Student Topology Seminar
Tuesday November 25, 2014
LCB322 — 10:45  11:35
Kishalaya Saha
Rotation on binary trees and hyperbolic geometry
Abstract:
Tree rotation is a wellknown technique to make a binary tree balanced. A
natural question is to find the smallest number d(n) such that given any two trees
with n nodes, we can obtain one from the other using at most d(n) rotations.
Sleator, Tarjan, and Thurston showed that finding a pair of trees with large
rotation distance is related to finding polyhedra that require many tetrahedra to
triangulate. They produced such polyhedra using hyperbolic geometry and proved
d(n)=2n6 for all sufficiently large n.
Tuesday November 18, 2014
LCB322 — 10:45  11:35
Radhika Gupta
Thompson's Group
Abstract:
This is a continuation of last weeks' talk.
Tuesday November 11, 2014
LCB322 — 10:45  11:35
Radhika Gupta
Thompson's Group
Abstract:
We define Thompson's group $F$ as a subgroup of the group of
all homeomorphisms from $[0,1]$ to itself. We look at tree diagram
representations and normal form of elements of $F$. Using these we prove
every proper quotient group of $F$ is abelian. We also prove that every
nonabelian subgroup of $F$ contains a free abelian group of infinite
rank. This talk is based on 'Notes on Richard Thompson's group $F$ and
$T$' by Cannon, Floyd and Parry.
Tuesday October 28, 2014
LCB322 — 10:45  11:35
Derrick Wigglesworth
The Rips Complex and Some Applications (cont.)
Abstract:
This is a continuation of last weeks' talk.
Tuesday October 21, 2014
LCB322 — 10:45  11:35
Derrick Wigglesworth
The Rips Complex and Some Applications
Abstract:
We will define the Rips complex of a finitely generated group $G$ and prove Rips' Theorem that if G is hyperbolic, then the Rips complex is contractible for sufficiently large n. We will then use this fact to prove three theorems. First, hyperbolic groups have finitely many conjugacy classes of torsion elements. Second, that hyperbolic groups have trivial rational homology for k sufficiently large. Last, we will show that hyperbolic groups are finitely presented.
Tuesday October 7, 2014
LCB322 — 10:45  11:35
James Farre
Some Remarks on Bounded Cohomology and Stable Commutator Length
Abstract:
We give a definition of the bounded cohomology of a discrete group G, and
explore some of its basic properties. In particular, we give a construction to show
that if G is a free group, then bounded cohomology in degree two is an infinite
dimensional Banach space. We use this construction to show that the commutator
length of free groups is unbounded.
Tuesday September 30, 2014
LCB322 — 10:45  11:35
Radhika Gupta
BNS Invariants
Abstract:
In this talk we define BNS invariant of a discrete group. We
compute the invariant for some groups like free abelian group of rank two,
free nonabelian group of rank two and the Baumslag Solitaire group
$BS(1,2)$. We also prove openness of the invariant for finitely generated
groups. This talk is based on 'Notes on Sigma Invariants' by Ralph
Strebel.
Tuesday September 23, 2014
LCB322 — 10:45  11:35
Nicholas Cahill
Groups that Act Freely on Homology nspheres (Part 2)
Abstract:
Finite groups acting on spheres, their 'group theoretic' properties (e.g., the cyclic subgroup property) and the
classification of finite groups which can act orthogonally on the 3sphere.
Tuesday September 16, 2014
LCB322 — 10:45  11:35
Kishalaya Saha
Groups that Act Freely on Homology nspheres
Abstract:
We discuss the proof of a result by Milnor that says that if a group G
acts freely on a manifold with the same mod two homology as that of a sphere, then
any element of order two in G is central.
Tuesday September 9, 2014
LCB322 — 10:45  11:35
Dawei Wang
A Topological Proof of Grushko's Theorem on Free Products
Abstract:
Grushko's theorem states that the rank of a free product of two groups is
equal to the sum of the ranks of the two free factors. We give a
topological proof that uses simple topology and combinatorial arguments.
It has the advantage that there is no complicated cancellation procedure.
