Mini-Workshop 2550c at Mathematisches Forschungsinstitut Oberwolfach
December 7 - 12, 2025
Classical hyperbolic geometry
Norbert Peyerimhoff, Durham
Hyperbolic volume
Ruth Kellerhals, Fribourg
Boolean models in hyperbolic space
If you are interested in the slides, write an e-mail to Daniel Hug (KIT)
Alesker, S., Fu, J.H.G.: Integral Geometry and Valuations
Fu uses Alesker’s theory of smooth valuations to derive the kinematic formulas in all Riemannian space forms, treating the negatively curved case via analytic continuation.
Anderson, J.W: Hyperbolic Geometry
A very gentle introduction into basics of hyperbolic geometry
Beardon, A.F.: Geometry of Discrete Groups
This book is an introduction to PSL(2,C) and its discrete subgroups and the two- and three-dimensional hyperbolic spaces on which they act as groups of isometries. The first three chapters are written in the more general context of Möbius transformations of hyperbolic n-space. The next three chapters treat complex matrices, discrete and discontinuous groups, Jorgensen's inequality and Riemann surfaces. A long Chapter Seven contains the analytical geometry of the hyperbolic plane and its isometries. Chapters Eight through Ten deal with Fuchsian groups, fundamental domains and Poincare's theorem, and finitely generated groups. Chapter Eleven discusses uniform discreteness and other universal inequalities for discrete groups.
Benedetti, R., Petronio, C.: Lectures on Hyperbolic Geometry
Bergeron, N.: The Spectrum of Hyperbolic Surfaces
This book investigates the Laplace spectrum of compact and finite area non-compact hyperbolic surfaces with special focus on the non-compact case, the Selberg Trace Formula, arithmetic surfaces, and the Jacquet Langlands Correspondence.
Bernig, A., Faifman, D., Solanes, G.: Crofton Formulas in pseudo-Riemannian space forms
The most recent and comprehensive treatment of smooth valuation theory and kinematic formulas in simply connected, complete pseudo-Riemannian space forms of constant curvature.
Böhm, J., Hertel, E.: Polyedergeometrie in Räumen konstanter Krümmung. Birkhäuser, 1981
This book gives a detailed treatment of the metrical aspects of the theory of n-dimensional polytopes (and n-orthoschemes) in Euclidean and non-Euclidean spaces. Chapter 1 gives a short introduction to the description of polytopes in the standard models on non-Euclidean geometries. Chapters 2 and 3 together give the theory of volume based on equivalences of several kinds. Chapter 4 gives a very detailed discussion of orthoschemes and the metric relations as well as of regular simplices.
Bonahon, F.: Low-Dimensional Geometry. From Euclidean Surfaces to Hyperbolic Knots
The book provides a smooth introduction to the idea of employing geometry to face matters of topology in low dimensions, starting from elementary facts in two dimensions and eventually approaching the deepest ones in three dimensions. The book assumes very little mathematical knowledge.
Buser, P.: Geometry and Spectra of compact Riemann surfaces
A beautiful geometrically motivated introduction into compact hyperbolic surfaces with focus on trigonometry, length spectra, the Laplace spectrum and relations between them.
Chavel, I.: Eigenvalues in Riemannian Geometry
A classic into relevant background from spectral geometry (also relevant for researchers in spectral aspects hyperbolic geometry)
Coxeter, H.S.M.: Twelve Geometric Essays
In this book one finds Coxeter's formula for spherical 3-orthoschemes in terms of the infinite series S(a,b,c) and the relation to the hyperbolic counterpart by multiplication with i.
Gray, A.: Tubes
Presents Weyl’s tube formula, the Steiner formula, and the Gauss–Bonnet theorem from the viewpoint of differential geometry.
Hejal, D.A.: The Selberg Trace Formula and the Riemann Zeta Function
An encyclopedic introduction into the Selberg Trace formula
Howard, R.: The Kinematic Formula in Riemannian Homogeneous Spaces
Uses the structure of homogeneous spaces as Lie group quotients to derive Weyl’s tube formula and the kinematic formulas.
Iversen, B.: Hyperbolic Geometry
This book is focuses on 2- and 3-dimensional real hyperbolic geometry. The treatment avoids differential geometry as far as possible, and instead defines key geometric concepts such as geodesics and isometries in the context of metric spaces. Riemannian concepts such as curvature are not introduced. The text instead emphasizes linear algebra. For example, in addition to the more common Klein and Poincaré models for H^2, the text focuses on the sl2(R) model, where sl2(R) with the Killing form is the underlying vector space of type (−2,1), and H^2 is taken to be a sheet of the locus of vectors with norm 1. The advantage of this model is a rich vector calculus.
Katok, S.: Fuchsian Groups
Introduction into Fuchsian groups and 2-dim hyperbolic geometry
Kohlmann, P.: Curvature measures and Steiner formulae in space forms
Provides a very general proof of the Steiner formula and the Gauss–Bonnet theorem in Riemannian space forms using Federer's tools from geometric measure theory.
Maclachlan, C., Reid, A.W.: The Arithmetic of Hyperbolic 3-Manifolds
This book fills a real void in the literature, providing an accessible introduction to the useful and beautiful world of arithmetic hyperbolic 3-manifolds and orbifolds. The list of references is extensive.
McKean, H.P.: Selberg's Trace Formula as Applied to a Compact Riemann Surface
The classical source to learn about the Selberg Trace formula in the special case of hyperbolic surfaces relating the Laplace spectrum and closed geodesics, the Selberg Zeta Function; it contains the wrong statement that compact hyperbolic surfaces do not have small eigenvalues
Milnor, J.: Collected Papers Vol. 1 Geometry
This volume contains geometrical papers covering a wide variety of topics Including hyperbolic volume), and includes several previously unpublished works. One paper provides a proof of Schläfli’s volume differential formula.
Otal, J.-P., Rosas, E.: Pour toute surface hyperbolique de genre g, lambda_{2g-2}>1/4
The proof of a longstanding conjecture that a genus g closed surface has at most 2g-2 small eigenvalues
Parker, J.R.: Hyperbolic Spaces
Survey over models of real hyperbolic spaces in 2 dimensions, 3 dimensions (using SL(2,C)), 4 dimensions (based on Quaternions) and beyond (based on Clifford algebras)
Ramsay, A., Richtmyer, R.D.: Introduction to Hyperbolic Geometry
Randol, B.: The Length Spectrum of a Riemann Surface is Always of Unbounded Multiplicity
This 2 page long paper proves that every hyperbolic surface has arbitrarily large families of non-isomorphic closed geodesics of the same length
Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds
The book is *the* standard book about hyperbolic spaces and manifolds. It is divided into three parts. The first part is concerned with hyperbolic geometry and discrete groups. The main results are the characterization of hyperbolic reflection groups and Euclidean crystallographic groups. The second part is devoted to the theory of hyperbolic manifolds. The main results are Mostow's rigidity theorem and the determination of the global geometry of hyperbolic manifolds of finite volume. The third part integrates the first two parts in a development of the theory of hyperbolic orbifolds. The main result is Poincaré's fundamental polyhedron theorem. The list of references is very extensive. The historical notes are valuable.
Rudenko, D.: On the Goncharov depth and a formula for the volumes of orthoschemes
This fairly long paper aims at proving a part of Goncharov's depth conjecture for multiple polylogarithms, which gives a criterion for when they can be expressed using polylogarithms of lower depths. As a byproduct, Rudenko gives a mysterious “formula” expressing the volume of a hyperbolic orthoscheme of dimension 2n−1 or 2n via multiple polylogarithms of weight n evaluated at algebraic functions in exponents of dihedral angles of the orthoscheme.
Santalo, L.A.: Integral Geometry and Geometric Probability
A classical and foundational reference on integral geometry, including extensive treatments of the spherical and hyperbolic settings.
Solanes, G.: Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces
Part of a series of works in which Solanes develops integral geometric formulas in hyperbolic spaces, following Santaló’s ideas and employing Cartan’s method of moving frames.
Vinberg, E.B., Shvartsman, O.V.: Geometry II
The book provides a systematic exposition of hyperbolic geometry, including hyperbolic volume, and the theory of discrete groups of isometries in Euclidean and hyperbolic spaces. The book makes the subject accessible from the viewpoints of both elementary geometry and group theory.