Mutations and short geodesics in hyperbolic 3-manifolds
release_ixuy2sedajb2lmkx3t3cmfux4e
by
Christian Millichap
2015
Abstract
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 commensurabiltiy 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 area surfaces could intersect short geodesics in a hyperbolic 3-manifold.
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