1. Vertex numbers of simplicial complexes with free abelian fundamental groupFlorian Frick, Matt Superdock, 2025, original scientific article Abstract: We show that the minimum number of vertices of a simplicial complex with fundamental group ℤn is at most O(n) and at least Ω(n3/4). For the upper bound, we use a result on orthogonal 1-factorizations of K2n. For the lower bound, we use a fractional Sylvester–Gallai result. This application of extremal results in discrete geometry seems to be new. We also prove that any group presentation ⟨S|R⟩ ≅ ℤn whose relations are of the form gahbic for g, h, i ∈ S has at least Ω(n3/2) generators.
Keywords: simplicial complex, fundamental group, incidence geometry
Published in RUP: 18.09.2025; Views: 297; Downloads: 5
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