This book, "Geometry and Spectra of Compact Riemann Surfaces" by Peter Buser, is a comprehensive exploration of two key areas in mathematics: the geometric theory of compact Riemann surfaces of genus greater than one and the Laplace operator and its relationship with the geometry of these surfaces. It is a reissue of the 1992 edition, published by Birkhäuser, and is part of the Progress in Mathematics series. The book is intended for students, scholars, and researchers, providing a detailed and accessible introduction to the subject. The first part of the book focuses on the geometry of compact Riemann surfaces, using hyperbolic geometry and the technique of cutting and pasting. It covers topics such as hyperbolic structures, trigonometry, and the construction of compact Riemann surfaces through the pasting of geodesic polygons. The second part of the book delves into the spectrum of the Laplace operator, exploring how the geometry of a compact Riemann surface is reflected in the spectrum of the Laplacian. It discusses the relationship between the spectrum and geometric properties, including the use of the heat kernel and isoperimetric techniques. The book includes a detailed discussion of inverse spectral problems, where the goal is to determine the geometric properties of a surface from its spectrum. It presents results such as Huber's theorem, which states that two compact Riemann surfaces have the same eigenvalues of the Laplace operator if and only if they have the same lengths of closed geodesics. The book also covers the construction of isospectral manifolds and provides examples of isospectral Riemann surfaces. The text is well-structured, with each chapter containing introductory remarks and historical context. It includes appendices that provide additional information on topics such as curves and isotopies, and the theorems of Baer-Epstein-Zieschang. The book is supported by a bibliography and an index, making it a valuable resource for those interested in the geometry and spectra of compact Riemann surfaces.This book, "Geometry and Spectra of Compact Riemann Surfaces" by Peter Buser, is a comprehensive exploration of two key areas in mathematics: the geometric theory of compact Riemann surfaces of genus greater than one and the Laplace operator and its relationship with the geometry of these surfaces. It is a reissue of the 1992 edition, published by Birkhäuser, and is part of the Progress in Mathematics series. The book is intended for students, scholars, and researchers, providing a detailed and accessible introduction to the subject. The first part of the book focuses on the geometry of compact Riemann surfaces, using hyperbolic geometry and the technique of cutting and pasting. It covers topics such as hyperbolic structures, trigonometry, and the construction of compact Riemann surfaces through the pasting of geodesic polygons. The second part of the book delves into the spectrum of the Laplace operator, exploring how the geometry of a compact Riemann surface is reflected in the spectrum of the Laplacian. It discusses the relationship between the spectrum and geometric properties, including the use of the heat kernel and isoperimetric techniques. The book includes a detailed discussion of inverse spectral problems, where the goal is to determine the geometric properties of a surface from its spectrum. It presents results such as Huber's theorem, which states that two compact Riemann surfaces have the same eigenvalues of the Laplace operator if and only if they have the same lengths of closed geodesics. The book also covers the construction of isospectral manifolds and provides examples of isospectral Riemann surfaces. The text is well-structured, with each chapter containing introductory remarks and historical context. It includes appendices that provide additional information on topics such as curves and isotopies, and the theorems of Baer-Epstein-Zieschang. The book is supported by a bibliography and an index, making it a valuable resource for those interested in the geometry and spectra of compact Riemann surfaces.