The Ihara zeta function for infinite graphs

Lenz, Daniel; Pogorzelski, Felix; Schmidt, Marcel

doi:10.1090/tran/7508

The Ihara zeta function for infinite graphs
HTML articles powered by AMS MathViewer

by Daniel Lenz, Felix Pogorzelski and Marcel Schmidt PDF

Trans. Amer. Math. Soc. 371 (2019), 5687-5729 Request permission

Abstract:

We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings, and percolation graphs. Making use of Connes’s noncommutative integration theory, we construct a zeta function and present a determinant formula for it. We further introduce a notion of weak convergence of measure graphs and show that our construction is compatible with it. The approximation of the Ihara zeta function via the normalized version on finite graphs in the sense of Benjamini and Schramm follows as a special case. Our framework not only unifies corresponding earlier results occurring in the literature. It likewise provides extensions to rich new classes of objects such as percolation graphs.

References

Heinz Bauer, Measure and integration theory, De Gruyter Studies in Mathematics, vol. 26, Walter de Gruyter & Co., Berlin, 2001. Translated from the German by Robert B. Burckel. MR 1897176, DOI 10.1515/9783110866209
Itai Benjamini, Russell Lyons, and Oded Schramm, Unimodular random trees, Ergodic Theory Dynam. Systems 35 (2015), no. 2, 359–373. MR 3316916, DOI 10.1017/etds.2013.56
Itai Benjamini and Oded Schramm, Recurrence of distributional limits of finite planar graphs, Electron. J. Probab. 6 (2001), no. 23, 13. MR 1873300, DOI 10.1214/EJP.v6-96
G. Chinta, J. Jorgenson, and A. Karlsson, Heat kernels on regular graphs and generalized Ihara zeta function formulas, Monatsh. Math. 178 (2015), no. 2, 171–190. MR 3394421, DOI 10.1007/s00605-014-0685-4
Bryan Clair and Shahriar Mokhtari-Sharghi, Zeta functions of discrete groups acting on trees, J. Algebra 237 (2001), no. 2, 591–620. MR 1816705, DOI 10.1006/jabr.2000.8600
Bryan Clair and Shahriar Mokhtari-Sharghi, Convergence of zeta functions of graphs, Proc. Amer. Math. Soc. 130 (2002), no. 7, 1881–1886. MR 1896018, DOI 10.1090/S0002-9939-02-06532-2
Bryan Clair, The Ihara zeta function of the infinite grid, Electron. J. Combin. 21 (2014), no. 2, Paper 2.16, 19. MR 3210650
Alain Connes, Sur la théorie non commutative de l’intégration, Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978) Lecture Notes in Math., vol. 725, Springer, Berlin, 1979, pp. 19–143 (French). MR 548112
Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
Anton Deitmar, Ihara zeta functions of infinite weighted graphs, SIAM J. Discrete Math. 29 (2015), no. 4, 2100–2116. MR 3416854, DOI 10.1137/140957925
Jacques Dixmier, von Neumann algebras, North-Holland Mathematical Library, vol. 27, North-Holland Publishing Co., Amsterdam-New York, 1981. With a preface by E. C. Lance; Translated from the second French edition by F. Jellett. MR 641217
Gábor Elek, On limits of finite graphs, Combinatorica 27 (2007), no. 4, 503–507. MR 2359831, DOI 10.1007/s00493-007-2214-8
Gábor Elek, $L^2$-spectral invariants and convergent sequences of finite graphs, J. Funct. Anal. 254 (2008), no. 10, 2667–2689. MR 2406929, DOI 10.1016/j.jfa.2008.01.010
Gábor Elek, Weak convergence of finite graphs, integrated density of states and a Cheeger type inequality, J. Combin. Theory Ser. B 98 (2008), no. 1, 62–68. MR 2368023, DOI 10.1016/j.jctb.2007.03.004
Bent Fuglede and Richard V. Kadison, Determinant theory in finite factors, Ann. of Math. (2) 55 (1952), 520–530. MR 52696, DOI 10.2307/1969645
Rostislav I. Grigorchuk and Andrzej Żuk, The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps, Random walks and geometry, Walter de Gruyter, Berlin, 2004, pp. 141–180. MR 2087782
Michel L. Lapidus and Machiel van Frankenhuijsen, Fractal geometry, complex dimensions and zeta functions, 2nd ed., Springer Monographs in Mathematics, Springer, New York, 2013. Geometry and spectra of fractal strings. MR 2977849, DOI 10.1007/978-1-4614-2176-4
Daniele Guido, Tommaso Isola, and Michel L. Lapidus, Ihara’s zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. 255 (2008), no. 6, 1339–1361. MR 2565711, DOI 10.1016/j.jfa.2008.07.011
Daniele Guido, Tommaso Isola, and Michel L. Lapidus, Ihara’s zeta function for periodic graphs and its approximation in the amenable case, J. Funct. Anal. 255 (2008), no. 6, 1339–1361. MR 2565711, DOI 10.1016/j.jfa.2008.07.011
Daniele Guido, Tommaso Isola, and Michel L. Lapidus, A trace on fractal graphs and the Ihara zeta function, Trans. Amer. Math. Soc. 361 (2009), no. 6, 3041–3070. MR 2485417, DOI 10.1090/S0002-9947-08-04702-8
Misha Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps. I, Math. Phys. Anal. Geom. 2 (1999), no. 4, 323–415. MR 1742309, DOI 10.1023/A:1009841100168
D. Lenz, F. Pogorzelski, and M. Schmidt, Topological measure graphs and inverse semigroups (in preparation).
Daniel Lenz, Norbert Peyerimhoff, and Ivan Veselić, Groupoids, von Neumann algebras and the integrated density of states, Math. Phys. Anal. Geom. 10 (2007), no. 1, 1–41. MR 2340531, DOI 10.1007/s11040-007-9019-2
Hanfeng Li and Andreas Thom, Entropy, determinants, and $L^2$-torsion, J. Amer. Math. Soc. 27 (2014), no. 1, 239–292. MR 3110799, DOI 10.1090/S0894-0347-2013-00778-X
László Lovász, Large networks and graph limits, American Mathematical Society Colloquium Publications, vol. 60, American Mathematical Society, Providence, RI, 2012. MR 3012035, DOI 10.1090/coll/060
W. Lück, Approximating $L^2$-invariants by their finite-dimensional analogues, Geom. Funct. Anal. 4 (1994), no. 4, 455–481. MR 1280122, DOI 10.1007/BF01896404
Alan L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics, vol. 170, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1724106, DOI 10.1007/978-1-4612-1774-9
Vladimir G. Pestov, Hyperlinear and sofic groups: a brief guide, Bull. Symbolic Logic 14 (2008), no. 4, 449–480. MR 2460675, DOI 10.2178/bsl/1231081461
Felix Pogorzelski, Convergence theorems for graph sequences, Internat. J. Algebra Comput. 24 (2014), no. 8, 1233–1251. MR 3296365, DOI 10.1142/S0218196714500556
F. Pogorzelski, Banach space-valued ergodic theorems and spectral approximation, 2014. Thesis (Ph.D.)–Friedrich-Schiller-Universität Jena.
Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266
Christoph Schumacher and Fabian Schwarzenberger, Approximation of the integrated density of states on sofic groups, Ann. Henri Poincaré 16 (2015), no. 4, 1067–1101. MR 3317792, DOI 10.1007/s00023-014-0342-4
Jean-Pierre Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke $T_p$, J. Amer. Math. Soc. 10 (1997), no. 1, 75–102 (French). MR 1396897, DOI 10.1090/S0894-0347-97-00220-8
H. M. Stark and A. A. Terras, Zeta functions of finite graphs and coverings, Adv. Math. 121 (1996), no. 1, 124–165. MR 1399606, DOI 10.1006/aima.1996.0050
Audrey Terras, Zeta functions of graphs, Cambridge Studies in Advanced Mathematics, vol. 128, Cambridge University Press, Cambridge, 2011. A stroll through the garden. MR 2768284
Joachim Weidmann, Linear operators in Hilbert spaces, Graduate Texts in Mathematics, vol. 68, Springer-Verlag, New York-Berlin, 1980. Translated from the German by Joseph Szücs. MR 566954
Benjamin Weiss, Sofic groups and dynamical systems, Sankhyā Ser. A 62 (2000), no. 3, 350–359. Ergodic theory and harmonic analysis (Mumbai, 1999). MR 1803462