Recognizing Euclidean Space Forms with Minimal Fundamental Tetrahedra

Abstract

We completely recognize the topological structure of the ten compact euclidean space forms with special minimal tetrahedra, constructed by face pairings in nice papers of Molnár [8-9]. From these polyhedral descriptions we derive special presentations with two generators for the fundamental groups of the considered manifolds. Our proofs also show that such group presentations completely characterize the euclidean space forms among closed connected $3$-manifolds. The results have also didactical importance.