1. Introduction ................................................. 1
2. Hyperbolic Surfaces .......................................... 7
2.1. The hyperbolic plane .................................... 8
2.2. Fuchsian groups ........................................ 13
2.3. Geometrically finite groups ............................ 18
2.4. Classification of hyperbolic ends ...................... 22
2.5. Gauss-Bonnet theorem ................................... 28
2.6. Length spectrum and Selberg's zeta function ............ 31
3. Compact and Finite-Area Surfaces ............................ 37
3.1. Selberg's trace formula for compact surfaces ........... 37
3.2. Consequences of the trace formula ...................... 42
3.3. Finite-area hyperbolic surfaces ........................ 45
4. Spectral Theory for the Hyperbolic Plane .................... 49
4.1. Resolvent .............................................. 49
4.2. Generalized eigenfunctions ............................. 52
4.3. Scattering matrix ...................................... 56
5. Model Resolvents for Cylinders .............................. 61
5.1. Hyperbolic cylinders ................................... 61
5.2 Funnels ................................................. 68
5.3. Parabolic cylinder ..................................... 70
6. The Resolvent ............................................... 75
6.1. Compactification ....................................... 75
6.2. Analytic Fredholm theorem .............................. 79
6.3. Continuation of the resolvent .......................... 81
6.4 Structure of the resolvent kernel ....................... 84
6.5. The stretched product .................................. 87
7. Spectral and Scattering Theory .............................. 93
7.1. Essential and discrete spectrum ........................ 93
7.2. Absence of embedded eigenvalues ........................ 95
7.3. Generalized eigenfunctions ............................ 102
7.4. Scattering matrix ..................................... 105
7.5. Scattering matrices for the funnel and cylinders ...... 114
8. Resonances and Scattering Poles ............................ 117
8.1. Multiplicities of resonances .......................... 118
8.2. Structure of the resolvent at a resonance ............. 119
8.3. Scattering poles ...................................... 124
8.4. Operator logarithmic residues ......................... 126
8.5. Half-integer points ................................... 131
8.6. Coincidence of resonances and scattering poles ........ 137
9. Upper Bound for Resonances ................................. 147
9.1. Resonances and zeros of determinants .................. 148
9.2. Singular value estimates .............................. 151
9.3. Upper bound ........................................... 154
9.4. Estimates on model terms .............................. 156
10. Selberg Zeta Function ..................................... 171
10.1. Relative scattering determinant ...................... 173
10.2. Regularized traces ................................... 175
10.3. The resolvent 0-trace calculation .................... 183
10.4. Structure of the zeta function ....................... 189
10.5. Order bound .......................................... 196
10.6. Determinant of the Laplacian ......................... 203
11. Wave Trace and Poisson Formula ............................ 207
11.1. Regularized wave trace ............................... 208
11.2. Model wave kernel .................................... 209
11.3. Wave 0-trace formula ................................. 211
11.4. Poisson formula ...................................... 215
12. Resonance Asymptotics ..................................... 223
12.1. Lower bound on resonances ............................ 223
12.2. Lower bound near the critical line ................... 226
12.3. Weyl formula for the scattering phase ................ 229
13. Inverse Spectral Geometry ................................. 237
13.1. Resonances and the length spectrum ................... 238
13.2. Hyperbolic trigonometry .............................. 239
13.3. Teichmuller space .................................... 242
13.4. Finiteness of isospectral classes .................... 248
14. Patterson-Sullivan Theory ................................. 259
14.1. A measure on the limit set ........................... 259
14.2. Ergodicity ........................................... 267
14.3. Hausdorff measure of the limit set ................... 274
14.4. The first resonance .................................. 278
14.5. Prime geodesic theorem ............................... 284
14.6. Refined asymptotics of the length spectrum ........... 289
15. Dynamical Approach to the Zeta Function ................... 297
15.1. Schottky groups ...................................... 298
15.2. Symbolic dynamics .................................... 300
15.3. Dynamical zeta function .............................. 303
15.4. Growth estimates ..................................... 308
A. Appendix ................................................... 315
A.1. Entire functions ...................................... 315
A.2. Distributions and Fourier transforms .................. 320
A.3. Spectral theory ....................................... 324
A.4. Singular values, traces, and determinants ............. 330
A.5. Pseudodifferential operators .......................... 336
References .................................................... 341
Notation Guide ................................................ 351
Index ......................................................... 353
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