Consider an m Mb (H) that is nontrivial. Use the classification of isometries in Mb(C) to...

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Consider an m Mb (H) that is nontrivial. Use the classification of isometries in Mb(C) to prove the following. (a) m is parabolic if, and only if m has one fixed point in R. Further- more, m is conjugate in Mb(H) to the map q(z) = z + 1. (b) m is elliptic if, and only if m has one fixed point in H. Further- more, m is conjugate in Mb* (H) to a rotation by 0 (i.e a map of the form cos sin sin 5), for some 8 R). cos (c) m is loxodromic if, and only if m has two fixed points in R. Fur- thermore, m is conjugate in Mb (H) to the map q(z) = kz, for some k > 0. (d) For each of the three types of isometries, derive conditions equiv- alent to those given in 2(a) - (c) in terms of Trace(m). Consider an m Mb (H) that is nontrivial. Use the classification of isometries in Mb(C) to prove the following. (a) m is parabolic if, and only if m has one fixed point in R. Further- more, m is conjugate in Mb(H) to the map q(z) = z + 1. (b) m is elliptic if, and only if m has one fixed point in H. Further- more, m is conjugate in Mb* (H) to a rotation by 0 (i.e a map of the form cos sin sin 5), for some 8 R). cos (c) m is loxodromic if, and only if m has two fixed points in R. Fur- thermore, m is conjugate in Mb (H) to the map q(z) = kz, for some k > 0. (d) For each of the three types of isometries, derive conditions equiv- alent to those given in 2(a) - (c) in terms of Trace(m).

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