If $G$ is a free group on $n>1$ generators, then $[G,G]$ is also free, being a subgroup of a free group. Is there a formula for the number of generators of this free group in terms of n?
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8$\begingroup$ Yes, it's infinite. $\endgroup$– Ryan BudneyCommented Aug 18, 2012 at 22:38
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1$\begingroup$ It's infinite. One way to see this is to write down the corresponding covering space of the wedge of $n$ circles and note that it is homotopy equivalent to a wedge of infinitely many circles. $\endgroup$– Qiaochu YuanCommented Aug 18, 2012 at 22:48
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1$\begingroup$ See mathoverflow.net/questions/74902/… $\endgroup$– Benjamin SteinbergCommented Aug 18, 2012 at 23:05
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