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Search Results: 1 - 10 of 326430 matches for " Genevieve S. Walsh "
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Great circle links and virtually fibered knots
Genevieve S. Walsh
Mathematics , 2004,
Abstract: We show that all two-bridge knot and link complements are virtually fibered. We also show that spherical Montesinos knot and link complements are virtually fibered. This is accomplished by showing that such knot complements are finitely covered by great circle link complements.
The bumping set and the characteristic submanifold
Genevieve S. Walsh
Mathematics , 2012, DOI: 10.2140/agt.2014.14.283
Abstract: We show here that the Nielsen core of the bumping set of the domain of discontinuity of a Kleinian group $\Gamma$ is the boundary of the characteristic submanifold of the associated 3-manifold with boundary. Some examples of interesting characteristic submanifolds are given. We also give a construction of the characteristic submanifold directly from the Nielsen core of the bumping set. The proofs are from "first principles", using properties of uniform domains and the fact that quasi-conformal discs are uniform domains.
Incompressible surfaces and spunnormal form
Genevieve S. Walsh
Mathematics , 2005,
Abstract: Suppose M is a cusped finite-volume hyperbolic 3-manifold and T is an ideal triangulation of M with essential edges. We show that any incompressible surface S in M that is not a virtual fiber can be isotoped into spunnormal form in T . The proof is based directly on ideas of W. Thurston.
Orbifolds and commensurability
Genevieve S. Walsh
Mathematics , 2010,
Abstract: These are notes based on a series of talks that the author gave at the "Interactions between hyperbolic geometry and quantum groups" conference held at Columbia University in June of 2009.
Virtually Haken fillings and semi-bundles
Daryl Cooper,Genevieve S Walsh
Mathematics , 2004, DOI: 10.2140/gt.2006.10.2237
Abstract: Suppose that M is a fibered three-manifold whose fiber is a surface of positive genus with one boundary component. Assume that M is not a semi-bundle. We show that infinitely many fillings of M along dM are virtually Haken. It follows that infinitely many Dehn-surgeries of any non-trivial knot in the three-sphere are virtually Haken.
Three-manifolds, virtual homology, and group determinants
Daryl Cooper,Genevieve S Walsh
Mathematics , 2006, DOI: 10.2140/gt.2006.10.2247
Abstract: We apply representation theory to study the homology of equivariant Dehn-fillings of a given finite, regular cover of a compact 3-manifold with boundary a torus. This yields a polynomial which gives the rank of the part of the homology carried by the solid tori used for Dehn-filling. The polynomial is a symmetrized form of the group determinant studied by Frobenius and Dedekind. As a corollary every such hyperbolic 3-manifold has infinitely many virtually Haken Dehn-fillings.
The big Dehn surgery graph and the link of S^3
Neil R. Hoffman,Genevieve S. Walsh
Mathematics , 2013,
Abstract: In a talk at the Cornell Topology Festival in 2005, W. Thurston discussed a graph which we call "The Big Dehn Surgery Graph", B. Here we explore this graph, particularly the link of S^3, and prove facts about the geometry and topology of B. We also investigate some interesting subgraphs and pose what we believe are important questions about B.
Coxeter groups, hyperbolic cubes, and acute triangulations
Sang-hyun Kim,Genevieve S. Walsh
Mathematics , 2013,
Abstract: Let $C(L)$ be the right-angled Coxeter group defined by an abstract triangulation $L$ of $\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a hyperbolic right-angled reflection group if and only if $L$ can be realized as an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A corollary is that an abstract triangulation of $\mathbb{S}^2$ can be realized as an acute triangulation exactly when it satisfies a combinatorial condition called "flag no-square". We also study generalizations of this result to other angle bounds, other planar surfaces and other dimensions.
The automorphism group of the free group of rank two is a CAT(0) group
Adam Piggott,Kim Ruane,Genevieve S. Walsh
Mathematics , 2008,
Abstract: We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.
Commensurability classes of 2-bridge knot complements
Alan W. Reid,Genevieve S. Walsh
Mathematics , 2006,
Abstract: We show that a hyperbolic 2-bridge knot complement is the unique knot complement in its commensurability class. We also discuss constructions of commensurable hyperbolic knot complements and put forth a conjecture on the number of hyperbolic knot complements in a commensurability class.
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