Sentences over Random Groups I: Existential Sentences

2024/06/21 by Massalha, Sobhi
#03C10 #20F06 #20F65 #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO)

paper · doi:10.48550/arxiv.2406.15080

Abstract

Random groups of density d<(1)/(2) are infinite hyperbolic, and of density d>(1)/(2) are finite. We prove that for any given system of equations Σ, all the solutions of Σover a random group of density d<(1)/(2) are projected from solutions of Σover the free group F_k, with overwhelming probability, where k is the rank of the group. We conclude that any given sentence in the Boolean algebra of universal sentences, is a truth sentence over F_k if and only if it is a truth sentence over random groups of density d<(1)/(2), with overwhelming probability.

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