Open Access. Powered by Scholars. Published by Universities.®
- Institution
Articles 1 - 5 of 5
Full-Text Articles in 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 …
Hidden Symmetries And Commensurability Of 2-Bridge Link Complements, Christian Millichap, William Worden
Hidden Symmetries And Commensurability Of 2-Bridge Link Complements, Christian Millichap, William Worden
Faculty Publications
In this paper, we show that any nonarithmetic hyperbolic 2-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic 2-bridge link complement cannot irregularly cover a hyperbolic 3-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a characterization of 3-manifolds with nontrivial JSJ-decomposition and rank-two fundamental groups. We also show that the only commensurable hyperbolic 2-bridge link complements are the figure-eight knot complement and the 622 link complement. Our work requires a careful analysis of the tilings of R2 that come from lifting the canonical triangulations of …
Growth Conditions For Uniqueness Of Smooth Positive Solutions To An Elliptic Model, Joon Hyuk Kang
Growth Conditions For Uniqueness Of Smooth Positive Solutions To An Elliptic Model, Joon Hyuk Kang
Faculty Publications
The uniqueness of positive solution to the elliptic model
∆u + u[a + g(u, v)] = 0 in Ω, ∆v + v[a + h(u, v)] = 0 in Ω, u = v = 0 on ∂Ω,
were investigated.
Factorial Growth Rates For The Number Of Hyperbolic 3-Manifolds Of A Given Volume, Christian Millichap
Factorial Growth Rates For The Number Of Hyperbolic 3-Manifolds Of A Given Volume, Christian Millichap
Faculty Publications
The work of Jørgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. In this paper, we construct examples showing that the number of hyperbolic knot complements with a given volume v can grow at least factorially fast with v. A similar statement holds for closed hyperbolic 3-manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of N(v) in terms of v for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with v …