Acylindricity of the action of right-angled Artin groups on extension graphs

The action of a right-angled Artin group on its extension graph is known to be acylindrical because the cardinality of the so-called r-quasi-stabilizer of a pair of distant points is bounded above by a function of r. The known upper bound of the cardinality is an exponential function of r. In this paper we show that the r-quasi-stabilizer is a subset of a cyclic group and its cardinality is bounded above by a linear function of r. This is done by exploring lattice theoretic properties of group elements, studying prefixes of powers and extending the uniqueness of quasi-roots from word length to star length. We also improve the known lower bound for the minimal asymptotic translation length of a right-angled Artin group on its extension graph.