## An asymptotic dimension for metric spaces, and the 0-th Novikov-Shubin invariant

Citations: | 6 - 5 self |

### BibTeX

@TECHREPORT{Guido_anasymptotic,

author = {Daniele Guido and Tommaso Isola},

title = {An asymptotic dimension for metric spaces, and the 0-th Novikov-Shubin invariant},

institution = {},

year = {}

}

### OpenURL

### Abstract

A nonnegative number d∞, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a class of open manifolds with bounded geometry the asymptotic dimension coincides with the 0-th Novikov-Shubin number α0 defined in a previous paper [D. Guido, T. Isola, J. Funct. Analysis, 176 (2000)]. Thus the dimensional interpretation of α0 given in the mentioned paper in the framework of noncommutative geometry is established on metrics grounds. Since the asymptotic dimension of a covering manifold coincides with the polynomial growth of its covering group, the stated equality generalises to open manifolds a result by Varopoulos. 0. Introduction.

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Citation Context ...y are the large scale counterpart of the spectral dimension, namely of the dimension as it is recovered by the Weyl asymptotics. On the other hand, recall that in Alan Connes’ noncommutative geometry =-=[4]-=-, a nontrivial singular trace, associated to some power of the resolvent of the Dirac operator, plays the role of integration over the noncommutative space, and such a power is the dimension of the sp... |

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Citation Context ...symptotic dimension for metric spaces. The main purpose of this section is to introduce an asymptotic dimension for metric spaces. Other notions of asymptotic dimension have been considered by Gromov =-=[9]-=- (see also the papers by Yu [23] and Dranishnikov [7]). Davies [6] proposed a definition in the case of cylindrical ends of a Riemannian manifold. We shall give a definition of asymptotic dimension in... |

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Citation Context ...oulos [21] for covering manifolds, namely the equality between α0 and the growth of the covering group. We recall that the Novikov-Shubin numbers [17] where introduced, after the definition by Atiyah =-=[2]-=- oftheL 2 -Betti numbers in terms of the von Neumann trace of the covering group, as finer invariants of the spectral behaviour of the p-Laplacian near zero, and where shown to be homotopy invariant b... |

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Citation Context ...ll regions U ⊂ B(x, r), λ1(U) ≥ α r2 ( ) β V (x, r) , vol (U) where λ1(U) is the first Dirichlet eigenvalue of ∆ in U. (iii) Assumption 3.1 is satisfied by all manifolds with positive Ricci curvature =-=[16]-=-, and covering manifolds whose group of deck transformations has polynomial growth [20]. Proposition 3.3. Let M be a complete Riemannian manifold of C∞-bound ed geometry, satisfying Assumption 3.1. −2... |

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Citation Context ...tric spaces, f : X → Y is said to be a rough isometry if there are a ≥ 1, b, ε ≥ 0 s.t. (i) a−1δX(x1,x2) − b ≤ δY (f(x1),f(x2)) ≤ aδX(x1,x2)+b, for all x1,x2 ∈ X, (ii) ⋃ x∈X BY (f(x),ε)=Y. Lemma 1.6 (=-=[3]-=-, Proposition 4.3). If f : X → Y is a rough isometry, there is a rough isometry f − : Y → X, with constants a, b− ,ε−, s.t. (i) δX(f − ◦ f(x),x) <cX, x ∈ X, (ii) δY (f ◦ f − (y),y) <cY, y ∈ Y . Theore... |

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Citation Context ... there are some connections between the asymptotic dimension of a manifold and the notion of dimension at infinity for semigroups (in our case the heat kernel semigroup) considered by Varopoulos (see =-=[22]-=-). The volume doubling property is a weak form of polynomial growth condition, and still guarantees the finiteness of the asymptotic dimension (for manifolds of bounded geometry). Proposition 3.5. Let... |

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Citation Context ...ition in the case of cylindrical ends of a Riemannian manifold. We shall give a definition of asymptotic dimension in the setting of metric dimension theory, based on the (local) Kolmogorov dimension =-=[15]-=-46 D. GUIDO AND T. ISOLA and state its main properties. We compare our definition with Davies’ in the next Section, and discuss its relations with Gromov’s in Remark 1.18. In the following, unless ot... |

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Citation Context ...spaces. The main purpose of this section is to introduce an asymptotic dimension for metric spaces. Other notions of asymptotic dimension have been considered by Gromov [9] (see also the papers by Yu =-=[23]-=- and Dranishnikov [7]). Davies [6] proposed a definition in the case of cylindrical ends of a Riemannian manifold. We shall give a definition of asymptotic dimension in the setting of metric dimension... |

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Citation Context ...Dirichlet eigenvalue of ∆ in U. (iii) Assumption 3.1 is satisfied by all manifolds with positive Ricci curvature [16], and covering manifolds whose group of deck transformations has polynomial growth =-=[20]-=-. Proposition 3.3. Let M be a complete Riemannian manifold of C∞-bound ed geometry, satisfying Assumption 3.1. −2 log pt(x,x) Then d∞(M) = lim supt→∞ log t , for any x ∈ M. Proof. Follows from Proposi... |

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Citation Context ... it is invariant under rough isometries. Finally we show that the asymptotic dimension of an open manifold with C∞-bounded geometry and satisfying an isoperimetric inequality introduced by Grigor’yan =-=[8]-=- coincides with the 0-th Novikov-Shubin number α0 as defined in [14]. On the one hand this strengthens the dimensional interpretation given in [14], and on the other it shows that the generalised limi... |

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Citation Context ...bers in terms of the von Neumann trace of the covering group, as finer invariants of the spectral behaviour of the p-Laplacian near zero, and where shown to be homotopy invariant by Gromov and Shubin =-=[10]-=-. It was observed by Roe [18] that, when the covering group is amenable, the von Neumann trace of an operator may be computed as an average of its integral kernel on the manifold w.r.t. a suitable exh... |

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Citation Context ...ann trace of the covering group, as finer invariants of the spectral behaviour of the p-Laplacian near zero, and where shown to be homotopy invariant by Gromov and Shubin [10]. It was observed by Roe =-=[18]-=- that, when the covering group is amenable, the von Neumann trace of an operator may be computed as an average of its integral kernel on the manifold w.r.t. a suitable exhaustion. Hence this procedure... |

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Citation Context ...n metric grounds the dimensional character of the 0-th Novikov-Shubin number. Finally we study the relation of d∞ with the notion of (metric) asymptotic dimension for cylindrical ends given by Davies =-=[6]-=-. Such definition is given in terms of the volume growth of the end, therefore when the end has bounded geometry Davies’ asymptotic dimension coincides with ours. Indeed Davies requires the growth to ... |

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Citation Context ...r>0, (3.1) V (x, 2r) ≤ AV (x, r) C V (x, √ r) ≤ pr(x, C x) ≤ ′ V (x, √ (3.2) r) where V (x, r) := vol (B(x, r)) and pt(x, y) is the heat kernel on M. Remark 3.2. (i) Inequality (3.1) is introduced in =-=[5]-=- and called the volume doubling property. (ii) A result of Coulhon-Grigor’yan ([5], Corollary 7.3) ([8], Proposition 5.2) states that the assumption above is equivalent to the following isoperimetric ... |

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Citation Context ...+ lim R→∞ log R R→∞ log R = d∞(M)+d∞(N). Remark 1.16. (a) Conditions under which the inequality in Theorem 1.8 (iii) becomes an equality are often studied in the case of (local) dimension theory (cf. =-=[1, 19]-=-). The previous Proposition gives such a condition for the asymptotic dimension. (b) As the asymptotic dimension is invariant under rough isometries, it is natural to substitute the continuous space w... |

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(Show Context)
Citation Context ...asymptotic dimension of a manifold with C ∞ -bounded geometry may be computed in terms of its volume growth, the equality between α0 and d∞ may be seen as a generalization of the result of Varopoulos =-=[21]-=- for covering manifolds, namely the equality between α0 and the growth of the covering group. We recall that the Novikov-Shubin numbers [17] where introduced, after the definition by Atiyah [2] oftheL... |

13 | T.: Analysis and geometry on groups (Cambridge Univ - Varopoulos, Saloff-Coste, et al. - 1992 |

6 | Singular traces, dimensions, and Novikov-Shubin invariants - Guido, Isola - 2000 |

4 | Noncommutative Riemann integration and singular traces for C ∗ - algebras - Guido, Isola |

4 |
theory and von Neumann II1-factors”, Dokl.Akad.Nauk SSSR 289
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Citation Context ...may be seen as a generalization of the result of Varopoulos [21] for covering manifolds, namely the equality between α0 and the growth of the covering group. We recall that the Novikov-Shubin numbers =-=[17]-=- where introduced, after the definition by Atiyah [2] oftheL 2 -Betti numbers in terms of the von Neumann trace of the covering group, as finer invariants of the spectral behaviour of the p-Laplacian ... |

4 |
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(Show Context)
Citation Context ...,R)) log R , x ∈ M, and analogously for N, then d∞(M × N) =d∞(M)+d∞(N). Proof. (i) Let us fix x0 ∈ M and R > 0, and consider xR ∈ M s.t. δ(x0,xR) =R, which exists because M is not compact, and let γ :=-=[0, 1]-=- → M be a minimizing geodesics between γ(0) = x0, γ(1) = xR. Clearly γ([0, 1)) ⊂ □ASYMPTOTIC DIMENSION 51 BM(x0,R), hence, if x1,... ,xk are the centres of a minimal covering by rballs of γ([0, 1)), ... |

3 | A semicontinuous trace for almost local operators on an open manifold - Guido, Isola |

2 | Entropy and "-capacity of sets in functional spaces - Kolmogoroff, Tihomirov - 1961 |

1 | Asymptotic topology, Uspekhi Mat - Dranishnikov |

1 |
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- traces, dimensions
- 1998
(Show Context)
Citation Context ...ticular, when M has a discrete group of isometries Γ with a compact quotient, the asymptotic dimension of the manifold coincides with the asymptotic dimension of the group, hence with its growth (cf. =-=[12]-=-). Therefore, by a result of Varopoulos [21], it coincides with the 0-th Novikov-Shubin invariant. We will generalise this result in Section 3. Let us conclude this Section with some examples. Other e... |

1 |
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(Show Context)
Citation Context ...mension of a covering manifold coincides with the polynomial growth of its covering group, the stated equality generalises to open manifolds a result by Varopoulos. 0. Introduction. In a recent paper =-=[14]-=-, we extended the notion of Novikov-Shubin numbers to amenable open manifolds and showed that they have a dimensional interpretation in the framework of noncommutative geometry. Here we introduce an a... |