Generalized word metrics and a Mazur–Ulam-type theorem for gyrogroups

We investigate the metric geometry of gyrogroups, a class of group-like structures whose binary operation is generally nonassociative. In particular, we extend the notion of the word metric from finitely generated groups to gyrogroups. This extension enables any gyrogroup to be viewed as a metric space, providing a suitable framework for proving a Mazur–Ulam-type theorem and for analyzing its algebraic and combinatorial structure via the associated right Cayley graph in a manner analogous to the classical setting of groups.