The divisor of Selberg’s zeta function for Kleinian groups. Appendix A by Charles Epstein. (English) Zbl 1012.11083
Selberg’s zeta-function is defined in terms of the length spectrum of a compact or, more generally, a finite volume hyperbolic surface. Its divisor is determined by the eigenvalues and scattering poles of the Laplacian and the Euler characteristic of the surface. The authors note “this relationship may be regarded as one form of Selberg’s trace formula for a compact or finite volume surface and as a particularly precise quantization of an ergodic system, namely, the geodesic flow”.
The authors extend this relationship to convex cocompact hyperbolic manifolds in all dimensions. They show the divisor of the Selberg zeta-function is determined by the eigenvalues and scattering poles of the Laplacian together with the Euler characteristic of the corresponding compactified manifold with boundary. In even dimensions, this uses work of Bunke and Olbrich.
The contents of the paper are as follows: §1 Introduction. §2 Symbolic dynamics and Selberg’s zeta-function. §3 Geometry at infinity (coordinate neighborhoods, envelopes of horospheres, normal flow). §4 Review of scattering theory (model space, resolvant, eigenfunctions, operator). §5 The ‘logarithmic derivative’ of the scattering operator.§6 The logarithmic derivative of the zeta-function. §7 Computation of the divisor (spectral term, topological term, residues in odd and even dimensions).
Appendix A: An asymptotic volume formula for convex cocompact hyperbolic manifolds (by C. Epstein).
Appendix B: The scattering operator and zeta-function for a class of cylindrical manifolds.
The authors extend this relationship to convex cocompact hyperbolic manifolds in all dimensions. They show the divisor of the Selberg zeta-function is determined by the eigenvalues and scattering poles of the Laplacian together with the Euler characteristic of the corresponding compactified manifold with boundary. In even dimensions, this uses work of Bunke and Olbrich.
The contents of the paper are as follows: §1 Introduction. §2 Symbolic dynamics and Selberg’s zeta-function. §3 Geometry at infinity (coordinate neighborhoods, envelopes of horospheres, normal flow). §4 Review of scattering theory (model space, resolvant, eigenfunctions, operator). §5 The ‘logarithmic derivative’ of the scattering operator.§6 The logarithmic derivative of the zeta-function. §7 Computation of the divisor (spectral term, topological term, residues in odd and even dimensions).
Appendix A: An asymptotic volume formula for convex cocompact hyperbolic manifolds (by C. Epstein).
Appendix B: The scattering operator and zeta-function for a class of cylindrical manifolds.
Reviewer: Peter B.Gilkey (Eugene)
MSC:
| 11M36 | Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 22E40 | Discrete subgroups of Lie groups |
| 37C30 | Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. |
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
Keywords:
geodesic flow; convex cocompact hyperbolic manifolds; divisor; Selberg zeta-function; eigenvalues; scattering poles; Laplacian; Euler characteristic; symbolic dynamicsReferences:
| [1] | S. Agmon, “On the spectral theory of the Laplacian on noncompact hyperbolic manifolds” in Journées “Équations aux dérivées partielles” (Saint Jean de Monts, 1987), École Polytechnique, Palaiseau, 1987, exp. no. 17. · Zbl 0636.58037 |
| [2] | M. Babillot and M. Peigné, Closed geodesics in homology classes on hyperbolic manifolds with cusps, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 901–906. · Zbl 0884.58078 · doi:10.1016/S0764-4442(97)86966-2 |
| [3] | E. W. Barnes, The theory of the \(G\)-function, Quart. J. Pure Appl. Math. 31 (1899), 264–314. · JFM 30.0389.02 |
| [4] | D. Borthwick, Scattering theory and deformations of asymptotically hyperbolic metrics, preprint, http://www.arXiv.org/abs/dg-ga/9711016. |
| [5] | D. Borthwick and P. Perry, Scattering poles for hyperbolic manifolds, submitted to Trans. Amer. Math. Soc. · Zbl 1009.58021 |
| [6] | R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429–460. JSTOR: · Zbl 0282.58009 · doi:10.2307/2373793 |
| [7] | U. Bunke and M. Olbrich, Fuchsian groups of the second kind and representations carried by the limit set, Invent. Math. 127 (1997), 127–154. · Zbl 0880.30035 · doi:10.1007/s002220050117 |
| [8] | –. –. –. –., Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group, Ann. of Math. (2) 149 (1999), 627–689. JSTOR: · Zbl 0969.11019 · doi:10.2307/120977 |
| [9] | ——–, On the cohomology of Kleinian groups with coefficients in representations carried by the limit set, preprint, Humboldt-Universität zu Berlin, Institut für Reine Mathematik, 1995. |
| [10] | S.-S. Chern, On the curvature integra in a Riemannian manifold, Ann. of Math. (2) 46 (1945), 674–684. JSTOR: · Zbl 0060.38104 · doi:10.2307/1969203 |
| [11] | J. Elstrodt, F. Grunewald, and J. Mennicke, Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory, Springer Monogr. Math., Springer, Berlin, 1998. · Zbl 0888.11001 |
| [12] | C. Epstein, Envelopes of horospheres and Weingarten surfaces in hyperbolic \(3\)-space, preprint, Princeton Univ., 1984. |
| [13] | ——–, private communication. · Zbl 1207.68077 |
| [14] | D. B. A. Epstein and A. Marden, “Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces” in Analytical and Geometric Aspects of Hyperbolic Space (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser. 111 , Cambridge Univ. Press, Cambridge, 1987, 113–253. · Zbl 0612.57010 |
| [15] | D. Fried, The zeta functions of Ruelle and Selberg, I, Ann. Sci. École Norm. Sup. (4) 19 (1986), 491–517. · Zbl 0609.58033 |
| [16] | R. Froese and P. Hislop, in preparation. |
| [17] | R. Froese, P. Hislop, and P. Perry, The Laplace operator on hyperbolic three manifolds with cusps of nonmaximal rank, Invent. Math. 106 (1991), 295–333. · Zbl 0763.58028 · doi:10.1007/BF01243915 |
| [18] | R. Gangolli, Zeta functions of Selberg’s type for compact space forms of symmetric spaces of rank one, Illinois J. Math. 21 (1977), 1–41. · Zbl 0354.33013 |
| [19] | I. C. Gohberg and E. I. Sigal, An operator generalization of the logarithmic residue theorem and the theorem of Rouché, Math. USSR-Sb. 13 (1971), 603–625. · Zbl 0254.47046 · doi:10.1070/SM1971v013n04ABEH003702 |
| [20] | L. Guillopé, Théorie spectrale de quelques variétés à bouts, Ann. Sci. École Norm. Sup. (4) 22 (1989), 137–160. · Zbl 0682.58049 |
| [21] | –. –. –. –., “Fonctions zêta de Selberg et surfaces de géométrie finie” in Zeta Functions in Geometry (Tokyo, 1990), Adv. Stud. Pure Math. 21 , Kinokuniya, Tokyo, 1992, 33–70. · Zbl 0794.58044 |
| [22] | L. Guillopé and M. Zworski, Polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity, Asymptotic Anal. 11 (1995), 1–22. · Zbl 0859.58028 |
| [23] | –. –. –. –., Upper bounds on the number of resonances for non-compact Riemann surfaces, J. Funct. Anal. 129 (1995), 364–389. · Zbl 0841.58063 · doi:10.1006/jfan.1995.1055 |
| [24] | –. –. –. –., Scattering asymptotics for Riemann surfaces, Ann. of Math. (2) 145 (1997), 597–660. JSTOR: · Zbl 0898.58054 · doi:10.2307/2951846 |
| [25] | –. –. –. –., The wave trace for Riemann surfaces, Geom. Funct. Anal. 9 (1999), 1156–1168. · Zbl 0947.58022 · doi:10.1007/s000390050110 |
| [26] | D. Hejhal, The Selberg Trace Formula for \(¶SL(2,\mathbbR)\), Vol. 1, Lecture Notes in Math. 548 , Springer, Berlin, 1976; Vol. 2, Lecture Notes in Math. 1001 , Springer, Berlin, 1983. |
| [27] | P. D. Hislop, “The geometry and spectra of hyperbolic manifolds” in Spectral and Inverse Spectral Theory (Bangalore, 1993), Proc. Indian Acad. Sci. Math. Sci. 104 (1994), 715–776. · Zbl 0832.58005 |
| [28] | L. Hörmander, The Analysis of Linear Partial Differential Operators, III: Pseudodifferential Operators, Grundlehren Math. Wiss. 274 , Springer, Berlin, 1985. · Zbl 0601.35001 |
| [29] | M. Joshi and A. Sá Barreto, Inverse scattering on asymptotically hyperbolic manifolds, Acta Math. 184 (2000), 41–86. · Zbl 1142.58309 · doi:10.1007/BF02392781 |
| [30] | T. Kato, Perturbation Theory for Linear Operators, 2d ed., Grundlehren Math. Wiss. 132 , Springer, Berlin, 1976. · Zbl 0342.47009 |
| [31] | P. Lax and R. S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Funct. Anal. 46 (1982), 280–350. · Zbl 0497.30036 · doi:10.1016/0022-1236(82)90050-7 |
| [32] | –. –. –. –., Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces, I, Comm. Pure Appl. Math. 37 (1984), 303–328.; II, 37 (1984), 779–813.; III, 38 (1985), 179–208. · Zbl 0544.10024 · doi:10.1002/cpa.3160370304 |
| [33] | N. Mandouvalos, Scattering opertator, Eisenstein series, inner product formula and “Maass-Selberg” relations for Kleinian groups, Mem. Amer. Math. Soc. 78 (1989), no. 400. · Zbl 0673.10023 |
| [34] | –. –. –. –., Relativity of the spectrum and discrete groups on hyperbolic spaces, Trans. Amer. Math. Soc. 350 (1998), 559–569. JSTOR: · Zbl 1044.11577 · doi:10.1090/S0002-9947-98-01803-0 |
| [35] | A. Manning, Axiom A diffeomorphisms have rational zeta functions, Bull. London Math. Soc. 3 (1971), 215–220. · Zbl 0219.58007 · doi:10.1112/blms/3.2.215 |
| [36] | R. Mazzeo, The Hodge cohomology of a conformally compact metric, J. Differential Geom. 28 (1988), 309–339. · Zbl 0656.53042 |
| [37] | –. –. –. –., Elliptic theory of differential edge operators, I, Comm. Partial Differential Equations 16 (1991), 1615–1664. · Zbl 0745.58045 · doi:10.1080/03605309108820815 |
| [38] | –. –. –. –., Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds, Amer. J. Math. 113 (1991), 25–45. JSTOR: · Zbl 0725.58044 · doi:10.2307/2374820 |
| [39] | R. Mazzeo and R. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), 260–310. · Zbl 0636.58034 · doi:10.1016/0022-1236(87)90097-8 |
| [40] | R. B. Melrose, Geometric Scattering Theory, Stanford Lectures, Cambridge Univ. Press, Cambridge, 1995. · Zbl 0849.58071 |
| [41] | R. B. Melrose and M. Zworski, Scattering metrics and geodesic flow at infinity, Invent. Math. 124 (1996), 389–436. · Zbl 0855.58058 · doi:10.1007/s002220050058 |
| [42] | L. B. Parnovskiĭ, The Selberg trace formula and the Selberg zeta function for cocompact discrete subgroups of \(\SO_+(1,n)\) (in Russian), Funktsional. Anal. i Prilozhen. 26 (1992), 55–64.; English translation in Funct. Anal. Appl. 26 (1992), 196–203. |
| [43] | S. J. Patterson, The Laplacian operator on a Riemann surface, I, Compositio Math. 31 (1975), 83–107.; II, 32 (1976) 71–112.; III, 33 (1976), 227–259. · Zbl 0321.30020 |
| [44] | –. –. –. –., The exponent of convergence of Poincaré series, Monatsh. Math. 82 (1976), 297–315. · Zbl 0349.30012 · doi:10.1007/BF01540601 |
| [45] | –. –. –. –., The limit set of a Fuchsian group, Acta. Math. 136 (1976), 241–273. · Zbl 0336.30005 · doi:10.1007/BF02392046 |
| [46] | –. –. –. –., “The Selberg zeta-function of a Kleinian group” in Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, 1989, 409–441. · Zbl 0668.10036 |
| [47] | –. –. –. –., “On Ruelle’s zeta-function” in Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday (Ramat Aviv, 1989), Part II, Israel Math. Conf. Proc. 3 , Weizmann, Jeruslaem, 1990, 163–184. |
| [48] | P. A. Perry, The Laplace operator on a hyperbolic manifold, II: Eisenstein series and the scattering matrix, J. Reine Angew. Math. 398 (1989), 67–91. · Zbl 0677.58044 · doi:10.1515/crll.1989.398.67 |
| [49] | –. –. –. –., The Selberg zeta function and a local trace formula for Kleinian groups, J. Reine Angew. Math. 410 (1990), 116–152. · Zbl 0697.10027 · doi:10.1515/crll.1990.410.116 |
| [50] | –. –. –. –., The Selberg zeta function and scattering poles for Kleinian groups, Bull. Amer. Math. Soc. (N.S.) 24 (1991), 327–333. · Zbl 0723.11028 · doi:10.1090/S0273-0979-1991-16024-6 |
| [51] | –. –. –. –., “The Selberg zeta function and scattering poles for Kleinian groups” in Mathematical Quantum Theory, II: Schrödinger Operators (Vancouver, B.C., 1993), CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, 1995, 243–251. · Zbl 0824.11037 |
| [52] | –. –. –. –., A trace-class rigidity theorem for Kleinian groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 251–257. · Zbl 0861.57017 |
| [53] | –. –. –. –., “Meromorphic continuation of the resolvent for Kleinian groups” in Spectral Problems in Geometry and Arithmetic (Iowa City, Iowa, 1997), Contemp. Math. 237 Amer. Math. Soc., Providence, 1999, 123–147. · Zbl 0942.58033 |
| [54] | ——–, A Poisson summation formula and lower bounds for resonances in hyperbolic manifolds, to appear in J. Funct. Anal. · Zbl 1035.58020 · doi:10.1155/S1073792803212241 |
| [55] | I. R. Porteous, The normal singularities of a submanifold, J. Differential Geom. 5 (1971), 543–564. · Zbl 0226.53010 |
| [56] | M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972. · Zbl 0242.46001 |
| [57] | ——–, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, 1978. · Zbl 0401.47001 |
| [58] | A. Rocha, Meromorphic extension of the Selberg zeta function for Kleinian groups via thermodynamic formalism, Math. Proc. Cambridge Philos. Soc. 119 (1996), 179–190. · Zbl 0855.30037 · doi:10.1017/S0305004100074065 |
| [59] | D. Ruelle, Zeta-functions for expanding maps and Anosov flows, Invent. Math. 34 (1976), 231–242. · Zbl 0329.58014 · doi:10.1007/BF01403069 |
| [60] | P. Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), 113–120. · Zbl 0618.10023 · doi:10.1007/BF01209019 |
| [61] | H. Schlichtkrull, Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Progr. Math. 49 , Birkhäuser, Boston, 1984. · Zbl 0555.43002 |
| [62] | D. R. Scott, Selberg-type zeta functions for the group of complex two by two matrices of determinant one, Math. Ann. 253 (1980), 177–194. · Zbl 0449.22013 · doi:10.1007/BF02352871 |
| [63] | R. T. Seeley, “Complex powers of an elliptic operator” in Singular Integrals (Chicago, 1966), Proc. Sympos. Pure Math. 10 , Amer. Math. Soc., Providence, 1967, 288–307. · Zbl 0159.15504 |
| [64] | A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. · Zbl 0072.08201 |
| [65] | M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer Ser. Soviet Math., Springer, Berlin, 1987. · Zbl 0616.47040 |
| [66] | B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Note Ser. 35 , Cambridge Univ. Press, Cambridge, 1979. · Zbl 0423.47001 |
| [67] | J. Sjöstrand and M. Zworski, Lower bounds on the number of scattering poles, Comm. Partial Differential Equations 18 (1993), 847–857. · Zbl 0784.35070 · doi:10.1080/03605309308820953 |
| [68] | D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984), 259–277. · Zbl 0566.58022 · doi:10.1007/BF02392379 |
| [69] | M.-F. Vignéras, “L’équation fonctionnelle de la fonction zêta de Selberg du groupe modulaire \(¶SL(2,\mathbbZ)\)” in Journées Arithmétiques de Luminy (Luminy, 1978), Astérisque 61 , Soc. Math. France, Montrouge, 1979, 235–249. |
| [70] | M. Wakayama, Zeta functions of Selberg’s type associated with homogeneous vector bundles, Hiroshima Math. J. 15 (1985), 235–295. · Zbl 0592.22012 |
| [71] | –. –. –. –., A note on the Selberg zeta function for compact quotients of hyperbolic spaces, Hiroshima Math. J. 21 (1991), 539–555. · Zbl 0829.22019 |
| [72] | M. Zworski, “Counting scattering poles” in Spectral and Scattering Theory (Sanda, Japan, 1992), Lecture Notes in Pure and Appl. Math. 161 , Dekker, New York, 1994, 301–331. · Zbl 0823.35139 |
| [73] | –. –. –. –., Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces, Invent. Math. 136 (1999), 353–409. · Zbl 1016.58014 · doi:10.1007/s002220050313 |
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