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

2024/03/21 by Paige Hillen, Hillen, Paige
Engineering · #Advanced Numerical Analysis Techniques #FOS: Mathematics #Geometric Topology (math.GT)

paper · pdf · doi:10.48550/arxiv.2403.14081

openalex publication_date 2024/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let d be a square free positive integer and ℚ(√(d)) a totally real quadratic field over ℚ. We show there exists an arithmetic lattice L in SL(8,ℝ) with entries in the ring of integers of ℚ(√(d)) and a sequence of lattices Γ_n commensurable to L such that the systole of the locally symmetric finite volume manifold Γ_n \diagdown SL(8,ℝ) \diagup SO(8) goes to infinity as n → ∞, yet every Γ_n contains the same hyperbolic 3-manifold group Π, 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.

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