Geometry over finite local rings: Rigidity and Isospectrality
Yishai Lavi, Ori Parzanchevski
TL;DR
This work introduces and analyzes free projective spaces $\mathbb{P}_{fr}^{d-1}(\mathcal{O}_{r})$ over finite local rings, arising as $r$-geodesic spheres in Bruhat-Tits buildings. It proves two rigidity phenomena—automorphic rigidity (all automorphisms come from the underlying algebraic group and ring automorphisms) and subgraph rigidity (the sparse $\mathcal{X}_{m,n}$ determine the whole complex and extend automorphisms)—and conducts a detailed spectral study of the induced subgraphs $\mathcal{X}_{1,n}$, showing they are excellent expanders with spectra calculable in terms of $q$, $d$, $n$, and $r$. The paper further demonstrates isospectrality across rings with the same residue field, and establishes that, for suitable parameters, graphs coming from $\mathbb{Z}/p^{r}$ and $\mathbb{F}_{p}[t]/(t^{r})$ are isospectral yet non-isomorphic, thereby separating spectral and graph-theoretic structure in these geometric-combinatorial objects. These results extend the understanding of high-dimensional expanders and illuminate isospectral phenomena beyond fields to finite local rings.
Abstract
We study the simplicial order complexes obtained from free modules over finite local rings. These complexes arise naturally as geodesic spheres in Bruhat-Tits buildings over non-archimedean local fields. We establish two forms of rigidity, showing that their automorphism groups arise from the underlying algebraic group, and that they are determined by sparse induced subgraphs. We compute the spectra of these subgraphs and show that they form excellent expanders, which results in expansion for geodesic powers of Bruhat-Tits buildings. The computation also reveals that local rings with the same residue order give rise to isospectral induced subgraphs. Combining this with our rigidity results we show that the graphs arising from $n$-spaces over $\mathbb{Z}/p^{r}$ and $\mathbb{F}_{p}[t]/(t^{r})$ are isospectral and non-isomorphic.