Geometric and arithmetic properties of Löbell polyhedra

Bogachev, Nikolay; Douba, Sami

doi:10.2140/agt.2025.25.2281

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Geometric and arithmetic properties of Löbell polyhedra

Nikolay Bogachev and Sami Douba

Algebraic & Geometric Topology 25 (2025) 2281–2295

DOI: 10.2140/agt.2025.25.2281

Abstract

The Löbell polyhedra form an infinite family of compact right-angled hyperbolic polyhedra in dimension $3$ . We observe, through both elementary and more conceptual means, that the “systoles” of the Löbell polyhedra approach $0$ , so that these polyhedra give rise to particularly straightforward examples of closed hyperbolic $3$ -manifolds with arbitrarily small systole, and constitute an infinite family even up to commensurability. By computing number-theoretic invariants of these polyhedra, we refine the latter result, and also determine precisely which of the Löbell polyhedra are quasi-arithmetic.

Keywords

geometry, low-dimensional topology, hyperbolic 3-manifolds, Coxeter groups

Mathematical Subject Classification

Primary: 57K32

Secondary: 22E40, 57M50

References

Publication

Received: 19 July 2023

Accepted: 10 March 2024

Published: 11 August 2025

Authors

Nikolay Bogachev
Department of Computer and Mathematical Sciences University of Toronto Scarborough Toronto, ON Canada	Institute for Information Transmission Problems Moscow Russia

Sami Douba
Institut des Hautes Études Scientifiques Bures-sur-Yvette France

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Volume 25, issue 4 (2025)