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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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    @ 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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Here's another way to rephrase the question. Every subgroup of $\pi_1(M)$ corresponds to a connected cover of $M$. Further, $\pi_1(M)$ characterizes closed surfaces, in the sense that if two closed surfaces have isomorphic fundamental groups, then they are diffeomorphic. (This is something very special to closed surfaces!)

So, another way to recast your question is the following: Which closed surfaces cover themselves in a nontrivial way? (Nontrivial means more than 1-sheeted).

As a further hint: study the Euler Characteristic.

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    By $M\cong N$ I meant that $M$ and $N$ are diffeomorphic. You're right about the Euler characteristic - try to come up wit ha strong argument ;-). Incidentally, you must still prove that the two surfaces of Euler characteristic $0$ (the torus and Klein bottle) actually have such nontrivial coverings.2012-11-29