, 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

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