Efficient cycles of hyperbolic manifolds

Roberto Frigerio, Ennio Grammatica and Bruno Martelli

Vol. 332 (2024), No. 1, 115–145

DOI: 10.2140/pjm.2024.332.115

Abstract

Let $N$ be a complete finite-volume hyperbolic $n$-manifold. An efficient cycle for $N$ is the limit (in an appropriate measure space) of a sequence of fundamental cycles whose ${\ell }^1$-norm converges to the simplicial volume of $N$. Gromov and Thurston’s smearing construction exhibits an explicit efficient cycle, and Jungreis and Kuessner proved that, in dimension $n\ge 3$, such a cycle actually is the unique efficient cycle for a huge class of finite-volume hyperbolic manifolds, including all the closed ones. We prove that, for $n\ge 3$, the class of finite-volume hyperbolic manifolds for which the uniqueness of the efficient cycle does not hold is exactly the commensurability class of the figure-8 knot complement (or, equivalently, of the Gieseking manifold).

Keywords

simplicial volume, complexity, stable complexity, Gieseking manifold, Gieseking-like manifold, measure homology

Mathematical Subject Classification

Primary: 37D40, 57K32, 57N65

Milestones

Received: 2 November 2023

Revised: 28 August 2024

Accepted: 18 October 2024

Published: 20 November 2024