0

Let $G$ be a discrete subgroup of $M$ group of isometries of the plane whose translation group is not trivial. Prove that there is a point $p_0$ in the plane that is not fixed by any element of $G$ except the identity.

Could someone give me a hint on how I might start?

amir
  • 1,311

1 Answers1

1

Hints:

  1. How many fixed points does one single isometry have? So what kind of set do you get when you take a union of these fixed point sets over all the elements of $g$?
  2. The fact that $G$ has a non-trivial translation is not relevant.
Prahlad Vaidyanathan
  • 31,554
  • I think that each isometry has at least one fixed point. But I can't really see what the union is.. – amir Oct 22 '13 at 13:35