Let $F_2$ denote the free group of rank two and consider the group $G=\langle a,b,c \mathbin | a^2b^2c^2=1\rangle$ which is the fundamental group of the connected sum of three projective planes. Does $G$ have $\mathbb{Z} \times F_2$ as a subgroup? Thanks!
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1 Answer
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The answer is `no'. No hyperbolic group contains a copy of $\mathbb{Z}^2$. To give some more details, the action of $\Gamma=\pi_1(3\mathbb{R}P^2)$ on the hyperbolic plane is free, discrete, and every element acts loxodromically. Commuting elements must have a common axis, but $\mathbb{Z}^2$ cannot act freely and discretely on the real line.
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$\begingroup$ Hey Henry, thanks for the answer. Does $F_2 \times \mathbb{Z}$ have $\mathbb{Z}^2$ as a subgroup? $\endgroup$– danCommented Feb 11, 2011 at 19:32
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4$\begingroup$ @dan Yes, of course, because $F_2$ has $\mathbb{Z}$ as subgroup. $\endgroup$ Commented Feb 11, 2011 at 19:49