Efficient cycles of hyperbolic manifolds

TL;DR

This paper characterizes the uniqueness of efficient cycles in hyperbolic manifolds, showing that except for the figure-8 knot complement class, such cycles are unique in all other finite-volume hyperbolic manifolds.

Contribution

It proves that the figure-8 knot complement class is the only exception to the uniqueness of efficient cycles among finite-volume hyperbolic manifolds in dimensions three and higher.

Findings

Efficient cycles are unique for most hyperbolic manifolds.

The figure-8 knot complement class is the only non-unique case.

The result applies to all dimensions n ≥ 3.

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 $n\geq 3$, such cycle actually is the unique efficient cycle for a huge class of finite volume hyperbolic manifolds, including all the closed ones. In this paper we prove that, for $n\geq 3$, 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).