UC San Diego Group Actions Seminar
Thursdays 10:00 - 10:50 AM
In-person meetings are held in AP&M 7321 and Zoom meetings are held here.
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
Fall 2023 Speakers Past Speakers
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Oct. 3 (Tuesday): Amanda Wilkens (University of Texas, Austin)
Location: AP&M 7218 and Zoom
Title: Poisson-Voronoi tessellations and fixed price in higher rank
Abstract: We overview the cost of a group action, which measures how much information is needed to generate its induced orbit equivalence relation, and the ideal Poisson-Voronoi tessellation (IPVT), a new random limit with interesting geometric features. In recent work, we use the IPVT to prove all measure preserving and free actions of a higher rank semisimple Lie group on a standard probability space have cost 1, answering Gaboriau's fixed price question for this class of groups. We sketch a proof, which relies on some simple dynamics of the group action and the definition of a Poisson point process. No prior knowledge on cost, IPVTs, or Lie groups will be assumed. This is joint work with Mikolaj Fraczyk and Sam Mellick.
video
- Oct. 12: Petr Naryshkin (WWU M?nster)
Location: AP&M 7321 and Zoom
Title: Borel asymptotic dimension of the boundary action of a hyperbolic group
Abstract: We give a new short proof of the theorem due to Marquis and Sabok, which states that the orbit equivalence relation induced by the action of a finitely generated hyperbolic group on its Gromov boundary is hyperfinite. Our methods permit moreover to show that every such action has finite Borel asymptotic dimension. This is a joint work with Andrea Vaccaro.
- Oct. 19: Koichi Oyakawa (Vanderbilt University)
Location: AP&M 7321
Title: Hyperfiniteness of boundary actions of acylindrically hyperbolic groups
Abstract: A Borel equivalence relation on a Polish space is called hyperfinite if it can be approximated by equivalence relations with finite classes. This notion has long been studied in descriptive set theory to measure complexity of Borel equivalence relations. Although group actions on hyperbolic spaces don't always induce hyperfinite orbit equivalence relations on the Gromov boundary, some natural boundary actions were recently found to be hyperfinite. Examples of such actions include actions of hyperbolic groups and relatively hyperbolic groups on their Gromov boundary and acylindrical group actions on trees. In this talk, I will show that any acylindrically hyperbolic group admits a non-elementary acylindrical action on a hyperbolic space with hyperfinite boundary action.
- Oct. 26: Christopher Shriver (University of Texas, Austin)
Location: AP&M 7321
Title: Sofic entropy, equilibrium, and local limits of Gibbs states
Abstract: I will introduce some interacting particle systems on finite graphs and Cayley graphs of countable groups, and discuss how sofic entropy helps understand them.
More specifically, we consider two notions of statistical equilibrium: an "equilibrium state" maximizes a functional called the pressure while a "Gibbs state" satisfies a local equilibrium condition. On amenable groups (for example, integer lattices) these notions are equivalent, under some assumptions on the interaction. Barbieri and Meyerovitch have recently shown that one direction holds for general sofic groups: equilibrium states are always Gibbs.
I will show that the converse fails in the simplest nontrivial case: the free boundary Ising state on a free group (an infinite regular tree) is Gibbs but not equilibrium. I will also discuss what this says about Gibbs states on finite locally-tree-like graphs: it is well-known that their local statistics are described by some Gibbs state on the infinite tree, but in fact they must locally look like a mixture of equilibrium states. This constraint can be used to compute local limits of finitary Gibbs states for a few interactions.
- Nov. 2: Forte Shinko (UC Berkeley)
Location: AP&M 7321
Title: Hyperfiniteness of generic actions on Cantor space
Abstract: A countable discrete group is exact if it has a free action on Cantor space which is measure-hyperfinite, that is, for every Borel probability measure on Cantor space, there is a conull set on which the orbit equivalence relation is hyperfinite. For an exact group, it is known that the generic action on Cantor space is measure-hyperfinite, and it is open as to whether the generic action is hyperfinite; an exact group for which the generic action is not hyperfinite would resolve a long-standing open conjecture about whether measure-hyperfiniteness and hyperfiniteness are equivalent. We show that for any countable discrete group with finite asymptotic dimension, its generic action on Cantor space is hyperfinite. This is joint work with Sumun Iyer.
- Nov. 9: Itamar Vigdorovich (Weizmann Institute)
Location: Zoom
Title: Stationary dynamics on character spaces and applications to arithmetic groups
Abstract: To any group G is associated the space Ch(G) of all characters on G. After defining this space and discussing its interesting properties, I'll turn to discuss dynamics on such spaces. Our main result is that the action of any arithmetic group on the character space of its amenable/solvable radical is stiff, i.e, any probability measure which is stationary under random walks must be invariant. This generalizes a classical theorem of Furstenberg for dynamics on tori. Relying on works of Bader, Boutonnet, Houdayer, and Peterson, this stiffness result is used to deduce dichotomy statements (and 'charmenability') for higher rank arithmetic groups pertaining to their normal subgroups, dynamical systems, representation theory and more. The talk is based on a joint work with Uri Bader.
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- Nov. 16: Elyasheev Leibtag (Weizmann Institute)
Location: Zoom
Title: Images of algebraic groups and mixing properties
Abstract: Let G be an algebraic group over a local field. We will show that the image of G under an arbitrary continuous homomorphism into any (Hausdorff) topological group is closed if and only if the center of G is compact. We will show how mixing properties for unitary representations follow from this topological property.
video
- Nov. 30: Ferrán Valdez (National Autonomous University of Mexico, Morelia)
Location: AP&M 7321
Title: Big mapping class groups
Abstract: In this talk we will introduce big mapping class groups and compare them to classical mapping class groups. The goal of the talk is to convince you that big MCGs form an interesting class of Polish groups.
- Dec. 7: Zuo Lin (UCSD)
Location: AP&M 7321
- Jan. 11: Pieter Spaas (University of Copenhagen)
- Jan. 18:
- Jan. 25: Jane Wang (University of Maine)
- Feb. 1 (4:00 pm): Jinho Jeoung (Seoul National University)
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- March 14: