Hypertopes with prescribed diagram symmetries

Abstract

Let $\mathcal{G}$ be a finite connected simple graph with at least two vertices. We construct a finite regular hypertope $Γ$ whose diagram is the graph $\mathcal{G}$ with every edge labelled by $4$, and for which $Cor(Γ)/Aut(Γ) \cong Aut(\mathcal{G})$. This shows that any possible group of diagram symmetries can be realized by the correlations of a hypertope. The construction uses a group generated by involutions, with nilpotency class two, exponent four and commutator relations dictated by the defining graph.