Gaps in probabilities of satisfying some commutator-like identities
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GAPS IN PROBABILITIES OF SATISFYING SOME COMMUTATOR-LIKE IDENTITIES BY
Costantino Delizia Dipartimento di Matematica, University of Salerno Via Giovanni Paolo II, 132 - 84084 Fisciano, Italy e-mail: [email protected] AND
Urban Jezernik Departamento de Matem´ aticas, University of the Basque Country Apartado 644, 48080 Bilbao, Spain e-mail: [email protected] AND
Primoˇ z Moravec Faculty of mathematics and physics, University of Ljubljana Jadranska ulica 19, SI-1000 Ljubljana, Slovenia e-mail: [email protected] AND
Chiara Nicotera Dipartimento di Matematica, University of Salerno Via Giovanni Paolo II, 132 - 84084 Fisciano, Italy e-mail: [email protected] The first and the fourth authors have been supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA – INdAM). The second author has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 748129. He has also been supported by the Spanish Government grant MTM2017-86802-P and by the Basque Government grant IT974-16. The third author has been partially supported by the Slovenian Research Agency (research core funding No. P1-0222, and projects No. J1-8132, J1-7256 and N10061). Received September 9, 2018 and in revised form April 19, 2019
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C. DELIZIA ET AL.
Isr. J. Math.
ABSTRACT
We show that there is a positive constant δ < 1 such that the probability of satisfying either the 2-Engel identity [X1 , X2 , X2 ] = 1 or the metabelian identity [[X1 , X2 ], [X3 , X4 ]] = 1 in a finite group is either 1 or at most δ.
1. Introduction It is an old, elegant, well-known, and at the same time somewhat surprising result that the probability that two randomly chosen elements commute in a nonabelian finite group can not be arbitrarily close to 1. To be more precise (see [5]), the commuting probability of no finite group can belong to the interval ( 58 , 1), and so there is a gap in the possible probability values. Following on this, many other, deeper results on the structure of the set of all possible values of the probability of satisfying the commutator identity have since emerged (see [2] and the references therein). Recently, more general word maps on finite groups have been explored from a standpoint of a similar probabilistic flavor. Here, a word map on a group G is a map w : Gd → G induced by substitution from a word w ∈ Fd belonging to a free group of rank d. For a fixed element g ∈ G of a finite group G, set Pw=g (G) =
|w−1 (g)| |G|d
to be the probability that w(g1 , g2 , . . . , gd ) = g in G, where g1 , g2 , . . . , gd are chosen independently according to the uniform probability distribution on G. Following recent breakthroughs on the values of these probabilities for finite simple groups (see [6], [10] and [1]), applications have been developed also for infinite groups (see [7] for an approach via the Hausdorff dimension for residually finite groups), indicating how these probabilities of finite quotients of a gi
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