# Geometry and Topology Seminar

Current contacts: Dave Futer, Matthew Stover, or Sam Taylor.

The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 of Wachman Hall.

• Wednesday January 25, 2023 at 15:30, Wachman 617
Residually finite central extensions of lattices

Matthew Stover (Temple)

The preimage of PSL(2, Z) in any connected cover of PSL(2, R) is residually finite, and one can prove this very explicitly using nilpotent quotients. For n ≥ 2, Deligne famously proved using the congruence subgroup property that the central extension of Sp(2n, Z) by Z determined by its preimage in the universal cover of Sp(2n, R) is not residually finite. I will describe joint work with Domingo Toledo that develops methods, generalizing one interpretation of the argument for PSL(2, Z), to prove residual finiteness (in fact, linearity) of cyclic central extensions of fundamental groups of aspherical manifolds with residually finite fundamental group. I will then describe how this generalization applies to prove residual finiteness of cyclic central extensions of certain arithmetic lattices in PU(n, 1).

• Wednesday February 1, 2023 at 14:30, Wachman 617

Will Worden (Holy Family University)

Fully augmented links (FALs) are a class of hyperbolic links having especially nice geometric and combinatorial structures. A large subclass of these links, called octahedral FALs, have complements that are arithmetic manifolds with trace field $\mathbb{Q}(\sqrt{-1})$. Apart from these, the only other known example of an arithmetic FAL is the minimally twisted 8-chain link, shown to have trace field $\mathbb{Q}(\sqrt{-2})$ by Meyer—Millichap—Trapp. We’ll discuss work joint with Neil Hoffman that shows that these are in fact the only two possible trace fields for arthmetic FALs.

• Friday February 10, 2023 at 14:30, Wachman 617
Homeomorphism groups of (weakly) self-similar 2-manifolds

Nicholas Vlamis, CUNY Queens College

PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)

Abstract: The study of homeomorphism groups and mapping class groups of infinite-type 2-manifolds is in its infancy, and the deeper we dive into their structure we see that the class of infinite-type surfaces cannot be studied all at once.  Recently, Mann and Rafi have introduced a very useful way to partition this class into several nice subclasses whose homeomorphism groups/mapping class groups often share many properties.  In this talk, we will discuss recent results regarding the structure of homeomorphism groups of the class of weakly self-similar 2-manifolds, and how these can be viewed as natural extensions of results regarding the homeomorphism groups of the 2-sphere, the plane, and the open annulus.

In the morning background talk (at 9:30am), I will give an overview of some fundamental results and tools regarding the algebraic and topological structure of homeomorphism groups of compact (two-)manifolds (with a focus on the sphere).  We will touch on work of Anderson from the 50s, Fisher from the 60s, Kirby from the 60s/70s, and Calegari--Freedman from the aughts.

• Friday February 10, 2023 at 16:00, Wachman 617
From embedded contact homology to surface dynamics

Jo Nelson, Rice University and IAS

PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)

Abstract: I will discuss work in progress with Morgan Weiler on knot filtered embedded contact homology (ECH) of open book decompositions of $S^3$ along $T(2,q)$ torus knots to deduce information about the dynamics of symplectomorphisms of the genus $(q-1)/2$ pages which are freely isotopic to rotation by $1/(2q)$ along the boundary. I will explain the interplay between the topology of the open book, its presentation as an orbi-bundle, and our computation of the knot filtered ECH chain complex.  I will describe how knot filtered ECH realizes the relationship between the action and linking of Reeb orbits and its application to the study of the Calabi invariant and periodic orbits of symplectomorphisms of the pages.

In the morning background talk (at 11:00 am), I will give an introduction to Floer theories and Reeb dynamics for contact manifolds. I will give some background on this subject, including motivation from classical mechanics. I will then explain how to construct Floer theoretic contact invariants, illustrated by numerous graphics.

• Wednesday February 15, 2023 at 14:30, Wachman 617
Limits of asymptotically Fuchsian surfaces in a closed hyperbolic 3-manifold

Fernando Al Assal (Yale)

Let M be a closed hyperbolic 3-manifold and let Gr(M) be its 2-plane Grassmann bundle. We will discuss the following result: the weak-* limits of the probability area measures on Gr(M) of pleated or minimal closed connected essential K-quasifuchsian surfaces as K goes to 1 are all convex combinations of the probability area measures of the immersed closed totally geodesic surfaces of M and the probability volume (Haar) measure of Gr(M).

• Wednesday March 15, 2023 at 14:30, Wachman 617
TBD

Joseph Maher (CUNY)

TBD

• Wednesday March 22, 2023 at 14:30, Wachman 617
TBD