Non-Uniform Lattices of Large Systole Containing a Fixed 3-Manifold Group

Let $d$ be a square free positive integer and $\mathbb{Q}(\sqrt{d})$ a totally real quadratic field over $\mathbb{Q}$. We show there exists an arithmetic lattice L in $SL(8,\mathbb{R})$ with entries in the ring of integers of $\mathbb{Q}(\sqrt{d})$ and a sequence of lattices ${\mathrm{\Gamma }}_n$ commensurable to L such that the systole of the locally symmetric finite volume manifold ${\mathrm{\Gamma }}_n\mathrm{╲}SL(8,\mathbb{R})\mathrm{╱}SO(8)$ goes to infinity as $n\to \mathrm{\infty }$, yet every ${\mathrm{\Gamma }}_n$ contains the same hyperbolic 3-manifold group $\mathrm{\Pi }$, a finite index subgroup of the arithmetic hyperbolic 3-manifold vol3. Notably, such an example does not exist in rank one, so this is a feature unique to higher rank lattices.