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Let $\Sigma_g$ be a surface of genus $g \geq 2$. Let $H$ be a subgroup of $\pi_{1}(\Sigma_g)$ generated by homologically trivial loops in $\pi_1(\Sigma_g)$. What is known about this subgroup?

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    $\begingroup$ This is the kernel of the map $\pi_1(\Sigma_g)\to H_1(\Sigma;\mathbb{Z})$, and is therefore an infinite index subgroup which is infinitely generated and free. In fact, the induced covering space will have only one end, and will have infinite genus. Is there something more specific you're looking for? $\endgroup$
    – Ian Agol
    Jun 28, 2012 at 18:34

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It is the commutator subgroup of the surface group in question. Googling "commutator surface of surface group" or "universal abelian covering surface" will bring up a number of references -- abelian covering spaces have been particularly studies by people in dynamics (Babillot-Ledrappier comes to mind). A nice paper which comes up for the first search is Pollicott-Sharp's "Growth series...". If, as @Agol suggests, you ask a more specific question, you might get a more specific answer.

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