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

Great circle links in the three-sphere

We investigate great circle links in the three-sphere, the class of links where each component is a great circle. Using the geometry of their complements, we classify such links up to five components. For any two-bridge knot complement, there is a finite cover that is the complement of a link of great circles in $S^3$. We show that for many two-bridge knots, this cover contains a closed incompressible surface. Infinitely many fillings of the two-bridge knot lift to fillings of great circle link where the incompressibility of this surface is preserved. Using this, we show that infinitely many fillings of an infinite class of two-bridge knot complements are virtually Haken.Comment: Dissertation for degree awarded June 17, 2003 from UC Davis, 50 pages, 18 figure

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

Double bubbles in the 3-torus

We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic 3-torus. For comparable small volumes, we prove that an area minimizing double bubble in the 3-torus is the standard double bubble from R^3.Comment: 13 pages, 4 figures. Prepared on behalf of the participants in the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces, held at the Mathematical Sciences Research Institute in Berkeley, California, Summer 200