, Show that SL(2, C) acts on H 1 by isometries via M · A = M A M * . What is the stabilizer of I 2 ? Recover that Isom + (H 3 ) ? PSL(2, C) and H 3 ? PSL

, Hyperbolic subspace Propose a definition of a hyperbolic subspace of a hyperbolic space X = H n , and describe the hyperbolic subspaces in all the different models of H n

, Going back to H 3 , write a different proof using matrices. Prove in fact that loxodromic elements are dense in Isom

, A baby character variety Let us work in the Poincaré half-space model H ? C of the hyperbolic plane H 2 . We denote G = Isom + (H) the group of orientation-preserving isometries

, Show that f 0 = id ? G is in the closure of the conjugacy class C ? Isom + (H) of some/any parabolic isometry

, Let G act on itself by conjugation. Derive from the previous question that the quotient R is not Hausdorff

, We recall that an element of G is called semisimple (or completely reducible, or polystable, depending on context) if it is not parabolic. Let X ? R denote the subset of conjugacy classes of semisimple elements

, Exercise 10.8. Trace relations We let G = SL(2, C) in this exercise

, Show that for any A, B ? G, tr(AB) + tr(AB ?1 ) = tr A tr B

, Show that the trace of any element of the subgroup of G generated by A and B can be expressed as a polynomial in tr A, tr B, and tr AB with integer coefficients

. Optional, Show that any polynomial function of (A, B) ? G×G that is invariant by conjugation (that is, invariant by (A, B) ? (g Ag ?1 , gBg ? 1 ) for all g ? G) can be expressed as a polynomial function of tr A, tr B, and tr AB

, Classification in O + (n, 1) Recall that Isom(H n ) ? O + (n, 1), e.g. via the hyperboloid model. Using linear algebra, find a characterization of elliptic, loxodromic

, Exercise 11.5. Area of hyperbolic polygons How would you define a hyperbolic polygon? Find a formula for the area of any hyperbolic polygon

, Gromov hyperbolicity of hyperbolic space Let n 2. The goal of this exercise is to show that hyperbolic space H n is Gromov hyperbolic

, Consider the ideal triangle with vertices A = 0, B = ?, and C = 1 in the Poincaré half-plane

, Show that the distance from p to (BC) is achieved at p = (1, 1 + y 2 )

A. , B. , and C. ,

, Conclude that d(p, (BC) ? (C A)) ? where ? = arsinh(1) and conclude the exercise

N. Aydin and L. Hammoudi, Al-K?sh?'s Mift?h . al-H . isab, p.161, 2019.

L. V. Ahlfors, An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics, p.98, 1978.

A. D. Aleksandrov, A theorem on triangles in a metric space and some of its applications, In Trudy Mat. Inst. Steklov, vol.38, p.120, 1951.

A. Norbert-a'campo and . Papadopoulos, On Klein's so-called non-euclidean geometry, Sophus Lie and Felix Klein: the Erlangen program and its impact in mathematics and physics, vol.23, p.65, 2015.

D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol.33, p.118, 2001.

A. F. Beardon, Corrected reprint of the 1983 original, Graduate Texts in Mathematics, vol.91, p.89, 1995.

E. Beltrami, Saggio di interpretazione della geometria Non-Euclidea

E. Beltrami, Teoria fondamentale degli spazii di curvatura costante, vol.2, pp.232-255, 1868.

R. Martin, A. Bridson, and . Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, vol.319

, Mathematical Sciences

. Springer-verlag, , vol.118, p.154, 1999.

A. Beutelspacher and U. Rosenbaum, Projective geometry: from foundations to applications, p.50, 1998.

A. Beutelspacher and U. Rosenbaum, Projektive Geometrie. Vieweg Studium: Aufbaukurs Mathematik. [Vieweg Studies: Mathematics Course

. Friedr and W. Vieweg-&-sohn, Von den Grundlagen bis zu den Anwendungen, p.50, 2004.

D. Calegari, Blog post on the math web blog Geometry and the imagination, p.89, 2013.

É. Cartan, Leçons sur la géométrie des espaces de Riemann. Les Grands Classiques Gauthier-Villars

J. Éditions and . Gabay, , p.138, 1988.

A. Cayley, A sixth memoir upon quantics, Philos. Trans. Roy. Soc. London Ser. A, vol.149, p.65, 1859.

M. Coornaert, T. Delzant, and A. Papadopoulos, Les groupes hyperboliques de Gromov, Lecture Notes in Mathematics, vol.1441, p.118, 1990.
URL : https://hal.archives-ouvertes.fr/hal-00126300

J. W. Cannon, W. J. Floyd, R. Kenyon, and W. R. Parry, Hyperbolic geometry, Flavors of geometry, vol.31, p.2, 1997.

, Regularity of conformal maps, Dap, p.89

T. Das, D. Simmons, and M. Urba?ski, Geometry and dynamics in Gromov hyperbolic metric spaces, Mathematical Surveys and Monographs, vol.218, p.136, 2017.

B. Duchesne, Groups acting on spaces of non-positive curvature, Handbook of group actions, vol.III, p.120, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01288346

N. V. Efimov, Generation of singularites on surfaces of negative curvature, Mat. Sb, vol.64, issue.106, p.26, 1964.

E. , I: Introduction and Books I, II, The thirteen books of Euclid's Elements translated from the text of Heiberg, vol.II, 1956.

F. Fillastre and A. Seppi, Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions, Eighteen essays in non-Euclidean geometry, vol.29, p.75, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01675487

É. Ghys and P. Harpe, Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Mathematics. Birkhäuser Boston, vol.83, p.136, 1990.

É. Ghys, Poincaré and his disk, The scientific legacy of Poincaré, vol.36, p.7, 2010.

J. Gaster, B. Loustau, and L. Monsaingeon, Computing harmonic maps between riemannian manifolds, p.22, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02320952

M. Greenberg, Euclidean and non-Euclidean geometries, p.7, 1993.

M. Gromov, Hyperbolic groups, Essays in group theory, vol.8, p.119, 1987.

F. J. Herranz, R. Ortega, and M. Santander, Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry, J. Phys. A, vol.33, issue.24, p.75, 2000.

T. Iwaniec and G. Martin, Geometric function theory and non-linear analysis, Oxford Mathematical Monographs. The Clarendon Press, p.89, 2001.

V. Kapovitch, Conformally-flat. MathOverflow, p.114

F. Klein, Ueber die sogenannte nicht-euklidische geometrie, Mathematische Annalen, vol.4, issue.4, p.65, 1871.

F. Klein, Ueber die sogenannte nicht-euklidische geometrie, Mathematische Annalen, vol.6, issue.2, p.65, 1873.

J. M. Lee, Introduction to Riemannian manifolds, Graduate Texts in Mathematics, vol.176, p.41, 2018.

F. Matt, What is needed to prove the consistency of tarski's euclidean geometry? MathOverflow, p.13, 2017.

T. Klotz-milnor, Efimov's theorem about complete immersed surfaces of negative curvature, Advances in Math, vol.8, p.26, 1972.

. Mvg, Cosine law duality in hyperbolic trigonometry. Mathematics Stack Exchange, p.163

R. Nelson, Isometry classes of hyperbolic 3-space. Website, p.151, 2020.

A. Papadopoulos, Metric spaces, convexity and non-positive curvature, IRMA Lectures in Mathematics and Theoretical Physics, vol.6, p.82, 2014.

H. Poincare, 1882. First article in a series published in the same volume, Acta Math, vol.1, issue.1, pp.11-104

J. G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol.149, p.2, 2006.

W. F. Reynolds, Hyperbolic geometry on a hyperboloid, Amer. Math. Monthly, vol.100, issue.5, p.38, 1993.

J. Richter-gebert, A guided tour through real and complex geometry, p.75, 2011.

B. R. Bernhard-riemann, Über die Hypothesen, welche der Geometrie zu Grunde liegen". Klassische Texte der Wissenschaft

S. Spektrum, Historical and mathematical commentary by Jürgen Jost, p.22, 2013.

W. P. Thurston, Geometry and topology of 3-manifolds, p.3, 1980.

W. P. Thurston, Three-dimensional geometry and topology, vol.1, p.177, 1997.

, Wikipedia contributors. Cross-ratio -Wikipedia, the free encyclopedia, vol.10, p.58, 2019.