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 F = F2 by Krishnan, to the Brown–Thompson groups Fp, where p is any integer greater than or equal to 2. The non-commutative probability space considered is the group algebra ℂ[Fp], equipped with the canonical trace. The random variables in question are an := (xn + xn−1)/√2, where {xi}i≥0 represents the standard family of infinite generators. Analogously to the case of F = F2, it is established that the limit distribution of sn = (a0 + ⋯ + an−1)/√n converges to the standard normal distribution. Furthermore, it is demonstrated that for a state corresponding to Jones’s oriented subgroup is denoted by (Figure presented), such a Central Limit Theorem does not hold.