An extension of Krishnan's central limit theorem to the Brown-Thompson groups

Abstract

We extend a central limit theorem, recently established for the Thompson group $F=F_2$ by Krishnan, to the Brown-Thompson groups $F_p$, where $p$ is any integer greater than or equal to $2$. The non-commutative probability space considered is the group algebra $\mathbb{C}[F_p]$, equipped with the canonical trace. The random variables in question are $a_n:=(x_n+x_n^{-1})/\sqrt{2}$, where $\{x_i{\}}_{i\ge 0}$ represents the standard family of infinite generators. Analogously to the case of $F=F_2$, it is established that the limit distribution of $s_n=(a_0+\dots +a_{n-1})/\sqrt{n}$ converges to the standard normal distribution. Furthermore, it is demonstrated that for a state corresponding to Jones's oriented subgroup $\vec{F}$, such a central limit theorem does not hold.

arXiv commentAccepted for publication in Infinite Dimensional Analysis, Quantum Probability and Related Topics

References in corpus (6)

• On the Alexander Theorem for the oriented Thompson group $\vec{F}$
• On spectral measures for certain unitary representations of R. Thompson's group F
• Jones representations of Thompson's group $F$ arising from Temperley-Lieb-Jones algebras
• Arborescence of positive Thompson links
• On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$
• On the oriented Thompson subgroup ${\vec{F}}_3$ and its relatives in higher Brown-Thompson groups