Teichmuller/Moduli Space of some surface

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I am going through Thurston's book and have just started Teichmuller Theory.

I have computed the teichmuller and moduli space of the pair of pants and I am now trying to do the one for the open annulus.

I am trying to follow the same approach as for the pair of pants, i.e. cutting it along some geodesic to get a region bounded by two disjoint geodesics but those are all isometric hence the dimension of $\mathcal T(A)$ is $1$, is this correct?

If instead of the open annulus I take the open mobius band can I do the same thing?

Thanks in advance.