High-dimensional expanders from Kac--Moody--Steinberg groups
High-dimensional expanders are a generalization of the notion of expander graphs to simplicial complexes and give rise to a variety of applications in computer science and other fields. We provide a general tool to construct families of bounded degree high-dimensional spectral expanders. Inspired by...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
10.01.2024
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Subjects | |
Online Access | Get full text |
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Summary: | High-dimensional expanders are a generalization of the notion of expander
graphs to simplicial complexes and give rise to a variety of applications in
computer science and other fields. We provide a general tool to construct
families of bounded degree high-dimensional spectral expanders. Inspired by the
work of Kaufman and Oppenheim, we use coset complexes over quotients of
Kac-Moody-Steinberg groups of rank $d+1$, $d$-spherical and purely
$d$-spherical. We prove that infinite families of such quotients exist provided
that the underlying field is of size at least 4 and the Kac-Moody-Steinberg
group is 2-spherical, giving rise to new families of bounded degree
high-dimensional expanders. In the case the generalized Cartan matrix we
consider is affine, we recover the construction of O'Donnell and Pratt from
2022, (and thus also the one of Kaufman and Oppenheim) by considering Chevalley
groups as quotients of affine Kac-Moody-Steinberg groups. Moreover, our
construction applies to the case where the root system is of type
$\tilde{G}_2$, a case that was not covered in earlier works. |
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DOI: | 10.48550/arxiv.2401.05197 |