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## IHARA’S ZETA FUNCTION FOR PERIODIC GRAPHS AND ITS APPROXIMATION IN THE AMENABLE CASE (2008)

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Citations: | 10 - 5 self |

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96 | A survey on spectra of infinite graphs
- Mohar, Woess
- 1989
(Show Context)
Citation Context ...ctor, is the determinant of a deformed Laplacian on the graph. We first need some technical results. Let us denote by A the adjacency matrix of X, i.e. (Af)(v) = ∑ f(w), f ∈ ℓ 2 (V X). Then (by [23], =-=[24]-=-) ‖A‖ ≤ d := sup v∈V X deg(v) < ∞, and it is easy to see that A ∈ N0(X, Γ). Introduce (Qf)(v) := (deg(v) − 1)f(v), v ∈ V X, f ∈ ℓ 2 (V X), and ∆(u) := I − uA + u 2 Q ∈ N0(X, Γ), for u ∈ C. Let us reca... |

92 |
The Ihara-Selberg zeta function of a tree lattice
- Bass
(Show Context)
Citation Context ...able group action is the limit of the zeta functions of a suitable sequence of finite subgraphs. 0. Introduction The zeta functions associated to finite graphs by Ihara [20], Hashimoto [15, 16], Bass =-=[4]-=- and others, combine features of Riemann’s zeta function, Artin L-functions, and Selberg’s zeta function, and may be viewed as analogues of the Dedekind zeta functions of a number field. They are defi... |

91 |
Tree lattices
- Bass, Lubotzky
- 2001
(Show Context)
Citation Context ...inant formula was proved. They deduce this result as a specialization of the treatment of group actions on trees (the so-called theory of tree lattices, as developed by Bass, Lubotzky and others, see =-=[5]-=-). We mention [13] for a recent review of some results on zeta functions for finite or periodic simple graphs, and [12, 7, 8, 9] for the computation of the Ihara zeta function of several periodic simp... |

91 |
Zeta functions of finite graphs and coverings.
- Stark, Terras
- 1996
(Show Context)
Citation Context ...function satisfying a functional equation. They can be expressed as the determinant of a perturbation of the graph Laplacian and, for Ramanujan graphs, satisfy a counterpart of the Riemann hypothesis =-=[28]-=-. Other relevant papers are [31, 17, 18, 27, 25, 10, 21, 29, 30, 19, 3, 22]. In differential geometry, researchers have first studied compact manifolds, then infinite covers of those, and finally, non... |

78 |
Determinant theory in finite factors
- Fuglede, Kadison
- 1952
(Show Context)
Citation Context ...f not necessarily normal operators in a von Neumann algebra with a finite trace. The results obtained are used in Section 4 to prove a determinant formula for the zeta function. In a celebrated paper =-=[11]-=-, Fuglede and Kadison defined a positive-valued determinant for finite factors (i.e. von Neumann algebras with trivial center and finite trace). Such a determinant is defined on all invertible element... |

75 |
On discrete subgroups of the two by two projective linear group over p-adic fields
- Ihara
- 1966
(Show Context)
Citation Context ...f a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs. 0. Introduction The zeta functions associated to finite graphs by Ihara =-=[20]-=-, Hashimoto [15, 16], Bass [4] and others, combine features of Riemann’s zeta function, Artin L-functions, and Selberg’s zeta function, and may be viewed as analogues of the Dedekind zeta functions of... |

74 |
Zeta functions of finite graphs and representations of p-adic groups. Automorphic forms and geometry of arithmetic varieties,
- Hashimoto
- 1989
(Show Context)
Citation Context ...ph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs. 0. Introduction The zeta functions associated to finite graphs by Ihara [20], Hashimoto =-=[15, 16]-=-, Bass [4] and others, combine features of Riemann’s zeta function, Artin L-functions, and Selberg’s zeta function, and may be viewed as analogues of the Dedekind zeta functions of a number field. The... |

70 |
Répartition asymptotique des valeurs propres de l’opérateur de Hecke Tp
- Serre
- 1997
(Show Context)
Citation Context ... equation. They can be expressed as the determinant of a perturbation of the graph Laplacian and, for Ramanujan graphs, satisfy a counterpart of the Riemann hypothesis [28]. Other relevant papers are =-=[31, 17, 18, 27, 25, 10, 21, 29, 30, 19, 3, 22]-=-. In differential geometry, researchers have first studied compact manifolds, then infinite covers of those, and finally, noncompact manifolds with greater complexity. Likewise, in the graph setting, ... |

44 |
L2-cohomology and group cohomology, Topology
- Cheeger, Gromov
- 1986
(Show Context)
Citation Context ... 4.1 (Determinant formula). We have ZX,Γ(u) −1 = (1 − u 2 ) −χ(2) (X) detΓ(∆(u)), for |u| < 1 α , where χ (2) (X) := ∑ 1 1 ∑ 1 − |Γv| 2 |Γe| is the L2-Euler characteristic of (X, Γ), as introduced in =-=[6]-=-. v∈F0 e∈F1 This theorem was first proved in [7] and is based on formula (iv) in Theorem 2.2 and the equality detΓ(I − uT) = (1 − u2 ) −χ(2) (X) detΓ(∆(u)), for |u| < 1 α . The main difference with th... |

39 | Zeta functions of finite graphs.
- Kotani, Sunada
- 2000
(Show Context)
Citation Context ... equation. They can be expressed as the determinant of a perturbation of the graph Laplacian and, for Ramanujan graphs, satisfy a counterpart of the Riemann hypothesis [28]. Other relevant papers are =-=[31, 17, 18, 27, 25, 10, 21, 29, 30, 19, 3, 22]-=-. In differential geometry, researchers have first studied compact manifolds, then infinite covers of those, and finally, noncompact manifolds with greater complexity. Likewise, in the graph setting, ... |

30 |
Counting paths in graphs. Enseign
- Bartholdi
- 1999
(Show Context)
Citation Context ... equation. They can be expressed as the determinant of a perturbation of the graph Laplacian and, for Ramanujan graphs, satisfy a counterpart of the Riemann hypothesis [28]. Other relevant papers are =-=[31, 17, 18, 27, 25, 10, 21, 29, 30, 19, 3, 22]-=-. In differential geometry, researchers have first studied compact manifolds, then infinite covers of those, and finally, noncompact manifolds with greater complexity. Likewise, in the graph setting, ... |

25 |
The spectrum of an infinite graph
- Mohar
- 1982
(Show Context)
Citation Context ...o a factor, is the determinant of a deformed Laplacian on the graph. We first need some technical results. Let us denote by A the adjacency matrix of X, i.e. (Af)(v) = ∑ f(w), f ∈ ℓ 2 (V X). Then (by =-=[23]-=-, [24]) ‖A‖ ≤ d := sup v∈V X deg(v) < ∞, and it is easy to see that A ∈ N0(X, Γ). Introduce (Qf)(v) := (deg(v) − 1)f(v), v ∈ V X, f ∈ ℓ 2 (V X), and ∆(u) := I − uA + u 2 Q ∈ N0(X, Γ), for u ∈ C. Let u... |

24 | A combinatorial proof of Bass’s evaluation of the Ihara-Selberg zeta function for graphs
- Foata, Zeilberger
- 1999
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21 |
Density of states in spectral geometry
- Adachi, Sunada
- 1993
(Show Context)
Citation Context ...menable group, where a space is said to be amenable if it possesses a regular exhaustion. Such a result was stated by Cheeger and Gromov in [6] for CW-complexes and was proved by Adachi and Sunada in =-=[2]-=- for covering manifolds. We give here a proof in the case of covering graphs. Throughout this section, X is a connected, countably infinite graph, and Γ is a countable discrete amenable group of autom... |

19 |
On zeta and L-functions of finite graphs
- Hashimoto
- 1990
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18 |
The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps, in: “Random Walks and Geometry
- Grigorchuk, A
- 2004
(Show Context)
Citation Context ...lgebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara zeta function. Moreover, we answer in the affirmative one of the questions raised in =-=[12]-=- by Grigorchuk and ˙ Zuk. Accordingly, we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs. 0... |

16 |
A note on the Følner condition for amenability
- Adachi
(Show Context)
Citation Context ...we have detτ((I + uA)(I + uB)) = detτ(I + uA)detτ(I + uB). Proof. The proof is inspired by that of Lemma 3 in [11]. Let us write a := log(I + uA), b := log(I + uB) ∈ A, and let c(t) := e ta e b , t ∈ =-=[0, 1]-=-. As ‖a‖ ≤ − log(1 − |u|‖A‖), and ‖b‖ ≤ − log(1 − |u|‖B‖), we get ‖c(t) − 1‖ = ‖e ta − e −b ‖‖e b ‖ ( ‖b‖ ≤ e ( 1 ≤ 1 − |u|‖B‖ e ‖a‖ + e ‖b‖ − 2 ) 1 1 − |u|‖A‖ + ) 1 − 2 < 1, 1 − |u|‖B‖ for all t ∈ [0... |

15 | What are zeta functions of graphs and what are they good for
- Horton, Stark, et al.
- 2006
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14 |
Selberg–Ihara’s zeta function for p-adic discrete groups, in “Automorphic Forms and Geometry of Arithmetic Varieties
- Hashimoto, Hori
- 1989
(Show Context)
Citation Context ...ph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs. 0. Introduction The zeta functions associated to finite graphs by Ihara [20], Hashimoto =-=[15, 16]-=-, Bass [4] and others, combine features of Riemann’s zeta function, Artin L-functions, and Selberg’s zeta function, and may be viewed as analogues of the Dedekind zeta functions of a number field. The... |

13 | A trace on fractal graphs and the Ihara zeta function
- Guido, Isola, et al.
(Show Context)
Citation Context ...eral functional equations for various possible completions of the zeta function. In the final section, we prove the approximation result mentioned above. In closing this introduction, we note that in =-=[14]-=- we define and study the Ihara zeta functions attached to a new class of infinite graphs, called self-similar fractal graphs, which have greater complexity than the periodic ones. The contents of this... |

12 |
A note on the zeta function of a graph
- Northshield
- 1998
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12 |
L-functions in geometry and some applications, Curvature and topology of Riemannian manifolds (Katata
- Sunada
- 1985
(Show Context)
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11 | Zeta functions of discrete groups acting on trees
- Clair, Mokhtari-Sharghi
(Show Context)
Citation Context ...DIC GRAPHS AND ITS APPROXIMATION IN THE AMENABLE CASE DANIELE GUIDO, TOMMASO ISOLA, MICHEL L. LAPIDUS Abstract. In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi =-=[7]-=- on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara... |

11 | Ihara zeta functions for periodic simple graphs, to appear - Guido, Isola, et al. |

10 |
Artin type L-functions and the density theorem for prime cycles on finite graphs
- Hashimoto
- 1992
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9 | L -cohomology and group cohomology, Topology 25 - Cheeger, Gromov - 1986 |

6 | Convergence of zeta functions of graphs - Clair, Mokhtari-Sharghi |

3 |
Bartholdi zeta functions of some graphs
- Mizuno, Sato
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2 |
Zeta functions of graphs with Z actions, preprint
- Clair
- 2006
(Show Context)
Citation Context ...so-called theory of tree lattices, as developed by Bass, Lubotzky and others, see [5]). We mention [13] for a recent review of some results on zeta functions for finite or periodic simple graphs, and =-=[12, 7, 8, 9]-=- for the computation of the Ihara zeta function of several periodic simple graphs. Date: August 9, 2006. 2000 Mathematics Subject Classification. Primary 05C25,11M41, 46Lxx; Secondary 05C38, 11M36, 30... |

1 | 13 [6]J.Cheeger,M.Gromov,L2-cohomology and group cohomology, Topology 25 - Boston, Boston - 2001 |