## Geometry of growth: approximation theorems for L 2 invariants (1998)

Citations: | 21 - 1 self |

### BibTeX

@MISC{Farber98geometryof,

author = {Michael Farber},

title = {Geometry of growth: approximation theorems for L 2 invariants},

year = {1998}

}

### OpenURL

### Abstract

Abstract. In this paper we study the problem of approximation of the L 2-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of Lück [L], dealing with towers of finitely sheeted normal coverings. We prove approximation theorems, establishing relations between the homological invariants, corresponding to infinite dimensional representations and sequences of finite dimensional representations, assuming that their normalized characters converge. Also, we find an approximation theorem for residually finite p-groups (p is a prime), where we use the homology with coefficients in a finite field Fp. We view sequences of finite dimensional flat bundles of growing dimension as examples of growth processes. We study a von Neumann category with a Dixmier type trace, which allows to describe the asymptotic invariants of growth processes. We introduce a new invariant of torsion objects, the torsion dimension. We show that the torsion dimension appears in general as an additional correcting term in the approximation theorems; it vanishes under some arithmeticity assumptions. We also show that the torsion dimension allows to establish non-triviality of the Grothendieck group of torsion

### Citations

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Citation Context ...rmal) trace; the construction of this trace uses universal summation machines of von Neumann [vN]. Note that Dixmier type traces play a very important role in the noncommutative geometry of A. Connes =-=[C]-=-. We show in §2, that not normal traces allow to define a dimension type function for the torsion objects of the extended category. We call this function the torsion dimension. Its main property is th... |

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Citation Context ... M) is denoted by H∗(X, M). Being an object of E(CA), it is a direct sum of its projective and torsion parts. The projective part of the extended homology coincides with the reduced L 2 homology, cf. =-=[A]-=- (defined by dividing the space of infinite L 2 chains by the closure of L 2 boundaries). The torsion part of the extended homology is responsible for the ”almost cycles” or ”asymptotic cycles” as the... |

189 |
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(Show Context)
Citation Context ... found as the limits of the normalized Betti numbers of finitely sheeted normal coverings. Before Lück it was only known that there is an inequality (called Kazhdan’s inequality [Ka], cf. also Gromov =-=[Gr]-=-, pages 13 and 153). One of the goals of the present paper is to generalize the Lück’s theorem in two directions. First, instead of finitely sheeted normal coverings we consider flat vector bundles of... |

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Citation Context ... the same as the homology of the covering space ˜ Xk with coefficients twisted by ρk (here ˜ Xk → X denotes covering corresponding to Γk). ¿From the general properties of induced representations (cf. =-=[CR]-=-, §10) we see that the conditions B, C, D, E of section 9.1 are satisfied. We only need to check that the sequence of normalized characters of νk converge to χ0 : π → C, where χ0(g) = 0 for g ∈ π, g ̸... |

133 |
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Citation Context ...Fk), which is invariant under the action of π and such that the form 〈 , 〉k restricted on Lk assumes values in ok. D. Denote by LD k the dual lattice L D k = {w ∈ Wk; 〈w, x〉 ∈ ok for any x ∈ Lk}, cf. =-=[FT]-=-, page 122. Then Lk ⊂ LD k and the factor LD k /Lk is a finite group. We suppose that one can choose the lattices Lk such that that there is an integer M > 0 (independent of k) with M · LD k /Lk = 0. ... |

106 |
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Citation Context ...sed CA-submodule H1 ⊂ H which is isomorphic to H in CA, coincides with H. This property is equivalent to the requirement that the von Neumann algebra HomCA (H, H) of endomorphisms of H is finite. Cf. =-=[Di]-=-, part III, chapter 8, §1. A von Neumann category CA is called finite if all its objects are finite. 1.3. Trace and dimension. Let CA be a von Neumann category. Definition. A trace on category CA is a... |

55 |
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Citation Context ... the case of normal traces the torsion dimension is always zero. Also, the trace tr is not supposed to be faithful. Not normal traces are usually called Dixmier type traces, cf. [C], since J. Dixmier =-=[D]-=- was the first who constructed such traces. Dixmier type traces play very important role in the non-commutative geometry of A. Connes [C]. 2.1. First we will show that any non-normal trace determines ... |

48 |
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Citation Context ...ero quantity cannot be too small. The proof of Theorem 9.2 is given in section 12. For the definition of the notion determinant class (which appears in the statement (iii) of Theorem 9.2) we refer to =-=[BFKM]-=-. Cf. also [CFM], section 3.8, where it is24 GEOMETRY OF GROWTH explained why the condition of being of determinant class depends only on the torsion part of the extended homology. It is natural to a... |

20 |
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Citation Context ...logy Hi(X, M) is of determinant class assuming that the character χM of M admits an arithmetic approximation . This result generalizes a theorem proven by D. Burghelea, L. Friedlander and T. Kappeler =-=[BFK]-=- for the case M = ℓ ( π). The proof presented in this paper (cf. §12), is quite similar to the proof suggested in [BFK]. 9.1. Definition (Arithmetic approximation). Suppose that χ : π → C is a positiv... |

9 |
von Neumann categories and extended L2-cohomology’, K-Theory 15
- Farber
- 1998
(Show Context)
Citation Context ...s of this type can be found in §9, cf. Theorem 9.6. 0.6. In this paper we use the language of von Neumann categories, which provides a natural environment for developing the L 2 -homology theory, cf. =-=[F]-=-. We review this material briefly in §1. Traces on von Neumann categories play an important role; the traces allow to assign dimensions to objects of the von Neumann category, which generalize the von... |

1 |
Determinant lines, von Neumann algebras, and L2-torsion, Journal für reine und angewandte Mathematik
- Carey, Mathai
(Show Context)
Citation Context ...ot be too small. The proof of Theorem 9.2 is given in section 12. For the definition of the notion determinant class (which appears in the statement (iii) of Theorem 9.2) we refer to [BFKM]. Cf. also =-=[CFM]-=-, section 3.8, where it is24 GEOMETRY OF GROWTH explained why the condition of being of determinant class depends only on the torsion part of the extended homology. It is natural to ask for which gro... |

1 |
Approximating L2-invariants of amenable coverings: a combinatorial approach
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(Show Context)
Citation Context ...is the following. Suppose that we have an infinite polyhedron and the finite polyhedra K n with K n ⊂ K n+1 form its exhaustion. Fig. 2 Growth process of this type was considered in a recent preprint =-=[DM]-=- of J. Dodziuk and V. Mathai. Another example of a growth process provides a sequence of smaller and smaller polyhedral approximations to a given compact Riemannian manifold. Let us return now to the ... |

1 | Homological algebra of the Novikov - Farber - 1996 |

1 |
Théorie des caractères
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(Show Context)
Citation Context ...s representation is the given function χ. We will see that there is a canonical construction for this purpose. This construction is very similar to the classical constructions (cf. [N], §30, and also =-=[G]-=-); therefore we will be very brief. First, we will associate a Hilbert space Hχ with a given self-adjoint non-negative class function χ : π → C. We will denote by Jχ the following two-sided ideal of C... |