Ali Khezeli, Products of infinite countable groups have fixed price one скачать в хорошем качестве

Ali Khezeli, Products of infinite countable groups have fixed price one

The cost of a countable group G is, roughly speaking, the infimum number of edges per vertex, required to connect all elements of G in a stationary manner, using additional randomness. The cost of a probability-measure-preserving (pmp) action of G is defined with a similar idea (by requiring the graph to be a factor of the action). A longstanding problem, posed by Gaboriau, is whether the direct product of any two infinite countable groups has fixed price one; i.e., all pmp actions of the group have cost one. In this work, we resolve this problem affirmatively by constructing a random graph with arbitrarily low cost as a weak factor of i.i.d.; i.e., a weak limit of a sequence of factors of i.i.d. (it is known that weak factors of i.i.d. have the maximum cost). For finitely generated groups with nice growth behavior, the construction is based on a weak limit of a sequence of Poisson processes of balls (i.e., Boolean models) with small intensity and large constant radius. It is shown that a Poisson horoball process is obtained in the limit. Then, a graph is constructed using this horoball process and using the infinite touching property of the resulting horoballs, if the metric on the group is chosen carefully. For general finitely generated groups, we perturb the balls carefully from the beginning, in a way to ensure the infinite touching property of the limiting horoballs.