Nonuniform lattices of large systole containing a fixed 3-manifold group

Hillen, Paige

doi:10.2140/agt.2025.25.3089

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Nonuniform lattices of large systole containing a fixed $3$-manifold group

Paige Hillen

Algebraic & Geometric Topology 25 (2025) 3089–3102

DOI: 10.2140/agt.2025.25.3089

Abstract

Let $d \geq 2$ be a squarefree integer and $ℚ (\sqrt{d})$ 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 $ℚ (\sqrt{d})$ and a sequence of lattices $Λ_{n}$ commensurable to $ℒ$ such that the systole of the locally symmetric finite volume manifold $Λ_{n} ∖ SL (8, ℝ) ∕ SO (8)$ goes to infinity as $n \to \infty$ , yet every $Λ_{n}$ 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.

Keywords

lattice, arithmetic lattice, systole, systolic genus, hyperbolic manifold

Mathematical Subject Classification

Primary: 57R19, 22E40

References

Publication

Received: 22 March 2024

Revised: 29 June 2024

Accepted: 14 August 2024

Published: 17 September 2025

Authors

Paige Hillen
Department of Mathematics University of Wisconsin, Madison Madison, WI United States
https://sites.google.com/view/paigehillen

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Volume 25, issue 5 (2025)