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Random triangular groups at density $1/3$

  • Sylwia Antoniuk (a1), Tomasz Łuczak (a2) and Jacek Świa̧tkowski (a3)

Let ${\rm\Gamma}(n,p)$ denote the binomial model of a random triangular group. We show that there exist constants $c,C>0$ such that if $p\leqslant c/n^{2}$ , then asymptotically almost surely (a.a.s.) ${\rm\Gamma}(n,p)$ is free, and if $p\geqslant C\log n/n^{2}$ , then a.a.s. ${\rm\Gamma}(n,p)$ has Kazhdan’s property (T). Furthermore, we show that there exist constants $C^{\prime },c^{\prime }>0$ such that if $C^{\prime }/n^{2}\leqslant p\leqslant c^{\prime }\log n/n^{2}$ , then a.a.s. ${\rm\Gamma}(n,p)$ is neither free nor has Kazhdan’s property (T).

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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