Vanishing rank gradient via fixed price and the Ideal Poisson Voronoi Tessellation
发布时间：2026年03月19日
浏览次数：202
发布者: Ruixin Li
主讲人： Sam Mellick (Jagiellonian University)
活动时间： 从 2026-03-24 16:00 到 17:00
场地： 北京国际数学研究中心，镜春园78号院（怀新园）77201室
Abstract:
Cost is a fundamental invariant associated to probability measure preserving actions of groups, generalising the notion of “rank” (minimum size of a generating set). A group is said to have "fixed price" if all of its actions have the same cost. In recent work, we have been able to show that "higher rank" groups (such as SL_3(R) and Aut(T) × Aut(T')) have fixed price one. This implies, for instance, that lattices in SL_3(R) have vanishing rank gradient (the minimum size of generating set is sublinear in the covolume), resolving a conjecture of Abert-Gelander-Nikolov. A key ingredient in the argument is analysis of a new object from probability theory, the "Ideal Poisson-Voronoi tessellation" (IPVT). In higher rank, this object has truly bizarre properties. I will give an overview of cost and sketch the structure of the argument. No prior familiarity with cost or the requisite probability theory will be assumed. Joint work with Mikolaj Fraczyk and Amanda Wilkens.
Bio-Sketch:
Sam Mellick is an Australian mathematician working between the fields of ergodic theory, probability, and homogeneous dynamics (measured group theory). He obtained his PhD at Central European University in Hungary, and has since worked at the École Normale Supérieure de Lyon, McGill University, and is now an Adiunkt at Jagiellonian University in Poland.
Cost is a fundamental invariant associated to probability measure preserving actions of groups, generalising the notion of “rank” (minimum size of a generating set). A group is said to have "fixed price" if all of its actions have the same cost. In recent work, we have been able to show that "higher rank" groups (such as SL_3(R) and Aut(T) × Aut(T')) have fixed price one. This implies, for instance, that lattices in SL_3(R) have vanishing rank gradient (the minimum size of generating set is sublinear in the covolume), resolving a conjecture of Abert-Gelander-Nikolov. A key ingredient in the argument is analysis of a new object from probability theory, the "Ideal Poisson-Voronoi tessellation" (IPVT). In higher rank, this object has truly bizarre properties. I will give an overview of cost and sketch the structure of the argument. No prior familiarity with cost or the requisite probability theory will be assumed. Joint work with Mikolaj Fraczyk and Amanda Wilkens.
Bio-Sketch:
Sam Mellick is an Australian mathematician working between the fields of ergodic theory, probability, and homogeneous dynamics (measured group theory). He obtained his PhD at Central European University in Hungary, and has since worked at the École Normale Supérieure de Lyon, McGill University, and is now an Adiunkt at Jagiellonian University in Poland.
