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 l1-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 ≥ 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 ≥ 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).
Efficient cycles of hyperbolic manifolds
Roberto Frigerio;Bruno Martelli
2024-01-01
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 l1-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 ≥ 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 ≥ 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).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
