##
**Groups acting on hyperbolic space. Harmonic analysis and number theory.**
*(English)*
Zbl 0888.11001

Springer Monographs in Mathematics. Berlin: Springer. xv, 524 p. (1998).

The subject of the book under review is the spectral theory of the Laplacian for discrete groups acting on hyperbolic 3-space. It is located at the intersection of number theory, geometry, spectral theory of differential operators and group theory (finite, discrete, and Lie groups). It is also a subject of interest to physicists and topologists. There are graph theoretic analogues which are useful in computer science. See the proceedings of the recent IMA (Institute for Mathematics and its Applications) Conference on Emerging Applications of Number Theory [D. Hejhal et al. (eds.), The IMA Volumes in Mathematics and its Applications. 109. New York, NY: Springer (1999; Zbl 0919.00047)].

The volume under review is a welcome addition to the literature. Hyperbolic 3-space \(\mathbb H\) may be viewed as a space of 3-component quaternions with the special linear group \(\text{SL}(2,\mathbb C)\) of \(2\times 2\) complex matrices of determinant 1 acting by fractional linear transformation. The subject has immediate parallels with the theory of the 2-dimensional Poincaré upper half plane with its non-Euclidean arc length given by \(ds^2 = y^{-2}(dx^2 + dy^2)\) invariant under the action of \(\text{SL}(2, \mathbb R)\). For hyperbolic 3-space, you just add an extra \(x\) coordinate, which we will call \(u\). Then in the formula for the square of the arc length element \(ds^2\) replace \(dx^2\) with \(dx^2 + du^2\). There is a corresponding Laplace operator and much of this book concerns the spectral theory of that operator on fundamental domains for discrete subgroups of \(\text{SL}(2, \mathbb C)\) acting on \(\mathbb H\). There is a Selberg trace formula, Selberg zeta function, etc. See chapter 3 of the reviewer’s books [Harmonic analysis on symmetric spaces and applications. I (1985; Zbl 0574.10029); II (1989; Zbl 0668.10033)] for a survey of some of the 2-dimensional theory and chapter 5 for some of the general background. Short discussions of the Selberg zeta-function in the 2-dimensional case can be found in D. A. Hejhal [Duke Math. J. 43, 441–482 (1976; Zbl 0346.10010)] or J. Elstrodt [Jahresber. Dtsch. Math.-Ver. 83, 45–77 (1981; Zbl 0453.10021)].

One is immediately led to consider discrete subgroups like \(\text{SL}(2, O _K)\), where \(O_K\) is the ring of integers of an imaginary quadratic number field \(K\). There is much number theoretical interest in this case. For example, the volume of the fundamental domain involves the Dedekind zeta function of the number field \(K\). There is a 1-1 correspondence (first observed by Bianchi in 1892) between the cusps of the fundamental domain and the elements of the ideal class group of the number field. It is conjectured that “the Artin \(L\)-functions of the 2-dimensional irreducible complex representations of the absolute Galois group of \(K\) should occur as Mellin transforms of certain eigenfunctions with eigenvalue 1 for \(-\Delta\) acting on \(L^2(\text{PSL}(2,\mathcal O)\backslash\mathbb H)\).”

Let us give a brief outline of the contents. Chapter 1 gives various models for 3-dimensional hyperbolic space \(\mathbb H\). In addition to the upper half-space model, one has the unit ball, hyperboloid, and Kleinian models.

Chapter 2 involves groups \(\Gamma\) acting discontinuously on \(\mathbb H\). The elements of \(\text{SL}(2, \mathbb C)\) are classified according to location of the trace. There will be elements that are loxodromic (non-real trace) in addition to those that occur in the Selberg trace formula for \(\text{SL}(2, \mathbb R)\). Lower bounds for volumes of fundamental domains for discrete subgroups \(\Gamma\) of \(\text{SL}(2, \mathbb C)\) acting on \(\mathbb H\) are considered. Note that the use of commas/periods in decimal notation is inconsistent (e.g., see pages 59 and 346).

In chapter 3 the reader will find automorphic functions, the Maass-Selberg relations, Fourier expansions, Eisenstein series. Chapter 4 concerns the spectral theory of the Laplacian on \(L^2(\Gamma\backslash \mathbb H)\) and the resolvent kernel.

The topic of chapter 5 is Selberg’s trace formula for compact fundamental domains. The result is given in terms of the trace of a product of two resolvents. On the left hand side is a sum over the eigenvalue spectrum of the Laplacian for compact \(M=\Gamma \backslash \mathbb H\). On the right hand side is a sum over the length spectrum of closed geodesics in \(M\). Following a suggestion of Marie-France Vignéras, the authors obtain an analogue of Huber’s theorem, which says that if two discrete co-compact groups in \(\text{PSL}(2, \mathbb C)\) have eigenvalue spectra that agree up to at most finitely many terms, then the eigenspectra, length spectra, volumes must agree. A similar result is obtained if the length spectra agree up to finitely many terms. The Selberg zeta-function is defined as a product over a maximal reduced system of primitive hyperbolic or loxodromic conjugacy classes \(\{\gamma\}\) in \(\Gamma\). The product involves the eigenvalues of \(\gamma\) in a somewhat similar way to the real case. It is more complicated as one needs complex conjugates and an inner product over pairs of positive integers satisfying a congruence condition modulo the number of elements of finite order in the centralizer of \(\gamma\). The Weyl law for the asymptotics of the eigenvalues of the Laplacian on \(L^2(M)\) is obtained, while the authors note that there is no explicit example of a non-zero eigenvalue known for any \(\Gamma\) satisfying the hypotheses of the chapter. Next is the prime geodesic theorem, which says that the number of primitive hyperbolic or loxodromic conjugacy classes in \(\Gamma\) which have norm \(\leq x\) is asymptotic to \[ \frac {x^2}{2\log x}\qquad\text{as}\quad x\rightarrow \infty. \]

Chapter 6 has as its topic the case that the fundamental domain is non-compact but has finite volume. Once again the Selberg trace formula is considered. The attempt to obtain a Weyl law falls short without more explicit knowledge of the function \(\phi\), which is the determinant of the scattering matrix coming from Fourier expansions of the Eisenstein series at the cusps of the fundamental domain. There is an interesting discussion of the history of the question as to whether one should expect a Weyl law for all cofinite subgroups \(\Gamma\). The work of Phillips and Sarnak leads one to conjecture that at least in the 2-dimensional case when \(\Gamma\) is not co-compact or arithmetic, then there may be only finitely many members of the discrete spectrum of the Laplacian. But “Sarnak somewhat excludes the hyperbolic spaces of dimensions \(n\geq 3\).” Here the prime geodesic theorem at least goes as in Chapter 5, except the factor of 1/2 seems to be missing.

The special case that \(\Gamma = \text{SL}(2, O_K)\), for \(K\) an imaginary quadratic field, is considered in Chapter 7. It is shown, for example, that, for this \(\Gamma\), the smallest positive eigenvalue of \(-\Delta\) on \(L^2(\Gamma \backslash\mathbb H)\) is \(\geq 3/4\). Selberg’s conjecture would replace 3/4 with 1. Tables of eigenvalues for \(K\) of discriminant \(-4\) and \(-8\) are given.

Chapter 8 concerns Eisenstein series for \(\text{SL}(2, O_K)\). It is shown that the determinant of the scattering matrix is related to the zeta function of the Hilbert class field of \(K\). Weyl’s asymptotic law is proved for this sort of \(\Gamma\).

Chapter 9 begins with reduction theory for integral binary hermitian forms. While computing the volume of the fundamental domain, an interesting error of G. Humbert is corrected. The mass formula for binary hermitian forms is proved. This is a result in the tradition of C. L. Siegel’s work on quadratic forms. The more general result of Hel Braun differs from that of this book.

Chapter 10 provides some interesting examples of discontinuous groups on \(\mathbb H\). Some come from quaternion groups and others from unit groups of quadratic forms. Arithmetic and non-arithmetic groups are considered. Some very simple discontinuous groups generated by reflections in the faces of tesselating tetrahedra are found in Section 10.4. In the spirit of M.-F. Vignéras [C. R. Acad. Sci., Paris, Sér. A 287, 47–49 (1978; Zbl 0387.10013)], the Corollary on page 455 says that for any positive integer \(n\) there are \(n\) pairwise non-conjugate co-compact \(\Gamma _i\) in \(\text{Iso}(\mathbb H)\) which are all isospectral.

To summarize, this excellent book covers a huge amount of material and is highly recommended for anyone interested in this subject.

The volume under review is a welcome addition to the literature. Hyperbolic 3-space \(\mathbb H\) may be viewed as a space of 3-component quaternions with the special linear group \(\text{SL}(2,\mathbb C)\) of \(2\times 2\) complex matrices of determinant 1 acting by fractional linear transformation. The subject has immediate parallels with the theory of the 2-dimensional Poincaré upper half plane with its non-Euclidean arc length given by \(ds^2 = y^{-2}(dx^2 + dy^2)\) invariant under the action of \(\text{SL}(2, \mathbb R)\). For hyperbolic 3-space, you just add an extra \(x\) coordinate, which we will call \(u\). Then in the formula for the square of the arc length element \(ds^2\) replace \(dx^2\) with \(dx^2 + du^2\). There is a corresponding Laplace operator and much of this book concerns the spectral theory of that operator on fundamental domains for discrete subgroups of \(\text{SL}(2, \mathbb C)\) acting on \(\mathbb H\). There is a Selberg trace formula, Selberg zeta function, etc. See chapter 3 of the reviewer’s books [Harmonic analysis on symmetric spaces and applications. I (1985; Zbl 0574.10029); II (1989; Zbl 0668.10033)] for a survey of some of the 2-dimensional theory and chapter 5 for some of the general background. Short discussions of the Selberg zeta-function in the 2-dimensional case can be found in D. A. Hejhal [Duke Math. J. 43, 441–482 (1976; Zbl 0346.10010)] or J. Elstrodt [Jahresber. Dtsch. Math.-Ver. 83, 45–77 (1981; Zbl 0453.10021)].

One is immediately led to consider discrete subgroups like \(\text{SL}(2, O _K)\), where \(O_K\) is the ring of integers of an imaginary quadratic number field \(K\). There is much number theoretical interest in this case. For example, the volume of the fundamental domain involves the Dedekind zeta function of the number field \(K\). There is a 1-1 correspondence (first observed by Bianchi in 1892) between the cusps of the fundamental domain and the elements of the ideal class group of the number field. It is conjectured that “the Artin \(L\)-functions of the 2-dimensional irreducible complex representations of the absolute Galois group of \(K\) should occur as Mellin transforms of certain eigenfunctions with eigenvalue 1 for \(-\Delta\) acting on \(L^2(\text{PSL}(2,\mathcal O)\backslash\mathbb H)\).”

Let us give a brief outline of the contents. Chapter 1 gives various models for 3-dimensional hyperbolic space \(\mathbb H\). In addition to the upper half-space model, one has the unit ball, hyperboloid, and Kleinian models.

Chapter 2 involves groups \(\Gamma\) acting discontinuously on \(\mathbb H\). The elements of \(\text{SL}(2, \mathbb C)\) are classified according to location of the trace. There will be elements that are loxodromic (non-real trace) in addition to those that occur in the Selberg trace formula for \(\text{SL}(2, \mathbb R)\). Lower bounds for volumes of fundamental domains for discrete subgroups \(\Gamma\) of \(\text{SL}(2, \mathbb C)\) acting on \(\mathbb H\) are considered. Note that the use of commas/periods in decimal notation is inconsistent (e.g., see pages 59 and 346).

In chapter 3 the reader will find automorphic functions, the Maass-Selberg relations, Fourier expansions, Eisenstein series. Chapter 4 concerns the spectral theory of the Laplacian on \(L^2(\Gamma\backslash \mathbb H)\) and the resolvent kernel.

The topic of chapter 5 is Selberg’s trace formula for compact fundamental domains. The result is given in terms of the trace of a product of two resolvents. On the left hand side is a sum over the eigenvalue spectrum of the Laplacian for compact \(M=\Gamma \backslash \mathbb H\). On the right hand side is a sum over the length spectrum of closed geodesics in \(M\). Following a suggestion of Marie-France Vignéras, the authors obtain an analogue of Huber’s theorem, which says that if two discrete co-compact groups in \(\text{PSL}(2, \mathbb C)\) have eigenvalue spectra that agree up to at most finitely many terms, then the eigenspectra, length spectra, volumes must agree. A similar result is obtained if the length spectra agree up to finitely many terms. The Selberg zeta-function is defined as a product over a maximal reduced system of primitive hyperbolic or loxodromic conjugacy classes \(\{\gamma\}\) in \(\Gamma\). The product involves the eigenvalues of \(\gamma\) in a somewhat similar way to the real case. It is more complicated as one needs complex conjugates and an inner product over pairs of positive integers satisfying a congruence condition modulo the number of elements of finite order in the centralizer of \(\gamma\). The Weyl law for the asymptotics of the eigenvalues of the Laplacian on \(L^2(M)\) is obtained, while the authors note that there is no explicit example of a non-zero eigenvalue known for any \(\Gamma\) satisfying the hypotheses of the chapter. Next is the prime geodesic theorem, which says that the number of primitive hyperbolic or loxodromic conjugacy classes in \(\Gamma\) which have norm \(\leq x\) is asymptotic to \[ \frac {x^2}{2\log x}\qquad\text{as}\quad x\rightarrow \infty. \]

Chapter 6 has as its topic the case that the fundamental domain is non-compact but has finite volume. Once again the Selberg trace formula is considered. The attempt to obtain a Weyl law falls short without more explicit knowledge of the function \(\phi\), which is the determinant of the scattering matrix coming from Fourier expansions of the Eisenstein series at the cusps of the fundamental domain. There is an interesting discussion of the history of the question as to whether one should expect a Weyl law for all cofinite subgroups \(\Gamma\). The work of Phillips and Sarnak leads one to conjecture that at least in the 2-dimensional case when \(\Gamma\) is not co-compact or arithmetic, then there may be only finitely many members of the discrete spectrum of the Laplacian. But “Sarnak somewhat excludes the hyperbolic spaces of dimensions \(n\geq 3\).” Here the prime geodesic theorem at least goes as in Chapter 5, except the factor of 1/2 seems to be missing.

The special case that \(\Gamma = \text{SL}(2, O_K)\), for \(K\) an imaginary quadratic field, is considered in Chapter 7. It is shown, for example, that, for this \(\Gamma\), the smallest positive eigenvalue of \(-\Delta\) on \(L^2(\Gamma \backslash\mathbb H)\) is \(\geq 3/4\). Selberg’s conjecture would replace 3/4 with 1. Tables of eigenvalues for \(K\) of discriminant \(-4\) and \(-8\) are given.

Chapter 8 concerns Eisenstein series for \(\text{SL}(2, O_K)\). It is shown that the determinant of the scattering matrix is related to the zeta function of the Hilbert class field of \(K\). Weyl’s asymptotic law is proved for this sort of \(\Gamma\).

Chapter 9 begins with reduction theory for integral binary hermitian forms. While computing the volume of the fundamental domain, an interesting error of G. Humbert is corrected. The mass formula for binary hermitian forms is proved. This is a result in the tradition of C. L. Siegel’s work on quadratic forms. The more general result of Hel Braun differs from that of this book.

Chapter 10 provides some interesting examples of discontinuous groups on \(\mathbb H\). Some come from quaternion groups and others from unit groups of quadratic forms. Arithmetic and non-arithmetic groups are considered. Some very simple discontinuous groups generated by reflections in the faces of tesselating tetrahedra are found in Section 10.4. In the spirit of M.-F. Vignéras [C. R. Acad. Sci., Paris, Sér. A 287, 47–49 (1978; Zbl 0387.10013)], the Corollary on page 455 says that for any positive integer \(n\) there are \(n\) pairwise non-conjugate co-compact \(\Gamma _i\) in \(\text{Iso}(\mathbb H)\) which are all isospectral.

To summarize, this excellent book covers a huge amount of material and is highly recommended for anyone interested in this subject.

Reviewer: Audrey A. Terras (La Jolla)

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |

11M36 | Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) |

11F03 | Modular and automorphic functions |

11E39 | Bilinear and Hermitian forms |

11H55 | Quadratic forms (reduction theory, extreme forms, etc.) |

35P25 | Scattering theory for PDEs |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |