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Full-Text Articles in Physical Sciences and Mathematics
Spectrally Similar Incommensurable 3-Manifolds, David Futer, Christian Millichap
Spectrally Similar Incommensurable 3-Manifolds, David Futer, Christian Millichap
Faculty Publications
Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3–manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for every n ≫ 0, we construct a pair of incommensurable hyperbolic 3–manifolds Nn and Nµn whose volume is approximately n and whose length spectra agree up to length n.
Both Nn and Nµn are built by gluing two standard submanifolds along a complicated pseudo-Anosov map, ensuring that …
Mutations And Short Geodesics In Hyperbolic 3-Manifolds, Christian Millichap
Mutations And Short Geodesics In Hyperbolic 3-Manifolds, Christian Millichap
Faculty Publications
In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurability classes by analyzing their cusp shapes.
The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial (complex) length spectrum. This work requires us to analyze when least …