# UC San Diego Group Actions Seminar

## Thursdays 10:00 - 10:50 AM

Talks will be given on zoom, if you have the password you can join here. Please send an email to one of the organizers to get the the password.

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.

Spring 2022 Speakers      Past Speakers

• 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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