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Let $E_{ij}(x)\in Mat_{7\times7}(\overline{\mathbb{F}}_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x)$, $b(y)=1+E_{23}(y)+E_{45}(y)+E_{67}(y)$, for two indeterminants $x,y$. Set $G_8=<a(x), b(y) | x, y\in\overline{\mathbb{F}}_2>$. Is $G_8$ connected?

For easier visualization, here are the matrices:

$$ a(x)=\begin{pmatrix} 1&x& & & & & \\ &1&0& & & & \\ & &1&x& & & \\ & & &1&0& & \\ & & & &1&x& \\ & & & & &1&0\\ & & & & & &1 \end{pmatrix} ,\ b(y)=\begin{pmatrix} 1&0& & & & & \\ &1&y& & & & \\ & &1&0& & & \\ & & &1&y& & \\ & & & &1&0& \\ & & & & &1&y\\ & & & & & &1 \end{pmatrix} $$

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