So I have this question which is somewhat directed to people knowing a bit about translation surfaces. I am sure it is only a technical issue.

I consider $f : (\Sigma, \omega) \longrightarrow (\Sigma, \omega)$ an affine automorphism of $(\Sigma, \omega)$, that is who writes down as an element of $\mathrm{SL}(2, \mathbb{R}) \ltimes \mathbb{R}^2$ in coordinates charts. I assume that $f$ is not the identity. Is $f$ necessarily non-trivial in $\mathrm{MCG}(\Sigma)$ the mapping class group of $\Sigma$?

The question is actually can $f$ be isotopic to the identity? The natural way to tackle the question seems to me to look at the action of $f$ on the saddle connections. This way one easily proves that if $f$ (or a suitable power that fixes the singular points) is isotopic to the identity *relatively to the set of singular points* then it must be the identity (because the image of any saddle connection must be itself).