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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.

Spotty
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  • What is your definition of T(A)? There are several competing definitions, they give real dimensions 0, 1 and $\infty$ respectively. – Moishe Kohan Jul 11 '15 at 17:33
  • space of structures (euclidean in this case) quotiented by diffeomorphism isotopic to the identity. – Spotty Jul 11 '15 at 22:49
  • Then, assuming your structures are complete, you get 1 dimensional space, parametrized by the length of the core loop. – Moishe Kohan Jul 12 '15 at 03:56

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