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

Eon-Kyung Lee; Sang-Jin Lee

doi:10.1142/s021819672350056x

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arXiv · 2212.02708

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

Eon-Kyung Lee·Sang-Jin Lee

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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.

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BibTeX RIS

Eon-Kyung Lee, Sang-Jin Lee. 2022-12-06. Acylindricity of the action of right-angled Artin groups on extension graphs. https://doi.org/10.1142/s021819672350056x

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