# GR group theory – quasi-homomorphism from integers to reals

Reference: Lemma 5.3 from "Groups That Affect Circles" by Etiyenne Ghys.

To let $$f: mathbb {Z} rightarrow mathbb {R}$$ be a quasi-homomorphism, i. $$| f (a + b) -f (a) -f (b) | leq D$$ $$forall$$ $$a$$ and $$b$$ in the $$mathbb {Z}$$ ($$mathbb {R}$$ and $$mathbb {Z}$$ are considered here as additive groups and so you see the plus sign). I have to prove that there is a unique number $$tau$$ $$epsilon$$ $$mathbb {R}$$ so that $$f (n) -n tau$$ is limited.

I have shown the part of the uniqueness separately and have found limits that work in the following limiting cases:
(i) If $$n$$ is positive, I've limited it up through $$f (0)$$ With $$tau = f (1) + D$$,
(ii) if $$n$$ is positive, I limited it down through $$f (0)$$ With $$tau = f (1) -D$$,
(iii) if $$n$$ is negative, I limited it up by $$f (0)$$ With $$tau = f (-1) + D$$,
(iv) If $$n$$ is negative, I limited it down through $$f (0)$$ With $$tau = f (-1) -D$$, I understand that I have always used $$f (0)$$ tying them, but that's just what I can see since I split $$n$$ as $$n$$ time the generator $$1$$ $$epsilon$$ $$mathbb {Z}$$, Please help me to bind this with only one of a kind $$tau$$,