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The original question was askes here.

I donot know how to apply or compute any example. I think a specified explanation will be helpful.

Let $M=\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and $f=A\in\mathrm{SL}(2,\mathbb{Z})$ be the quotient action on $\mathbb{T}^2$ induced from $f(x)=Ax$. Since $A$ preserves the lattice $\mathbb{Z}^2$, $f$ is well defined.

We know that $\pi_1(\mathbb{T}^2)=\langle\alpha,\beta\rangle\cong\mathbb{Z}^2$. What can we see about $\pi_1(M_f)$?

HJRW answered there that $\pi_1(M_f)\cong\pi_1(X)\rtimes_{f_*}\mathbb{Z}$. Would you write down explicitly the multiplication of this semiproduct in this example?

Thanks!

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    It is exactly as you expect: if $t$ generates your copy of $\mathbb{Z}$, then $t^{-1}\alpha t = A\alpha$, etc.2011-10-11

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