41 research outputs found
Orbifolds and commensurability
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.Comment: 11 page
Great circle links and virtually fibered knots
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.Comment: Minor changes. To appear in Topology. 12 pages, 7 figure
The big Dehn surgery graph and the link of S^3
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.Comment: 15 pages, 4 figures, 4 ancillary files. Reorganized and shortened
from previous versions, while correcting one error in the proof of Theorem
5.4. Also, ancillary files detailing our computations with the computer
program ORB have been provide
The automorphism group of the free group of rank two is a CAT(0) group
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.Comment: 7 pages, 2 figures. The manuscript has been modified in minor ways in
accordance with a referee's recommendations, and a misattribution of the
result "Aut F_2 is biautomatic" has been correcte
Three-manifolds, virtual homology, and group determinants
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.Comment: This is the version published by Geometry & Topology on 29 November
200
Virtually Haken fillings and semi-bundles
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.Comment: This is the version published by Geometry & Topology on 29 November
200