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Identify the compact surfaces $X$ for which there exist a proper subgroup $G$ of $\pi_1(X)$ such that $G\cong \pi_1(X)$.

EDIT: Suggestions?

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    one possibility is a torus2012-11-29
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    Sorry, but I would rather not identify such surfaces for you until you've shown where your own efforts broke down.2012-11-29
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    @ Neal: Yea sorry, I should have been more careful! I am not looking just for an answer at all, I want to know how to get to the answer. But I am having trouble figuring out how to tackle this question, so any hints would be appreciated?2012-11-29

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