[Binghamton Geometry Topology Seminar] This week: Edgar Bering

[sapir at math.binghamton.edu]

This week we will meet as usual on Thursday, November 7 at 2:50pm in
WH 100E.

We will go to lunch with the speaker at noon on Thursday (meet by my
office, WH213) and dinner after the talk at a location that is to be
determined.

Speaker: Edgar Bering (Temple University)
Title: Special covers of alternating links

Abstract: The “virtual conjectures” in low-dimensional topology, stated by
Thurston in 1982, postulated that every hyperbolic 3-manifold has finite
covers that are Haken and fibered, with large Betti numbers. These
conjectures were resolved in 2012 by Agol and Wise, using the machine of
special cube complexes. Since that time, many mathematicians have asked
how big a cover one needs to take to ensure one of these desired
properties.

We begin to give a quantitative answer to this question, in the setting of
alternating links in S3. If an alternating link L has a diagram with n
crossings, we prove that the complement of L has a special cover of degree
less than 72((n−1)!)2. As a corollary, we bound the degree of the
cover required to get Betti number at least k. We also quantify residual
finiteness, bounding the degree of a cover where a closed curve of length
k fails to lift. This is joint work with David Futer.