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!



Does the group $\langle a,b,c \mathbin | a^2b^2c^2=1\rangle $ have $\mathbb{Z} \times F_2$ as a subgroup?
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. 


