The Journal of Geometric Analysis Volume 5, Number 3, 1995 Spaces On and Beyond the Boundary of Existence By Peter Petersen, Frederick Wilhelm, and Shun-hui Zhu ABSTRACT. In this note we discuss various questions on whether or not quotients of Riemannian manifolds by Lie groups can be the Gromov-Hausdorff limits of manifolds with certain curvature bounds. In particular we show that any quotient of a manifold by a Lie group is a limit of manifolds with a lower curvature bound; this answers a question posed by Burago, Gromov, and Perelman. On the other hand, we prove that not all such spaces are limits of manifolds with absolute curvature bounds. We also give examples of spaces with curvature >_ 1 that are not limits of manifolds with curvature >_ 8 > 1/4. Introduction Since its introduction in [G], the Gromov-Hausdorff convergence technique has proved ex- tremely useful in Riemannian geometry (see, e.g., [FI ], [Pe]). In particular, the following problem seems to be very important: General Problem. Given an integer n > 2 and real numbers k < K, let Mk(n) (resp. .A4~ (n)) denote the class of complete, connected, pointed Riemannian n-manifolds with sectional curvatures > k (resp. > k and < K ). Describe the metric spaces in the Gromov-Hausdorff closures of A/[k(n) and AA~ (n) (denoted by Mk (n) and M ~ (n) respectively). In [GP1] it was proved that all spaces in Mk(n) are so-called Alexandrov spaces (see [BGP], where they are called FSCBB spaces) with curvature > k and dimension < n. In [F2] Fukaya proved that all spaces in .L4ff (n) look like quotient spaces N/O(n) where N is a Riemannian manifold and O (n) acts by isometries on N. Theorems 2 and 3 in this paper show that the converses of these results are not true. Our first result is concerned with a question brought up in [BGP]. Math Subject Classification Primary 53C20. Key Words and Phrases Gromov-Hausdorff convergence, Alexandrov spaces, orbifolds, Gromov-Hausdorff limits of Riemannian manifolds. Peter Petersen was supported in part by the National Science Foundation and Alfred P. Sloan Foundation, Frederick Wilhelm was supported by an Alfred P. Sloan doctoral dissertation fellowship, and Shun-hui Zhu was supported in part by the National Science Foundation. (~)1995 The Journal of Geometric Analysis ISSN 1050-6926