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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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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