UNIVERSITY OF CALIFORNIA Santa Barbara On the Fundamental Group of Noncompact Manifolds with Nonnegative Ricci Curvature A Dissertation submitted in partial satisfaction of the requirement for the degree of Doctor of Philosophy in Mathematics by William C. Wylie Committee in charge: Professor Guofang Wei, Chair Professor Xianzhe Dai Professor Daryl Cooper June 2006 UMI Number: 3218835 3218835 2006 UMI Microform Copyright All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 by ProQuest Information and Learning Company. The dissertation of William C. Wylie is approved Daryl Cooper Xianzhe Dai Guofang Wei, Committee Chairman May 2006 Dedication Dedicated to my mother and father for always supp orting me. Acknowledgments I would like to thank my advisor Guofang Wei for all of her guidance and encouragement. I could not have hoped for a better mentor. I would also like to thank Lisa for keeping me sane and providing me with constant patience and support. I am eternally grateful to Liam Donohoe, James Tattersall, and Joanna Su for spending so many hours introducing me to the beauty of mathematics, without them I never would have come to UCSB in the first place. I am also grateful to Xianzhe Dai, Daryl Cooper, and Rick Ye for many inspiring lectures and helpful discussions. Finally, I must thank all of my friends at UCSB who have made my time here so enjoyable. iii Vita of William C. Wylie Education Providence College Mathematics & Comp. Sci. B.S. 2001 Univ. of California at Santa Barbara Mathematics M.A. 2003 Univ. of California at Santa Barbara Mathematics Ph.D. 2006 Fields of Study Riemannian Geometry and Global Geometric Analysis. Publications William C. Wylie, Noncompact Manifolds with Nonnegative Ricci Curva- ture, To appear in Journal of Geometric Analysis. Appointments 2005-2006 Graduate Council Departmental Mentorship Award Fellow, Uni- versity of California at Santa Barbara. 2003-2005 (3 quarters) Teaching Associate, University of California at Santa Barbara. 2001-2005 (10 quarters) Teaching Assistant, University of California at Santa Barbara. 2004-2005 (3 quarters) Research Assistant, University of California at Santa Barbara. iv Abstract On the Fundamental Group of Noncompact Manifolds with Nonnegative Ricci Curvature by William C. Wylie We study the fundamental group of noncompact Riemannian manifolds with nonnegative Ricci curvature. We show that the fundamental group of a noncompact, complete, Rieman- nian manifold with nonnegative Ricci curvature and small lin- ear diameter growth is almost the fundamental group of a large ball. We make this precise by studying semi-local fundamen- tal groups. We also find relationships between the semi-local fundamental groups and special Gromov-Hausdorff limits of a manifold called tangent cones at infinity. As an application we show that any tangent cone at infinity of a complete open mani- fold with nonnegative Ricci curvature and small linear diameter growth is its own universal cover. We also derive bounds on the number of generators of the fun- damental group for some families of complete open manifolds with nonnegative Ricci curvature. In fact we show that the fun- damental group of these manifolds behaves somewhat like the fundamental group of a compact manifold. We also show there is a relationship between the volume growth of a manifold with nonnegative Ricci curvature and the length of a loop represent- ing an element of infinite order in π 1 (M). v Contents 1 Introduction 1 2 Background 6 2.1 The Fundamental Group of Manifolds with Nonnegative Ricci Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Gromov-Hausdorff Convergence . . . . . . . . . . . . . . . . 10 3 Semi-Local Fundamental Groups 16 3.1 Introduction and Statement of Results . . . . . . . . . . . . 16 3.2 Nullhomotopy Radius . . . . . . . . . . . . . . . . . . . . . . 23 3.3 The Halfway Lemma for G(p,r,R) . . . . . . . . . . . . . . . 28 3.4 Localized Uniform Cut Lemma . . . . . . . . . . . . . . . . 32 3.5 Proof of Theorem 3.1.5 . . . . . . . . . . . . . . . . . . . . . 37 3.6 Proof of Theorems 3.1.8 and 3.1.9 . . . . . . . . . . . . . . . 39 4 The Lo ops to Infinity Property and Diameter Growth 45 4.1 Introduction and Statement of Results . . . . . . . . . . . . 45 4.2 The Splitting Theorem and the Loops to Infinity Property . 48 4.3 The Loop Pulling Lemma . . . . . . . . . . . . . . . . . . . 51 vi 4.4 Manifolds with Sublinear Diameter and Large Volume Growth 55 4.5 α-Noncollapsing . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.6 Volume Growth and the Length of Homotopically Nontrivial Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 vii Chapter 1 Introduction There are many interesting relationships the b e tween the geometric and topological structures of a smooth complete Riemannian manifold. Roughly speaking, one can envision an n-dimensional manifold M as a subset of R k such that every point p has a neighborhood which is homeomorphic to an open subset of R n . The tangent space of M at p, T p M ⊂ R k is the space of all vectors v such that v = c  (0) where c : (−ε, ε) → M is a curve on M with c(0) = p. The Riemannian metric on M is the natural inner product on T p M, that is, the restriction of the dot product on R k to T p M. The Riemannian metric enables us consider M as a metric space by defining the distance as the infimum of lengths of paths between two points and allows us to define geometric concepts such as length, diameter, volume, and angle. M is a topological space so one can also study its topological properties such as compactness, connectedness, and homotopy and homology groups. In studying the interactions between the geometric and topological struc- tures of a manifold, curvature plays a pivotal role. To define curvature we 1 begin with the most simple manifolds. Let S be an orientable smooth sur- face embedded in R 3 . Since S is orientable we can choose a smooth normal vector field to S, n(x). The Gauss map g : S → S 2 is the map which takes x to n(x). The Gaussian curvature of S at x, κ(x), is the determinant of the differential of g at x. The magnitude of κ(x) measures how quickly the nor- mal vector n(x) is turning at x, or how curved the surface is. Moreover, if the surface is sphere-like around x then the gauss map preserves orientation and if S is saddle-like around x then g reverses orientation. Therefore, κ(x) is positive if S looks like a sphere around x and negative if S looks like a saddle around x. Clearly curvature is not a topological quantity. However, one of the most amazing results in geometry, the Gauss Bonnet theorem, shows that the integral of curvature is a topological quantity. Theorem 1.0.1 (Gauss-Bonnet). If S is a compact, orientable surface then  S κ = 2πχ(S) where χ(S) is the Euler characteristic of S. Since the only compact orientable surface with positive Euler charac- teristic is the sphere, the Gauss Bonnet Theorem shows that any compact orientable surface with κ(x) > 0 for all x ∈ S is homeomorphic to the sphere. As we have seen, the condition κ(x) > 0 me ans that S curves in on itself at all points, thus it is not surprising to find that there are not many possibilities for the topology of these surfaces. To generalize the definition of curvature to higher dimensions let v 1 and v 2 be two unit length tangent vectors to M and let σ(v 1 , v 2 ) be the surface 2 [...]... dissertation we study further the interaction between the geometry and the fundamental group of a manifold with nonnegative Ricci curvature There are three parts In the first part we review results about the fundamental group of manifolds with nonnegative Ricci curvature and background involving Gromov 4 Hausdorff convergence In the second part we show that, for a large class of manifolds with nonnegative Ricci. .. Background 2.1 The Fundamental Group of Manifolds with Nonnegative Ricci Curvature In this section we review some results concerning the fundamental group of manifolds with nonnegative Ricci curvature The most basic geometric tool in studying manifolds with RicM ≥ 0 is the Bishop-Gromov relative volume comparison theorem which bounds the volume of a metric ball in terms of the corresponding ball in... that manifolds with nonnegative Ricci curvature and linear volume growth satisfy the hypotheses of either theorem In [15] Menguy gives an example of a manifold with nonnegative Ricci curvature which is not asymptotically polar However, the conclusion of Theorem 3.1.8 holds for this manifold The author is unaware of any examples of manifolds of nonnegative Ricci curvature which do not satisfy the conclusion... depend on which sequence of rescalings is chosen In general, the tangent cone at infinity of a manifold with nonnegative Ricci curvature may not be a metric cone, as an example of Menguy shows [15] However, the fundamental group of this example is well behaved This leads to the following question Question 2.2.10 Is the revised fundamental group of the tangent cone at infinity of a manifold with nonnegative. .. easily be smoothed and keep the same semi-local fundamental groups The smoothed surface, however, does not have nonnegative curvature In fact, the geometric semi-local fundamental groups are well behaved for a manifold with nonnegative sectional curvature Lemma 3.1.3 If N is a complete noncompact manifold with nonnegative sectional curvature then, for any point p in the soul of S, there exists large enough... give bounds on the number of generators of the fundamental group under some natural geometric conditions In fact, we show that the fundamental group of these manifolds behaves like the fundamental group of a compact manifold See Section 4.1 All manifolds in this thesis are assumed to be complete, noncompact, and without boundary For clarity we will often omit explicitly stating this hypothesis but it... thus in the limit we see only the structure “at infinity” of X If X is a manifold then rescaling the metric also rescales curvature by a positive constant In particular, if X has nonnegative Ricci curvature then the entire sequence of rescalings has nonnegative Ricci curvature and the Gromov Precompactness Theorem implies that tangent cones at infinity exist In fact, if X has nonnegative sectional curvature. .. properties of the geometric semi-local fundamental groups and the nullhomotopy radius Just as in [27] and [34] the proofs of Theorems 3.1.5 and 3.1.8 are the result of two lemmas, the Halfway Lemma and the Uniform Cut Lemma In Sections 3.3 and 3.4 we give proofs of versions of these two lemmas for the geometric semi-local fundamental groups The main difficulty here is that we work on the universal cover of an... space with a universal cover, the revised fundamental group π 1 (Y ) is the group of deck transformations of the universal cover Sormani and Wei ([30], Corollary 4.7-4.9) go on to prove results similar to those in the previous section of this chapter for π 1 (Y ) for limit spaces of sequences of manifolds with nonnegative Ricci curvature We are concerned with special limit spaces called tangent cones... finitely generated almost nilpotent group, G, there is a manifold with positive Ricci curvature and fundamental group G Therefore a finitely generated group G is the fundamental group of some manifold with nonnegative Ricci curvature if and only if G is almost nilpotent A major open problem is the following conjecture of Milnor Conjecture 2.1.8 (Milnor, [18]) If M has Ric ≥ 0 then π1 (M ) is finitely generated . University of California at Santa Barbara. iv Abstract On the Fundamental Group of Noncompact Manifolds with Nonnegative Ricci Curvature by William C. Wylie We study the fundamental group of noncompact. UNIVERSITY OF CALIFORNIA Santa Barbara On the Fundamental Group of Noncompact Manifolds with Nonnegative Ricci Curvature A Dissertation submitted in partial satisfaction of the requirement for the degree. bounds on the number of generators of the fun- damental group for some families of complete open manifolds with nonnegative Ricci curvature. In fact we show that the fun- damental group of these manifolds