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Title: Non-orderability of random triangular groups by using random 3CNF formulas.

Abstract: A random group in the triangular binomial model Gamma(n,p) is given by the presentation <S|R>, where S is a set of n generators and R is a random set of cyclically reduced relators of length 3 over S, with each relator included in R independently with probability p. When n approaches infinity, the asymptotic properties of groups in Gamma(n,p) vary widely with the choice of p=p(n). By Antoniuk-艁uczak-艢wi膮tkowski and 呕uk, there exist constants C, C' such that a random triangular group is asymptotically almost surely (a.a.s.) free, if p < Cn^{-2}, and a.a.s. infinite, hyperbolic, but not free, if p \in (C'n^{-2}, n^{-3/2 - epsilon}). We generalize the second statement by finding a constant c such that, if p \in (cn^{-2}, n^{-3/2 - epsilon}), then a random triangular group is a.a.s. not left-orderable. We prove this by linking left-orderability of Gamma \in Gamma(n,p) to the satisfiability of a random propositional formula, constructed from the presentation of Gamma. The left-orderability of quotients will also be discussed.聽

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