Point in the plane that is not fix

Question

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?

Answer 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.