  We study the right-angled Artin group action on the extension graph. We show
that this action satisfies a certain finiteness property, which is a variation
of a condition introduced by Delzant and Bowditch. As an application we show
that the asymptotic translation lengths of elements of a given right-angled
Artin group are always rational and once the defining graph has girth at least
6, they have a common denominator. We construct explicit examples which show
the denominator of the asymptotic translation length of such an action can be
arbitrary. We also observe that if either an element has a small syllable
length or the defining graph for the right-angled Artin group is a tree then
the asymptotic translation lengths are integers.
