Vanishing rank gradient via fixed price and the Ideal Poisson Voronoi Tessellation
Time: 2026-03-12
Published By: Ruixin Li
Speaker(s): Sam Mellick (Jagiellonian University)
Time: 16:00-17:00 March 24, 2026
Venue: Room 77201, Jingchunyuan 78, BICMR
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.
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.
