1999/09/30 by Danny Calegari
Mathematics · #math.GT
published as Proc. Amer. Math. Soc. 129 (2001), no. 10, 3109-3119 · 10 pages, 5 figures; minor changes. Accepted for publication in PAMS
arxiv created 2000/02/23 · arxiv updated 2009/11/30
Napoleon's theorem in elementary geometry describes how certain linear operations on plane polygons of arbitrary shape always produce regular polygons. More generally, certain triangulations of a polygon that tiles R2 admit deformations which keep fixed the symmetry group of the tiling. This gives rise to isolation phenomena in cusped hyperbolic 3-manifolds, where hyperbolic Dehn surgeries on some collection of cusps leaves the geometric structure at some other collection of cusps unchanged.