Geometry/Topology Seminar
Spring 2020
Thursdays (and sometimes Tuesdays) 3:404:30pm, in
Ryerson 358 (currently on zoom)

 Tuesday May 05 at 3:404:30pm in ZOOM
 Kevin Schreve, University of Chicago
 Mod p and torsion homology growth in nonpositive curvature

Abstract: We compute the growth of mod p homology in
finite index normal subgroups of of rightangled Artin
groups. We give examples where it differs from the rational
homology growth, and for odd primes p construct
closed locally CAT(0) manifolds with nonzero mod
p L^{2}Betti numbers outside
the middle dimension.

 Thursday May 07 at 3:004:00pm in ZOOM
 Genevieve Walsh, Tufts University
 Incoherence of freebyfree and surfacebyfree groups.

Abstract: A group is said to be coherent if every
finitely generated subgroup is finitely presented. This
property is enjoyed by free groups, and the fundamental
groups of surfaces and 3manifolds. A group that is not
coherent is incoherent, and it is very interesting to try
and understand which groups are coherent. We will discuss
some of the geometric and topological aspects of this
question, particularly quasiconvexity and algebraic fibers.
We show that freebyfree and surfacebyfree groups are
incoherent, when the rank and genus are at least 2. The
proof uses an understanding of fibers and also the RFRS
property. This is joint work with Robert Kropholler and
Stefano Vidussi.

 Thursday May 14 at 2:003:00pm in ZOOM
 Matthew Stover, Temple University
 Superrigidity and arithmeticity in rank one

Abstract: Margulis famously proved superrigidity of
irreducible higher rank lattices, then deduced they are
arithmetic. Corlette and GromovSchoen later pushed this to
quaternionic and Cayley hyperbolic lattices. However,
superrigidity fails dramatically for many classes of real
and complex hyperbolic lattices. I will talk about work with
Uri Bader, David Fisher, and Nicholas Miller proving that
representations of real and complex hyperbolic lattices
satisfying certain natural geometric assumptions still
satisfy the conclusions of superrigidity. The main
application is to prove arithmeticity of finitevolume real
or complex hyperbolic manifolds containing infinitely many
properly immersed maximal totally geodesic submanifolds of
dimension at least two.

 Thursday May 21 at 2:003:00pm in ZOOM
 David Futer, Temple University
 Special covers of alternating links

Abstract: The "virtual conjectures" in
lowdimensional topology, stated by Thurston in 1982,
postulated that every hyperbolic 3manifold 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 for a quantitative
statement: just how big does a cover need to be in order to
ensure one of these desired properties? We begin to give a
quantitative answer to this question, in the setting of
alternating links in S^{3}. If a prime, alternating
link K has a diagram with n crossings, we prove that the
complement of K has a special cover of degree less than
(n!). Corollaries of this result include a quantification of
residual finiteness, explicit control on the growth of Betti
numbers in covers, and an explicit bound on the rank of a
Zmodule on which the link group acts faithfully. Part of
the fun is reducing the dimension of the problem from
3manifolds to 2dimensional square complexes to maps
between 1dimensional graphs. This is joint work with Edgar
Bering.

 Thursday July 23 at 2:003:00pm in ZOOM
 Randall Kamien, University of Pennsylvania, Physics
 Applied Fibrations: Liquid Crystals

Abstract: Solid, Liquid, Gas — it turns out that
the three phases of matter are not all that is out there. In
particular, materials known as liquid crystals have some
properties of both solids and fluids. One of the simplest is
the “smectic phase.” Abstractly, the smectic phase is a
foliation of space with certain singularities,
“topological defects.” There is a species of these
defects that require the foliations to degenerate onto
lines, changing the problem into the open book decomposition
of space with the defects being the bindings. I will
introduce these ideas from the ground up and will pose some
questions about the class of allowable knots and links of
the defect line components.
Due to the high number of requests, we are no longer accepting speakers via selfinvitations.
For questions, contact