An extension of Krishnan’s central limit theorem to the Brown–Thompson groups

In this paper, we extend a Central Limit Theorem, recently established for the Thompson group [Formula: see text] by Krishnan, to the Brown–Thompson groups [Formula: see text], where [Formula: see text] is any integer greater than or equal to [Formula: see text]. The non-commutative probability space considered is the group algebra [Formula: see text], equipped with the canonical trace. The random variables in question are [Formula: see text], where [Formula: see text] represents the standard family of infinite generators. Analogously to the case of [Formula: see text], it is established that the limit distribution of [Formula: see text] converges to the standard normal distribution. Furthermore, it is demonstrated that for a state corresponding to Jones’s oriented subgroup is denoted by [Formula: see text], such a Central Limit Theorem does not hold.