Date: Tuesday September 20th 2022
Tuesday, 2:30pm - 4pm in Room 6495
Title: Accessibility for groups
Speaker: Zhihao Mu (CUNY GC)
Abstract: Stallings showed that every finitely generated group
has more than one end if and only if it splits over some finite
group. A group is called accessible if splitting over finite subgroups
terminates after finitely many steps. Inspired by Kneser’s prime
decomposition of 3-manifolds, Dunwoody proved accessibility for all
finitely presented groups. In this talk, we will sketch Dunwoody’s
proof and discuss other splittings, for example JSJ-splittings, of
finitely presented groups and applications.
Date: Tuesday October 11th 2022
Tuesday, 2:30pm - 4pm in Room 6495
Title: All closed orientable 3-manifolds are central in trisections of 5-manifolds
Speaker: Carol Badre (CUNY GC)
Abstract: Gay and Kirby generalised the notion of Heegaard
splittings to the trisections of smooth 4-manifolds, which was
subsequently generalised to the trisections of piecewise-linear
5-manifolds M by Rubinstein and Tillmann. The construction of
Rubinstein and Tillmann takes advantage of a natural colouring on the
vertices of particular triangulations of M inherited from a
piecewise-linear map p: M -> D where D is a 2-simplex. An object of
interest which is a consequence of this construction is the central
submanifold, which is obtained as the preimage of the barycentre of
D. Each 5-simplex of M contains a 3-cube of the central
submanifold. We present a reversal of this process by first
constructing a cubing of every closed orientable 3-manifold as a
branched cover of the 3-sphere over the Borromean rings and show that
they are central submanifolds of trisections of piecewise-linear
5-manifolds. This is joint work with Stephan Tillmann.
Date: Tuesday October 18th 2022
Tuesday, 2:30pm - 4pm in Room 6495
Title: The isometry group can be determined by the topology (in some sense)
Speaker: Yushan Jiang (CUNY GC)
Abstract: For a topological space, some metrics can determine
the topology in many cases, e.g. hyperbolic metric. Hence, it is
interesting to investigate the reverse direction: whether a metric can
be determined by the topology in some sense (especially the isometry
group). In fact, the isometry group of the hyperbolic metric on a
closed manifold is a finite group, but what can we say for the other
Riemannian metrics on the same topological manifold? In this talk, I
will show the nonsymmetry of the metric (more concretely, the isometry
group is a finite group) is determined by the topology in some
situations, e.g. hyperbolic manifolds. We will see how a great tool
called the Gromov norm (defined on the homology group with real
coefficient), plays a central role in the proof. After considering the
3-dimensional case, we will also show that the Gromov norm will not be
an essential invariant which can determine the nonsymmetry of the
metric.
Date: Tuesday October 25th 2022
Tuesday, 2:30pm - 4pm in Room 6495
Title: Reidemeister torsion and Fox calculus
Speaker: Michael Marinelli (CUNY GC)
Abstract: Torsion invariants were introduced by Reidemeister
in 1935, and were the first non-homotopy invariants for
manifolds. Reidemeister defined the torsion invariants for closed
three dimensional manifolds and used it to obtain the complete (PL)
classification of lens spaces. In this talk, we will define the
Reidemeister torsion, show some computations, and prove that for
K(G,1) 3-manifolds with boundary, the torsion can be computed using
the technique of Fox calculus. We will also outline a theorem by
Milnor from 1962 which states that the classical knot invariant called
the Alexander polynomial can be obtained from the Reidemeister torsion
of a knot complement.
Date: Tuesday Nov 8th & 15th 2022
Tuesday, 2:30pm - 4pm in Room 6495
Title: A Brief Introduction to the famous Virtual Haken Conjecture
Speaker: Yushan Jiang (CUNY GC)
Abstract: I will begin with some 3-manifolds theory like the
definition of Haken 3-manifolds and explain why they are interesting
and important (the Haken hierarchy and Thurston's geometrization
theorem for Haken 3-manifolds). I will show define "virtual" and
discuss the motivation and reason why we consider this idea. Finally,
I will introduce the Virtual Haken Conjecture and some related famous
long standing conjectures (now theorems): Geometrization conjecture,
Surface Subgroup Conjecture and Virtual Fibering Conjecture.
This talk is mainly for telling the wonderful stories and briefly
illustrating how PDE, dynamics, geometric group theory and so
many other fields all get involved in this topological topic:
3-manifold groups.
Date: Tuesday Dec 6th 2022
Tuesday, 2:30pm - 4pm in Room 6495
Title: Mapping class group and curve graphs
Speaker: Zhihao Mu (CUNY GC)
Abstract: The curve complex of a finite type surface was
introduced by Harvey and proved to be Gromov hyperbolic by Masur and
Minsky. It becomes an essential role in studying the coarse geometry
of mapping class groups. In this talk, I will introduce curve
complexes, marking complexes(as a combinatorial model for mapping
class group), subsurface projections and the Behrstock's inequality
focusing on its application for finding the lower bound of the
distance formula.