Pairs of Full-Rank Lattices With Parallelepiped-Shaped Common Fundamental Domains

2015/07/15 by Heidi Burgiel, H. Burgiel, Burgiel, H. +3
Computer Science · Mathematics · #Advanced Algebra and Logic #FOS: Mathematics #Finite Group Theory Research #Metric Geometry (math.MG) #Rings, Modules, and Algebras #math.MG

paper · pdf · doi:10.48550/arxiv.1507.04671

arxiv created 2015/07/15 · openalex publication_date 2015/07/15 · arxiv updated 2015/07/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We provide a complete characterization of pairs of full-rank lattices in ℝ^d which admit common connected fundamental domains of the type N[ 0,1) ^d where N is an invertible matrix of order d. Using our characterization, we construct several pairs of lattices of the type ( Mℤ^d,ℤ^d) which admit a common fundamental domain of the type N[ 0,1) ^d. Moreover, we show that for d=2, there exists an uncountable family of pairs of lattices of the same volume which do not admit a common connected fundamental domain of the type N[ 0,1) ².

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