Math 145

Topics in Topology, Kleinian Groups

Homework Assignments

Due DateAssignment
Wed, Sept 12Read Chapter 1

Definition: An isometry of E^2 is a map that preserves distance.

  1. Show that if an isometry of E^2 fixes three non-colinear points then it is the identity.
  2. Show that every isometry of E^2 is the composition of either 1, 2, or 3 reflections.
  3. Show that every isometry of E^2 is either a reflection, rotation, translation, or glide reflection.

You may want to add some additional statements in there as intermediate steps, but the goal is to prove 3 and 1 and 2 are meant as an outline to get you to 3. (I can imagine a nice statement between 1 and 2 for example) Try to write it up nicely please!

Mon, Sept 17 Read Chapter 2. And be prepared to start talking about it in class on Wed, Sept. 12. I would like the class to split up the following problems between you. Each of you just needs to work on 1, alone or in a team. See if you can start presenting these in class on Friday. It will spill over into Monday, and your write-ups can be due on Monday.

  1. Define stereographic projection. The goal is to show that stereographic projections takes circles to "circles" and preserves angles. The second circles is in quotes becasue in the complex plance we will consider lines to be circles (of infinite radius). This makes the statement of this and many other theorems nicer. The book suggests two proofs: Look in the book by Hilbert and Cohn-Vossen for a gemetric proof, or an analytic proof. See Note 2.6
  2. Show that the inversion in the unit circle given by z goes to the reciprical of its conjugate (Sorry, but I can't type math notation very well here) is circle and angle preserving. HINT: conjugate a very simple map of the sphere to itself with stereographic projection and use the earlier exercise.
  3. Project 2.6. We should definitely use stereographic projection here and the picture on page 88. Analyse the whole thing from BOTH points of view. Describe the group. How many elements, what are the orders of the elements, what subgroups are there, what are some nice sets of generators, what is the multiplication table, etc.
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