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arXiv · 2610.22916

Signed dual graphs and face spectral gaps of Cayley 2-complexes

Bobo Hua·Yong Lin·Tao Wang·Haohang Zhang
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Abstract

We construct a weighted signed dual multigraph for the Cayley $2$-complex of a finite presentation and represent its $\ell^2$ face Laplacian as a signed graph Laplacian with a diagonal potential. We prove that the face spectral gap above the kernel vanishes for the Baumslag--Solitar groups $\operatorname{BS}(1,m)$, $m\ge2$, using explicit finitely supported face chains. Our main vanishing theorem gives the same conclusion for every finitely presented amenable group with ordinary first Betti number at least two.

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Bobo Hua, Yong Lin, Tao Wang, Haohang Zhang. 2026-10-16. Signed dual graphs and face spectral gaps of Cayley 2-complexes. https://arxiv.org/abs/2610.22916

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