Density of random subsets and applications to group theory

Abstract

Developing an idea of M. Gromov (1993), we study the intersection formula for random subsets with density. The density of a subset A in a finite set E is defined by dens A WD log jEj .jAj/. The aim of this article is to give a precise meaning of Gromov's intersection formula: "Random subsets" A and B of a finite set E satisfy dens.A \ B/ D dens A C dens B 1.As an application, we exhibit a phase transition phenomenon for random presentations of groups at density =2 for any 0 < < 1, characterizing the C 0 . /-small cancellation condition. We also improve an important result of random groups by G. Arzhantseva and A. Ol'shanskii (1996) from density 0 to density 0 Ä d < 1=.120m 2 ln.2m//.

Phase transition for the existence of van Kampen 2-complexes in random groups

“…In Section 3, we prove Theorem 1.5 using the multidimensional intersection formula for random subsets (Theorem 3.6, [Tsa21, Theorem 3.7]), which generalizes the proof for the C ′ (λ) phase transition in [Tsa21,Theorem 1.4]. We will see in Remark 3.3 that the second assertion of the theorem is equivalent to the following corollary.…”

“…The first detailed proof of such an existence is given in [BNW20, Theorem 2.1], using an analog of the probabilistic pigeonhole principle. Another proof is given in [Tsa21,Theorem 1.4]. An intuitive explanation using the "dimension reasoning" is given in [Oll05] p.30: The dimension of the set of couples R ℓ × R ℓ is 2dℓ.…”

“…Let us consider the permutation invariant density model of random groups introduced by Gromov in [Gro93,p. 272] and developed in [Tsa21]. Fix the set of generators X m = {x 1 , .…”