaces that are isometrically immersed for the Teichmüller metric. These curves arise from SL(2,R)-orbits of highly symmetric translation surfaces, and the underlying surfaces have remarkable dynamical and algebro-geometric properties. A Teichmüller curve is algebraically primitive if the trace field of its affine symmetry group has degree g. In genus 2, Calta and McMullen independently discovered an infinite family of algebraically primitive Teichmüller curves. However, in higher genus, such curves seem to be much rarer. We will discuss a result that shows that the regular 14-gon yields the unique algebraically primitive Teichmüller curve in genus 3 of a particular combinatorial type. All relevant notions will be explained during the talk.

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  • January 26: Samantha Fairchild (Max Planck Institute)

    Title: Shrinking rates of horizontal gaps for generic translation surfaces

    Abstract: A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most R, we obtain precise decay rates as R goes to infinity for the difference in angle between two almost horizontal saddle connections. This is based on joint work with Jon Chaika.

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  • February 2: Tina Torkaman (Harvard University)

    Title: Intersection number and intersection points of closed geodesics on hyperbolic surfaces

    Abstract: In this talk, I will discuss the (geometric) intersection number between closed geodesics on finite volume hyperbolic surfaces. Specifically, I talk about the optimum upper bound on the intersection number in terms of the product of hyperbolic lengths. I also talk about the equidistribution of the intersection points between closed geodesics.

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  • February 7 (Tuesday) at 11 AM: Jingyin Huang (Ohio State University)

    (Joint with Functional Analysis Seminar)

    Location: AP&M 6402

    Title: Integral measure equivalence versus quasi-isometry for some right-angled Artin groups

    Abstract: Recall that two finitely generated groups G and H are quasi-isometric, if they admit a topological coupling, i.e. an action of G times H on a locally compact topological space such that each factor acts properly and cocompactly. This topological definition of quasi-isometry was given by Gromov, and at the same time he proposed a measure theoretic analogue of this definition, called the measure equivalence, which is closely related to the notion of orbit equivalence in ergodic theory. Despite the similarity in the definition of measure equivalence and quasi-isometry, their relationship is rather mysterious and not well-understood. We study the relation between these two notions in the class of right-angled Artin groups. In this talk, we show if H is a countable group with bounded torsion which is integrable measure equivalence to a right-angled Artin group G with finite outer automorphism group, then H is finitely generated, and H and G are quasi-isometric. This allows us to deduce integrable measure equivalence rigidity results from the relevant quasi-isometric rigidity results for a large class of right-angled Artin groups. Interestly, this class of groups are rigid for a reason which is quite different from other cases of measure equivalence rigidity. We will also do a quick survey of relevant measure equivalence rigidity and quasi-isometric rigidity results of other classes of groups, motivating our choice of right-angled Artin groups as a playground. This is joint work with Camille Horbez.

  • February 9: Gil Goffer (UCSD)

    Location: AP&M 7218 and Zoom

    Title: compact URS and compact IRS

    Abstract: I will discuss compact uniformly recurrent subgroups and compact invariant random subgroups in locally compact groups, and present results from ongoing projects with Pierre-Emanuel Caprace and Waltraud Lederle, and with Tal Cohen.

  • February 16: Or Landesberg (Yale University)

    Location: AP&M 7218 and Zoom

    Title: Non-Rigidity of Horocycle Orbit Closures in Geometrically Infinite Surfaces

    Abstract: Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner's theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood. We consider $\mathbb{Z}$-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of non-trivial horocycle orbit closures is known. Based on joint work with James Farre and Yair Minsky.

  • February 23: Homin Lee (Northwestern University)

    Location: AP&M 7218 and Zoom

    Title: Higher rank lattice actions with positive entropy

    Abstract: We discuss about smooth actions on manifold by higher rank lattices. We mainly focus on lattices in SLnR (n is at least 3). Recently, Brown-Fisher-Hurtado and Brown-Rodriguez Hertz-Wang showed that if the manifold has dimension at most (n-1), the action is either isometric or projective. Both cases, we don't have chaotic dynamics from the action (zero entropy). We focus on the case when one element of the action acts with positive topological entropy. These dynamical properties (positive entropy element) significantly constrains the action. Especially, we deduce that if there is a smooth action with positive entropy element on a closed n-manifold by a lattice in SLnR (n is at least 3) then the lattice should be commensurable with SLnZ. This is the work in progress with Aaron Brown.

  • March 2: Félix Lequen (Cergy-Pontoise University)

    Title: Bourgain's construction of finitely supported measures with regular Furstenberg measure

    Abstract: The possible asymptotic distributions of a random dynamical system are described by stationary measures, and in this talk we will be interested in the properties of these measures — in particular, whether they are absolutely continuous. First, I will quickly describe the case of Bernoulli convolutions, which can be seen as generalisations of the Cantor middle third set, and then the case of random iterations of matrices in SL(2, R) acting on the real projective line, where the stationary measure is unique under certain conditions, and is called the Furstenberg measure. It had been conjectured that the Furstenberg measure is always singular when the random walk has a finite support. There have been several counter-examples, and the aim of the talk will be to describe that of Bourgain, where the measure even has a very regular density. I will explain why the construction works for any simple Lie group, using the work of Boutonnet, Ioana, and Salehi Golsefidy on local spectral gaps in simple Lie groups.

  • March 9: Zvi Shem-Tov (Institute for Advanced Study)

    Title: Arithmetic Quantum Unique Ergodicity for 3-dimensional hyperbolic manifolds

    Abstract: The Quantum Unique Ergodicity conjecture of Rudnick and Sarnak says that eigenfunctions of the Laplacian on a compact manifold of negative curvature become equidistributed as the eigenvalue tends to infinity. In the talk I will discuss a recent work on this problem for arithmetic quotients of the three dimensional hyperbolic space. I will discuss our key result that Hecke eigenfunctions cannot concentrate on certain proper submanifolds. Joint work with Lior Silberman.

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  • March 16: Emilio Corso (University of British Columbia, Vancouver)

    Title: Asymptotic behaviour of expanding circles on compact hyperbolic surfaces

    Abstract: Equidistribution properties of translates of orbits for subgroup actions on homogeneous spaces are intimately linked to the mixing features of the global action of the ambient group. The connection appears already in Margulis' thesis (1969), displaying its full potential in the work of Eskin and McMullen during the nineties. On a quantitative level, the philosophy underlying this linkage allows to transfer mixing rates to effective estimates for the rate of equidistribution, albeit at the cost of a sizeable loss in the exponent. In joint work with Ravotti, we instead resort to a spectral method, pioneered by Ratner in her study of quantitative mixing of geodesic and horocycle flows, in order to obtain the precise asymptotic behaviour of averages of regular observables along expanding circles on compact hyperbolic surfaces. The primary goal of the talk is to outline the salient traits of this method, illustrating how it leads to the relevant asymptotic expansion. In addition, we shall also present applications of the main result to distributional limit theorems and to quantitative error estimates on the corresponding hyperbolic lattice point counting problem; predictably, the latter fail to improve upon the currently best known bound, achieved via finer methods by Selberg more than half a century ago.

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Fall 2022
  • October 6: Andrei Alpeev (Euler International Mathematical Institute)

    Title: Amenabilty and random orders

    Abstract: An invariant random order is a shift-invariant measure on the space of all orders on a group. It is easy to show that on an amenable group, any invariant random order could be invariantly extended to an invariant random total order. Recently, Glaner, Lin and Meyerovitch showed that this is no longer true for SL_3(Z). I will explain, how starting from their construction, one can show that this order extension property does not hold for non-amenable groups, and discuss an analog of this result for measure preserving equivalence relations.

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  • October 13: Konrad Wrobel (McGill University)

    Title: Orbit equivalence and wreath products

    Abstract:     We prove various antirigidity and rigidity results around the orbit equivalence of wreath product actions. Let F be a nonabelian free group. In particular, we show that the wreath products A ≀ F and B ≀ F are orbit equivalent for any pair of nontrivial amenable groups A, B. This is joint work with Robin Tucker-Drob.

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  • October 20: Florent Ygouf (Tel Aviv University)

    Title: Horospherical measures in the moduli space of abelian differentials

    Abstract: The classification of horocycle invariant measures on finite volume hyperbolic surfaces with negative curvature is known since the work of Furstenberg and Dani in the seventies: they are either the Haar measure or are supported on periodic orbits. This problem fits inside the more general problem of the classification of horospherical measures in finite volume homogenous spaces.

    In this talk, I will explain how similar questions arise in the moduli space of abelian differentials (and more generally in any affine invariant manifolds) and will discuss a notion of horospherical measures in that context. I will then report on progress toward a classification of those horospherical measures and related topological results. This is a joint work with J. Smillie, P. Smillie and B. Weiss.

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  • October 27: Elad Sayag (Tel Aviv University)

    Title: Entropy, ultralimits and Poisson boundaries

    Abstract: In many important actions of groups there are no invariant measures. For example: the action of a free group on its boundary and the action of any discrete infinite group on itself. The problem we will discuss in this talk is 'On a given action, how invariant measure can be? '. Our measuring of non-invariance will be based on entropy (f-divergence).

    In the talk I will describe the solution of this problem for the Free group acting on its boundary and on itself. For doing so we will introduce the notion of ultra-limit of G-spaces, and give a new description of the Poisson-Furstenberg boundary of (G,k) as an ultra-limit of G action on itself, with 'Abel sum' measures. Another application will be that amenable groups possess KL-almost-invariant measures (KL stands for the Kullback-Leibler divergence). All relevant notions, including the notion of Poisson-Furstenberg boundary and the notion of Ultra-filters will be explained during the talk. This is a master thesis work under the supervision of Yehuda Shalom.

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  • November 3: Nachi Avraham-Re'em (Hebrew University of Jerusalem)

    Title: Symmetric Stable Processes Indexed by Amenable Groups - Ergodicity, Mixing and Spectral Representation

    Abstract: Stationary symmetric \alpha-stable (S\alpha S) processes is an important class of stochastic processes, including Gaussian processes, Cauchy processes and Lévy processes. In an analogy to that the ergodicity of a Gaussian process is determined by its spectral measure, it was shown by Rosinski and Samorodnitsky that the ergodicity of a stationary S\alpha S process is characterized by its spectral representation. While this result is known when the process is indexed by \mathbb{Z} or \mathbb{R}, the classical techniques fail when it comes to processes indexed by non-Abelian groups and it was an open question whether the ergodicity of stationary S\alpha S processes indexed by amenable groups admits a similar characterization.

    In this talk I will introduce the fundamentals of stable processes, the ergodic theory behind their spectral representation, and the key ideas of the characterization of the ergodicity for processes indexed by amenable groups. If time permits, I will explain how to use a recent construction of A. Danilenko in order to prove the existence of weakly-mixing but not strongly-mixing stable processes indexed by many groups (Abelian groups, Heisenberg group).

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  • November 10: Rogelio Niño (National Autonomous University of Mexico, Morelia)

    Title: Arithmetic Kontsevich-Zorich monodromies of origamis

    Abstract: We present families of origamis of genus 3 that have arithmetic Kontsevich-Zorich monodromy in the sense of Sarnak. It is known this is true for origamis of genus 2, however the techniques for higher genera should be different. We present an outline of the proof for the existence of these families.

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  • November 17: Jayadev Athreya (University of Washington)

    Location: AP&M 6402 and on Zoom

    Title: Variance bounds for geometric counting functions

    Abstract: Inspired by work of Rogers in the classical geometry of numbers, we'll describe how to obtain variance bounds for classical geometric counting problems in the settings of translation surfaces and hyperbolic surfaces, and give some applications to understanding correlations between special trajectories on these types of surfaces. Parts of this will be joint work with Y. Cheung and H. Masur; S. Fairchild and H. Masur; and F. Arana-Herrera, and all of this has been inspired by joint work with G. Margulis.

  • November 22 (Tuesday) at 4 PM: Ruixiang Zhang (University of California Berkeley)

    (Joint with the Combinatorics Seminar)

    Location: AP&M 5829

    Title: A nonabelian Brunn-Minkowski inequality

    Abstract: The celebrated Brunn-Minkowski inequality states that for compact subsets X and Y of R^d, m(X + Y)^(1/d) >= m(X)^(1/d) + m(Y)^(1/d) where m is the Lebesgue measure. We will introduce a conjecture generalizing this inequality to every locally compact group where the exponent is believed to be sharp. In a joint work with Yifan Jing and Chieu-Minh Tran, we prove this conjecture for a large class of groups (including e.g. all real linear algebraic groups). We also prove that the general conjecture will follow from the simple Lie group case. For those groups where we do not know the conjecture yet (one typical example being the universal covering of SL_2(R)), we also obtain partial results. In this talk I will discuss this inequality and explain important ingredients, old and new, in our proof.

  • November 29 (Tuesday) at 2 PM: Camille Horbez (CNRS, Laboratoire de Mathématiques d'Orsay)

    (Joint with the Functional Analysis Seminar)

    Location:     AP&M 7321

    Title: Measure equivalence rigidity among the Higman groups

    Abstract: The Higman groups were introduced in 1951 (by Higman) as the first examples of infinite finitely presented groups with no nontrivial finite quotient. They have a simple presentation, with k >= 4 generators, where two consecutive generators (considered cyclically) generate a Baumslag-Solitar subgroup. Higman groups have received a lot of attention and remain mysterious in many ways. We study them from the viewoint of measured group theory, and prove that many of them are superrigid for measure equivalence (a notion introduced by Gromov as a measure-theoretic analogue of quasi-isometry). I will explain the motivation and context behind this theorem, some consequences, both geometric (e.g. regarding the automorphisms of their Cayley graphs) and for associated von Neumann algebras. I will also present some of the tools arising in the proof. This is joint work with Jingyin Huang.

Spring 2022
  • March 31: Andy Zucker (University of California San Diego)

    Location: AP&M 6402 and on Zoom

    Title: Minimal subdynamics and minimal flows without characteristic measures

    Abstract: Given a countable group G and a G-flow X, a probability measure on X is called characteristic if it is Aut(X, G)-invariant. Frisch and Tamuz asked about the existence, for any countable group G, of a minimal G-flow without a characteristic measure. We construct for every countable group G such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group G and a collection F of infinite subgroups of G, when is there a faithful G-flow for which the action restricted to any member of F is minimal? Joint with Joshua Frisch and Brandon Seward.

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  • April 7: Frank Lin (Texas A&M University)

    Title: Entropy for actions of free groups under bounded orbit equivalence

    Abstract: Joint work with Lewis Bowen.  The f-invariant is a notion of entropy for probability measure preserving (pmp) actions of free groups.  It is invariant under measure conjugacy and has some similarities to Kolmogorov-Sinal entropy.  Two pmp actions are orbit equivalent if their orbits can be matched almost everywhere in a measurable fashion.  Although entropy in general is not invariant under orbit equivalence, we show that the f-invariant is invariant under the stronger notion of bounded orbit equivalence.

  • April 14: Srivatsa Srinivas (University of California San Diego)

    Location: AP&M 6402 and on Zoom

    Title: An Escaping Lemma and its implications

    Abstract: Let $\mu$ be a measure on a finite group $G$. We define the spectral gap of $\mu$ to be the operator norm of the map that sends $\phi \in L^2(G)^{\circ}$ to $\mu * \phi$. We say that $\mu$ is symmetric if $\mu(x) = \mu(x^{-1})$. Now fix $G = SL_2(\mathbb{Z}/n\mathbb{Z}) \times SL_2(\mathbb{Z}/n\mathbb{Z})$, with $n \in \mathbb{N}$ being arbitrary. Suppose that $\mu$ is a measure on $G$ such that it's pushforwards to the left and right component have spectral gaps lesser than $\lambda_0 < 1$ and $\mu$ takes a minimum of $\alpha_0$ on it's support. Further suppose that the support of $\mu$ generates $G$. Then we show that there are constants $L, \beta > 0$ depending only on $\lambda_0,\alpha_0$ such that $\mu^{(*)L\log |G|}(\Gamma) \leq \frac{1}{|G|^{\beta}}$, where $\Gamma$ is the graph of any automorphism of $SL_2(\mathbb{Z}/n\mathbb{Z})$. We will discuss this result and its implications. This talk is based on joint work with Professor Alireza Salehi-Golsefidy.

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  • April 21: Seonhee Lim (Seoul National University)

    Location: AP&M 6402 and on Zoom

    Title: Complex continued fractions and central limit theorem for rational trajectories

    Abstract: In this talk, we will first introduce the complex continued fraction maps associated with some imaginary quadratic fields (d=1,2,3,7,11) and their dynamical properties. Baladi-Vallee analyzed (real) Euclidean algorithms and proved the central limit theorem for rational trajectories and a wide class of cost functions measuring algorithmic complexity. They used spectral properties of an appropriate bivariate transfer operator and a generating function for certain Dirichlet series whose coefficients are essentially the moment generating function of the cost on the set of rationals. We extend the work of Baladi-Vallee for complex continued fraction maps mentioned above. (This is joint work with Dohyeong Kim and Jungwon Lee.)

  • April 28: Osama Khalil (University of Utah)

    Location: Zoom

    Title: Mixing, Resonances, and Spectral Gaps on Geometrically Finite Manifolds

    Abstract: I will report on work in progress showing that the geodesic flow on any geometrically finite, rank one, locally symmetric space is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure of maximal entropy. The method is coding-free and is instead based on a spectral study of transfer operators on suitably constructed anisotropic Banach spaces, ala Gouezel-Liverani, to take advantage of the smoothness of the flow. As a consequence, we obtain more precise information on the size of the essential spectral gap as well as the meromorphic continuation properties of Laplace transforms of correlation functions.

  • May 5: Matthew Welsh (University of Bristol)

    Location: AP&M 6402 and on Zoom

    Title: Bounds for theta sums in higher rank

    Abstract: In joint work with Jens Marklof, we prove new upper bounds for theta sums -- finite exponential sums with a quadratic form in the oscillatory phase -- in the case of smooth and box truncations. This generalizes results of Fiedler, Jurkat and Körner (1977) and Fedotov and Klopp (2012) for one-variable theta sums and, in the multi-variable case, improves previous estimates obtained by Cosentino and Flaminio (2015). Key inputs in our approach include the geometry of Sp(n, Z) \ Sp(n, R), the automorphic representation of theta functions and their growth in the cusp, and the action of the diagonal subgroup of Sp(n, R).

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  • May 12: Yair Hartman (Ben-Gurion University)

    Location: Zoom

    Title: Tight inclusions

    Abstract: We discuss the notion of "tight inclusions" of dynamical systems which is meant to capture a certain tension between topological and measurable rigidity of boundary actions, and its relevance to Zimmer-amenable actions. Joint work with Mehrdad Kalantar.

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  • May 19: Robin Tucker-Drob (University of Florida)

    Location: AP&M 6402 and on Zoom

    Title: Amenable subrelations of treed equivalence relations and the Paddle-ball lemma

    Abstract: We give a comprehensive structural analysis of amenable subrelations of a treed quasi-measure preserving equivalence relation. The main philosophy is to understand the behavior of the Radon-Nikodym cocycle in terms of the geometry of the amenable subrelation within the tree. This allows us to extend structural results that were previously only known in the measure-preserving setting, e.g., we show that every nowhere smooth amenable subrelation is contained in a unique maximal amenable subrelation. The two main ingredients are an extension of Carrière and Ghys's criterion for nonamenability, along with a new Ping-Pong-style argument we call the "Paddle-ball lemma" that we use to apply this criterion in our setting. This is joint work with Anush Tserunyan.

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  • May 26: Dami Lee (University of Washington)

    Location: AP&M 6402 and on Zoom

    Title: Computation of the Kontsevich--Zorich cocycle over the Teichmüller flow

    Abstract: In this talk, we will discuss the dynamics on Teichmüller space and moduli space of square-tiled surfaces. For square-tiled surfaces, one can explicitly write down the SL(2,R)-orbit on the moduli space. To study the dynamics of Teichmüller flow of the SL(2,R)-action, we study its derivative, namely the Kontsevich--Zorich cocycle (KZ cocycle). In this talk, we will define what a KZ cocycle is, and by following explicit examples, we will show how one can compute the KZ monodromy. This is part of an ongoing work with Anthony Sanchez.

  • June 2: Israel Morales Jimenez (Universidad Nacional Autónoma de México)

    Location: Zoom

    Title: Big mapping class groups and their conjugacy classes

    Abstract: The mapping class group, Map(S), of a surface S, is the group of all isotopy classes of homeomorphisms of S to itself. A mapping class group is a topological group with the quotient topology inherited from the quotient map of Homeo(S) with the compact-open topology. For surfaces of finite type, Map(S) is countable and discrete. Surprisingly, the topology of Map(S) is more interesting if S is an infinite-type surface; it is uncountable, topologically perfect, totally disconnected, and more importantly, has the structure of a Polish group. In recent literature, this last class of groups is called “big mapping class groups”. In this talk, I will give a brief introduction to big mapping class groups and explain our results on the topological structure of conjugacy classes. This was a joint work with Jesús Hernández Hernández, Michael Hrušák, Manuel Sedano, and Ferrán Valdez.

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Winter 2022