Geometry over finite local rings: Rigidity and Isospectrality

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.