Math. Ann. 295,223-237 (1993) Malhematische Annalen 9 Springer-Verlag1993 Rigidity of cusps in deformations of hyperbolic 3-orbifolds Walter D. Neumann 1 and Alan W. Reid 2 t Department of Mathematics, Ohio State University, Columbus, OH 43210, USA 2 Department of Mathematics, University of Texas, Austin, TX 78713, USA Received August 20, 1991; in revised form February 28, 1992 Mathematics Subject Classification (1991): 57M50, 30F40, 32G 15 1 Introduction Let Q be a complete orientable finite volume hyperbolic 3-orbifold with cusps. Then Q is the interior of a compact 3-orbifold with boundary consisting of tori or quotients of the torus by cyclic groups of orders 2, 3, 4 or 6. A cusp C of Q is called non-rigid if the corresponding boundary component is a torus or the quotient of the toms by the cyclic group of order 2. In the latter case the boundary component is a sphere with four cone points of cone angle 7r. We shall call a non-rigid cusp a torus cusp respectively a pillow cusp when the corresponding boundary component is a toms respectively a sphere with four cone points. As discussed in [NR1] for example, one can make sense of Dehn filling a pillow cusp of a complete orientable cusped hyperbolic 3-orbifold. In the following, "hyperbolic 3-orbifold" will always mean "complete orientable hyperbolic 3-orbifold" unless otherwise indicated. in a previous paper [NR2], we constructed infinitely many examples of 2- cusped hyperbolic 3-orbifolds, cusps of which were pillow cusps, with the property that deforming one of the ends by any hyperbolic Dehn filling while keeping the other end complete left the geometry of the complete end unaffected. That is, the Euclidean structure on a horospherical section of the complete end is invariant under deformations induced by hyperbolic Dehn filling of the other end. We say that the former end is geometrically isolated from the latter one. In these examples each end was geometrically isolated from the other. They were constructed by taking the double of a complete orientable hyperbolic 3-orbifold with geodesic boundary a sphere with three cone points of cone angle 7r, 27rip and 27v/q and with a single pillow cusp. The important point as far as the examples are concerned is that in the double there is a rigid 2-orbifold separating the ends, and it is this that shields the geometry of the one end from the other. The purpose of this article is to explore further this isolation property of cusps of hyperbolic 3-orbifolds, together with other refinements; see below and Sect. 4. For