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Random Triangular Groups

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Part of the Springer INdAM Series book series (SINDAMS,volume 12)

Abstract

Let \(\langle S\vert R\rangle\) denote a group presentation, where S is a set of n generators while R is a set of relations consisting of distinct cyclically reduced words of length three. The above presentation is called a triangular group presentation and the group it generates is called a triangular group. We study the following model \(\Gamma (n,p)\) of a random triangular group. The set of relations R in \(\Gamma (n,p)\) is chosen randomly, namely every relation is present in R independently with probability p. We study how certain properties of a random group \(\Gamma (n,p)\) change with respect to the probability p. In particular, we show that there exist constants c, C > 0 such that if \(p < \frac{c} {n^{2}}\), then a.a.s. a random group \(\Gamma (n,p)\) is a free group and if \(p > C \frac{\log n} {n^{2}}\), then a.a.s. this group has Kazhdan’s property (T). What is more interesting, we show that there exist constants c′, C′ > 0 such that if \(\frac{C'} {n} < p < c' \frac{\log n} {n^{2}}\), then a.a.s. a random group \(\Gamma (n,p)\) is neither free, nor has Kazhdan’s property (T). We prove the above statements using random graphs and random hypergraphs.

Keywords

  • Triangular Group
  • Group Presentation
  • Random Hypergraphs
  • Random Intersection Graphs
  • Giant Component

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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  • DOI: 10.1007/978-3-319-20155-9_4
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References

  1. Antoniuk, S., Łuczak, T., Świa̧tkowski, J.: Collapse of random triangular groups: a closer look. Bull. Lond. Math. Soc. 46(4), 761–764 (2014)

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  2. Antoniuk, S., Łuczak, T., Świa̧tkowski, J.: Random triangular groups at density 1/3. Compos. Math. 151(1), 167–178

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  3. Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric Group Theory. London Mathematical Society Lecture Note Series vol. 182, pp. 1–295. Cambridge University Press, Cambridge (1993)

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  4. Żuk, A.: Property (T) and Kazhdan constants for discrete groups. Geom. Funct. Anal. 13, 643–670 (2003)

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Correspondence to Sylwia Antoniuk .

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Antoniuk, S., Łuczak, T., Świa̧tkowski, J. (2015). Random Triangular Groups. In: Benedetti, B., Delucchi, E., Moci, L. (eds) Combinatorial Methods in Topology and Algebra. Springer INdAM Series, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-20155-9_4

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