4 HOMEWORK HYPERBOLIC GEOMETRY
DANNY CALEGARI
This homework is due November 3rd in Kathy Paurs mailbox. There will be no class that day, since Ill be talking at a conference at Columbia. Recall that D usually denotes the Poincar disk model of hyperbolic space, and that D denotes the ideal boundary e otherwise known as the circle at innity. Problem 1. Let l1 , l2 be two intersecting lines, in the Klein disk model, and let S be the boundary circle. Let t1 , t2 be the two (Euclidean) tangents to S at the endpoints of l1 . Show that l1 , l2 are perpendicular if and only if the three lines l2 , t1 , t2 are either parallel, or intersect in a single point. (Here we think of l2 as the Eudlidean straight line extending the hyperolic straight line also called l2 ). Deduce the theorem in Euclidean geometry that t1 , t2 , l2 are parallel or coincident if and only if s1 , s2 , l1 are parallel or coincident, where s1 , s2 are the tangents to S at the endpoints of l2 . Hint: to nd the hyperbolic angle between l1 , l2 , move over to the Poincar disk model, e where apparent angles are equal to hyperbolic angles. Problem 2. For four distinct points v1 , v2 , v3 , v4 R , dene the crossratio to be (v4 v2 )(v3 v1 ) c(v1 , v2 , v3 , v4 ) = (v2 v1 )(v4 v3 ) If P SL(2, R) acting on R in the usual way, then show c(v1 , v2 , v3 , v3 ) = c((v1 ), (v2 ), (v3 ), (v4 )) Problem 3. With notation as in the previous problem, if v1 , v2 , v4 are in anticlockwise circular order, show there is a unique choice of P SL(2, R) with : {v1 , v2 , v4 } {0, 1, } (the bijection induced by should be the orderpreserving one). Show that c(v1 , v2 , v3 , v4 ) = (v3 ) Problem 4. In the upper halfspace model, show that the set of points P at constant hyperbolic distance t away from the hyperbolic straight line x = 0 consists of a pair of Euclidean straight lines. What is the relationship between the distance t and the angle between the lines in P and the line x = 0? Problem 5. Let K be the group of matrices of the form cos() sin() 0 sin() cos() 0 0 0 1 and A the group of matrices of the form cosh() 0 sinh() 0 1 0 sinh() 0 cosh()
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DANNY CALEGARI
Show that every element of SO(2, 1) can be expressed as k1 ak2 for some k1 , k2 K and a A; that is, we can write SO(2, 1) = KAK. How unique is such an expression? Problem 6. Let K be the subgroup of P SL(2, R) consisting of matrices of the form cos() sin() s 0 and A the subgroup of matrices of the form . Find an sin() cos() 0 s1 isomorphism from SO(2, 1) to P SL(2, R) taking K to K and A to A . (Careful! The isomorphism K K might not be the one you rst think of . . .) Problem 7. Recall the K subgroups and A dened in the previous question. Let N denote 1t the group of matrices of the form . Show that every element of P SL(2, R) can be 01 expressed as kan for some k K , a A and n N . How unique is this expression? This is an example of what is known as the KAN or Iwasawa decomposition. Problem 8. For a group G, the commutator subgroup G1 is the group generated by elements of the form [g, h] for g, h G (remember [g, h] = ghg 1 h1 ). We write G1 = [G, G]. Dene Gi inductively as the group generated by elements of the form [g, gi ] for g G and gi Gi1 . We write Gi = [G, Gi1 ]. A group G is nilpotent (of order n) if Gn is trivial, for some n. Show N as in the previous problem is nilpotent (this motivates the standard notation). Problem 9. We inherit notation from the previous problems. In the Poincar disk model, e show that the orbits of points under the group K are circles contained in the interior of D. Show that the orbits of points under the group A are arcs of circles (or straight lines) which intersect the boundary of H transversely. Show that the orbits of points under the group N are circles which intersect D tangentially. Use this to show that no element of one of these subgroups is a conjugate of an element of another one of these subgroups unless they are the identity elements. These third kind of circles are called horocircles or sometimes horocycles. Problem 10 (Hard). Let be a smooth curve embedded in R2 , bounding a region E. Dene a metric on E as follows: Let f be a smooth nowhere zero function on E which is equal 1 to dist(p,) for all p s...