  We provide a new, dynamical criterion for the radial rapid decay property. We
work out in detail the special case of the group $\Gamma := \mathbf{SL}_2(A)$,
where $A := \mathbb{F}_q[X,X^{-1}]$ is the ring of Laurent polynomials with
coefficients in $\mathbb{F}_q$, endowed with the length function coming from a
natural action of $\Gamma$ on a product of two trees, to show that is has the
radial rapid decay (RRD) property and doesn't have the rapid decay (RD)
property. The criterion also applies to irreducible lattices in semisimple Lie
groups with finite center endowed with a length function defined with the help
of a Finsler metric. These examples answer a question asked by Chatterji and
moreover show that, unlike the RD property, the RRD property isn't inherited by
open subgroups.
