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

Abstract

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.

References in corpus (4)

• Translation numbers in a Garside group are rational with uniformly bounded denominators
• Path lifting properties and embedding between RAAGs
• Embeddability of right-angled Artin groups on complements of trees
• On the finiteness property of hyperbolic simplicial actions: the right-angled Artin groups and their extension graphs