Density of random subsets and applications to group theory

  • Tsung-Hsuan Tsai

    Institut de Recherche Mathématique Avancée, Strasbourg, France
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Abstract

Developing an idea of M. Gromov (1993), we study the intersection formula for random subsets with density. The density of a subset in a finite set is defined by . The aim of this article is to give a precise meaning of Gromov’s intersection formula: “Random subsets” and of a finite set satisfy .

As an application, we exhibit a phase transition phenomenon for random presentations of groups at density for any , characterizing the -small cancellation condition. We also improve an important result of random groups by G. Arzhantseva and A. Ol’shanskii (1996) from density to density .

Cite this article

Tsung-Hsuan Tsai, Density of random subsets and applications to group theory. J. Comb. Algebra 6 (2022), no. 3/4, pp. 223–263

DOI 10.4171/JCA/63