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Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical Ramanujan complexes

Lookup NU author(s): Dr Alina Vdovina

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This is the authors' accepted manuscript of an article that has been published in its final definitive form by World Scientific Publishing Co. Pte Ltd, 2019.

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Abstract

© 2019 World Scientific Publishing Company. We construct vertex transitive lattices on products of trees of arbitrary dimension d ≥ 1 based on quaternion algebras over global fields with exactly two ramified places. Starting from arithmetic examples, we find non-residually finite groups generalizing earlier results of Wise, Burger and Mozes to higher dimension. We make effective use of the combinatorial language of cubical sets and the doubling construction generalized to arbitrary dimension. Congruence subgroups of these quaternion lattices yield explicit cubical Ramanujan complexes, a higher-dimensional cubical version of Ramanujan graphs (optimal expanders).

Publication metadata

Author(s): Rungtanapirom N, Stix J, Vdovina A

Publication type: Article

Publication status: Published

Journal: International Journal of Algebra and Computation

Year: 2019

Volume: 29

Issue: 6

Pages: 951-1007

Print publication date: 01/09/2019

Online publication date: 10/05/2019

Acceptance date: 06/03/2019

Date deposited: 05/08/2019

ISSN (print): 0218-1967

ISSN (electronic): 1793-6500

Publisher: World Scientific Publishing Co. Pte Ltd

URL: https://doi.org/10.1142/S0218196719500371

DOI: 10.1142/S0218196719500371

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