Igor Belegradek
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Registered User
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Jun 15 |
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Volume comparison theorems One way to find a formula is to write the unit $n$-sphere as a warped product over $[0,\pi]$ with unit $(n-1)$ sphere as a fiber, and $\sin r$ as the warping function. Then use Fubini theorem for warped product and induction. Is this homework? |
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Jun 15 |
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Negative curvature in the middle of $R^{3}$ I think the paper mathnet.or.kr/mathnet/thesis_file/… in [Bull. Korean Math. Soc. 49 (2012), No. 3, pp. 581–588] does exactly what you want. |
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Jun 15 |
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Negative curvature in the middle of $R^{3}$ What do you mean by the "standard Euclidean metric outside"? Do you mean that after removing your $N$ there is an isometry to the standard $\mathbb R^3$ minus a compact set? BTW, in dimension 2 any smooh nonpositive function on $\mathbb R^2$ can be realized as the scalar (or equaivalently sectional) curvature of a complete Riemannian metric. One simply solves the Jacobi equation $f_{xx}+Kf=0$ where $K$ is the sectional curvature, and then the metric is $dx^2+f^2dy^2$. See page 217 of [Kazdan-Warner, "Curvature Functions for Open 2-Manifolds", Annals of Math. 99, No. 2, (1974), pp. 203-219]. |
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Jun 14 |
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Do there exist exotic 4-tori? As for curvature restrictions no exotic 4-torus has a metric of nonnegative Ricci curvature (this is due to Cheeger-Gromoll and is a corollary of their splitting theorem). |
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Jun 14 |
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Do there exist exotic 4-tori? My understanding is that this is open, and in fact no exotic closed aspherical 4-manifolds are known. |
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Jun 11 |
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Negative sectionnal curvature and constant curvature By "random groups with expanders" I meant the groups constructed by adding relations to a free group randomly according to edges of a expander. Sloppy language... |
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Jun 11 |
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Negative sectionnal curvature and constant curvature In fact, these hyperbolic groups aren't fundamental groups of complete nonpositivaly curved (Riemannian) manifolds. |
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Jun 11 |
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Negative sectionnal curvature and constant curvature Misha, there are torsion-free hyperbolic groups that aren't the fundamental group of a manifold in Class 1. Namely, Gromov's random groups with expanders, see Naor-Silberman's theorem 1.1 in arxiv.org/abs/1005.4084. |
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Jun 11 |
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Explanations for mathematicians, about the falsifiability (or not) of string theory Voted to closed since this is not a math research question. Specifically, I do not see how falsifability of the string theory affects mathematics. It it quite clear that the string theory has already had a huge impact on (some areas of) math regardless of whether it describes physical reality. |
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Jun 9 |
accepted | Negative sectionnal curvature and constant curvature |
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Jun 8 |
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3 manifolds with diffeomorphic unit tangent bundles In the case when the 3-manifolds are lens spaces some partial classification is due to Sadeeb Ottenburger, see hss.ulb.uni-bonn.de/2009/1820/1820.htm, and arxiv.org/find/math/1/au:+Ottenburger_S/0/1/0/all/…. I am not sure whether there is a clean simple statement in this case. If memeory serves, he can handle all fundamental groups $\mathbb Z_r$ where $r$ is coprime to $6$. The proofs are surgery theoretic. |
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Jun 8 |
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Negative sectionnal curvature and constant curvature deleted 4 characters in body |
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Jun 8 |
answered | Negative sectionnal curvature and constant curvature |
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May 27 |
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Universal covering of compact surfaces Lee Mosher: this depends on how you define a compact surface of genus $g$. If the definition is "connected sum of $g$ tori" no classification is needed. If the definition is "compact oriented 2-manifold of Euler characteristic $2-2g$, then it is needed. I think, the polygon argument is easy, while the classification of surfaces is less so. |
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May 26 |
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Universal covering of compact surfaces Voted to close because the answer is immediate from the classification of surfaces (as Daniele explains). |
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May 21 |
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Closed geodesic loops around points in compact manifolds Anton: the argument in my first comment surely works. The argument in the second comment does not give complete detail, but I think the point is that the gradient flow of the energy functional gives a deformation retraction of the manifold of broken geodesics loops onto a small neighborhood of a constant loop at $p$, and that small neighborhood is contractible. |
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May 20 |
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Closed geodesic loops around points in compact manifolds In fact, one needs very little from the Morse theory because there is no critical points (e.g. the index theorem is not needed); basically, replace the loop space by the finite dimensional manifold of piecewise geodesic loops, and note that the energy functional has no critical points, so the finite dimensional manifold is contractible. The rest is elementary algebraic topology. What could be easier? |
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May 20 |
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Closed geodesic loops around points in compact manifolds Suppose $p$ is totally convex, so there is no geodesic loop based at $p$. By the main theorem of the Morse theory the loop space $\Omega_p M$ has a homotopy type of a CW complex with cells correponding to geodesic loops at $p$, so $\Omega_p M$ is contractible, i.e. $\pi_i(M)=0$ for $i>0$. Since $M$ is connected, it is contractible. No closed manifold is contractible by Poincare diality. I do not think there is a proof without Morse theory in disguise. |
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May 19 |
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Volume growth of covers and growth of deck-transformation groups This is very helpful. I see now that I have had some silly confusions about growth. |
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May 19 |
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Volume growth of covers and growth of deck-transformation groups By "equal" I mean that the growth functions have the same growth type, i.e. they dominate each other, where $g$ dominates $f$ if and only if $f(t)\le Ag(At+B)+B$ for all $t$ and some constants $A, B$. |
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May 19 |
asked | Volume growth of covers and growth of deck-transformation groups |
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May 11 |
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The center of a derived subgroup in an amenable group Juschenko-Monod recently constructed a f.g. simple infinite amenable group $G$. Clearly it has trivial center and its derived subgroups all equal $G$. |
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May 9 |
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Converse to Milnor’s theorem on manifolds with nonnegative Ricci curvature. The introduction of arxiv.org/abs/math/0109167 lists a number of examples (maybe all known ones?). I have not been following the subject for the last several years but I do not recall any new examples since that paper was written. |
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May 1 |
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Does a *topological* manifold have an exhaustion by compact submanifolds with boundary? Misha, it wasn't me. |
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Apr 30 |
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Purely parabolic Kleinian groups Yes, It does extend to the Karlsson-Noskov setting. |
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Apr 30 |
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Purely parabolic Kleinian groups I see. This should also extend to the setting of Karlsson-Noskov paper on isometry groups of spaces with contractive bordifications, such as e.g. visibility spaces. |
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Apr 30 |
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Purely parabolic Kleinian groups The original question was actually a toy case of the same question for purely parabolic isometry groups of Hadamard manifolds. Neither argument extends to this setting, as far as I can see. |
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Apr 29 |
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Purely parabolic Kleinian groups Thanks, Misha! I do not understand what you mean by the Zariski closure argument (what's is special about the Zarisky closure of a purely parabolic group?), but the other one I get: the group generated by high powers of parabolic elements with disjoint fixed points at infinity is geometrically finite (it visibly has a fundamental polyhedron with 4 faces), and hence it contains a hyperbolic element. |
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Apr 29 |
asked | Purely parabolic Kleinian groups |
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Apr 24 |
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Visual boundaries of universal covers of finite-volume nonpositively curved manifolds Sven, I forgot to mention that the lower curvature bound is never needed. |
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Apr 24 |
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Visual boundaries of universal covers of finite-volume nonpositively curved manifolds (cont) The hyperbolic element is some word in the above hyperbolic elements. Doesn't this work? |
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Apr 24 |
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Visual boundaries of universal covers of finite-volume nonpositively curved manifolds I do not know enough of the locally symmetric case to see why the answer is yes (as Misha says), but if so, then it seems you can combine the above to conclude that the answer is always yes, i.e., use rank rigidity to decompose the universal cover as a product of locally symmetric or rank one factors. Fix a point at infinity $z$, pick a tangent vector $v$ in its direction, project it to factors, and approximate the projections by axis of hyperbolic elements, which commute and stabilize the product of axes, which is a flat. Arguing in that flat find a hyperbolic element with axis ending at $z$. |
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Apr 24 |
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Visual boundaries of universal covers of finite-volume nonpositively curved manifolds If $M$ has rank one, then the answer is yes, see Theorem III.3.4 in Ballmann's DMV book. |
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Apr 19 |
answered | What does a mathematician expect from mathematics education? |
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Apr 5 |
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noncompact manifold with two ends splits? Removing two points from a 2-dimensional torus gives a manifold with two ends, but it is not the product of $\mathbb R$ and another manifold because that other manifold would have to be a circle for dimension reasons. The claim that the Busemann function of a line has no critical points is false. |
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Mar 18 |
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Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Euclidean space with uniformly thick tubular neighborhood @Malte: completeness is implies by a lower injectivity radius bound (all geodesics extend by a definite amount). The graph of $sin(1/x)$ is a counterexample to your last sentence. |
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Mar 6 |
asked | Hyperbolic groups with infinitely generated commutator subgroups |
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Feb 20 |
asked | Non-tame 3-manifolds covered by the Euclidean space |
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Feb 15 |
awarded | ● Nice Question |
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Feb 14 |
asked | Linear groups that are nonlinear over the integers |
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Feb 12 |
answered | Connected sum of topological manifolds |
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Feb 7 |
asked | Hyperbolic 3-manifolds with no geometrically finite structure |
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Jan 12 |
accepted | Is the volume functional contiunuous for compact manifolds with lower bounds on volume? |
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Jan 12 |
revised |
Is the volume functional contiunuous for compact manifolds with lower bounds on volume? added 32 characters in body |
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Jan 12 |
answered | Is the volume functional contiunuous for compact manifolds with lower bounds on volume? |
