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

Abstract

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 $\Gamma_n $ commensurable to L such that the systole of the locally symmetric finite volume manifold $\Gamma_n \diagdown SL(8,\mathbb{R}) \diagup SO(8)$ goes to infinity as $n \rightarrow \infty$, yet every $\Gamma_n$ contains the same hyperbolic 3-manifold group $\Pi$, 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.