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Student Topology SeminarTuesday November 25, 2014LCB322 — 10:45 - 11:35 Rotation on binary trees and hyperbolic geometry Abstract: Tree rotation is a well-known 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)=2n-6 for all sufficiently large n. Tuesday November 18, 2014LCB322 — 10:45 - 11:35 Thompson's Group Abstract: This is a continuation of last weeks' talk. Tuesday November 11, 2014LCB322 — 10:45 - 11:35 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 non-abelian 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, 2014LCB322 — 10:45 - 11:35 The Rips Complex and Some Applications (cont.) Abstract: This is a continuation of last weeks' talk. Tuesday October 21, 2014LCB322 — 10:45 - 11:35 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, 2014LCB322 — 10:45 - 11:35 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, 2014LCB322 — 10:45 - 11:35 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 non-abelian 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, 2014LCB322 — 10:45 - 11:35
Nicholas Cahill Groups that Act Freely on Homology n-spheres (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 3-sphere. Tuesday September 16, 2014LCB322 — 10:45 - 11:35 Groups that Act Freely on Homology n-spheres 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, 2014LCB322 — 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. |
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