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On commensurability of right-angled Artin groups II: RAAGs defined by paths

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  12 December 2019

MONTSERRAT CASALS–RUIZ ,
ILYA KAZACHKOV  and
ALEXANDER ZAKHAROV
MONTSERRAT CASALS–RUIZ
Affiliation:
Ikerbasque - Basque Foundation for Science and Matematika Saila, UPV/EHU, Sarriena s/n, 48940, Leioa - Bizkaia, Spain. e-mails: montsecasals@gmail.com, ilya.kazachkov@gmail.com
ILYA KAZACHKOV
Affiliation:
Ikerbasque - Basque Foundation for Science and Matematika Saila, UPV/EHU, Sarriena s/n, 48940, Leioa - Bizkaia, Spain. e-mails: montsecasals@gmail.com, ilya.kazachkov@gmail.com
ALEXANDER ZAKHAROV
Affiliation:
Chebyshev Laboratory, St Petersburg State University, 14th Line 29B, Vasilyevsky Island, 199178, St.Petersburg, Russia The Russian Foreign Trade Academy, 4a Pudovkina street, 119285, Moscow, Russia. e-mail: zakhar.sasha@gmail.com
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Abstract

In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length n > 4 is commensurable to a RAAG defined by a tree of diameter 4 if and only if n ≡ 2 (mod 4). These results follow from the connection that we establish between the classification of RAAGs up to commensurability and linear integer-programming.

MSC classification

Primary: 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
Secondary: 20F10: Word problems, other decision problems, connections with logic and automata 20F65: Geometric group theory 20E99: None of the above, but in this section
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 3 , May 2021 , pp. 559 - 608
Copyright
© Cambridge Philosophical Society 2019

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