Napoleon in isolation

Abstract

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 R 2 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 leave the geometric structure at some other collection of cusps unchanged. 1. GEOMETRIC ISOLATION 1.1. Definition. Let M be a complete, finite volume hyperbolic 3-manifold with n torus cusps, which we denote c1, . . , cn. The following definition is found in [7]: Definition 1.1. A collection of cusps Cjj . ... ) Cjm is geometrically isolated from a collection cil, .. , cin if any hyperbolic Dehn surgery on any collection of the Cik leaves the geometric structure on all the cj1 invariant. Note that this definition is not symmetric in the collections cik and cj,, and in fact there are examples which show that a symmetrized definition is strictly stronger (see [7]). More generally, we can ask for some prescribed set of fillings on the Cik which leave the cj, invariant. Generalized (non-integral) hyperbolic surgeries on a cusp are a holomorphic parameter for the space of all (not necessarily complete) hyperbolic structures on M with a particular kind of allowable singularities (i.e. generalized cone structures) in a neighborhood of the complete structure. Moreover, the complex dimension of the space of geometric shapes on a complete cusp is 1. Consequently, for dimension reasons whenever n > m there will be families of generalized surgeries leaving the geometric structures on the cjl invariant. There is no particular reason to expect, however, that any of these points will correspond to an integral surgery on the Cik. When there is a 1 complex dimensional holomorphic family of isolated generalized surgeries which contains infinitely many integral surgeries, we say that we have an example of an isolation phenomenon. Neumann and Reid describe other qualities of isolation in [7] including the following: Definition 1.2. A collection of cusps cjl .... , Cjm is strongly isolated from a collection cil . .. . Cim if after any hyperbolic Dehn surgeries on ally collection of the cjl, Received by the editors June 15, 1999 and, in revised form, March 6, 2000. 2000 Mathematics Subject Classification. Primary 57M50, 57M25. (?)2001 American Mathematical Society