We prove a noncommutative generalisation of Mahler's theorem on interpolation series, a celebrated result of $p$-adic analysis. Mahler's original result states that a function from $\mathbb{N}$ to $\mathbb{Z}$ is uniformly continuous for the $p$-adic metric if and only if it can be uniformly approximated by polynomial functions. We prove an analogous result for functions from a free monoid to a free group, where the $p$-adic metric is replaced by the pro-$p$ metric.

This is Joint work with Christophe Reutenauer.