Characterization of positive definite, radial functions on free groups

TL;DR

This paper characterizes positive definite, radial functions on free groups with infinite generators, extending previous work and providing new formulas for length-based functions and negative definite functions.

Contribution

It offers new characterizations of radial functions on free groups with respect to different length metrics and derives Levi-Khintchine formulas for these functions.

Findings

Characterization of radial functions on free groups with respect to $\, ext{ell}^2$ length.

Extension of characterizations to $\, ext{ell}^p$ length for $0 < p \,\leq 2$.

Derivation of Levi-Khintchine formulas for length-radial conditionally negative functions.

Abstract

This article studies the properties of positive definite, radial functions on free groups following the work of Haagerup and Knudby . We obtain characterizations of radial functions with respect to the $\ell^{2}$ length on the free groups with infinite generators and the characterization of the positive definite, radial functions with respect to the $\ell^{p}$ length on the free real line with infinite generators for $0 < p \leq 2$. We obtain the Levi-Khintchine formulas for length-radial conditionally negative functions as well.