A non-embedding result for Thompson's Group V · 2013. 8. 12. · A non-embedding result for...

A non-embeddingresult for

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

A non-embedding result forThompson’s Group V

Nathan Corwin

University of Nebraska – Lincoln

Groups St Andrews 20137 August 2013

s-ncorwin1@math.unl.edu

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Overview

Theorem (C. 2013)

Z o Z2 does not embed into Thompson’s Group V .

Give some motivation.

Define wreath product.

Define Thompson’s Group V.

Briefly discuss dynamics in the group.

Briefly discuss the proof of the theorem.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Overview

Theorem (C. 2013)

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Overview

Theorem (C. 2013)

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Overview

Theorem (C. 2013)

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Overview

Theorem (C. 2013)

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Overview

Theorem (C. 2013)

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

C Fand coC F

In the mid-1980’s Muller and Schupp showed that theclass of all of groups that have a context free wordproblem (denoted C F ) is equivalent to the the class ofgroups that are virtually free.

A natural generalization of C F is the class coC F : allgroups for which the coword problem is context free.

This class was first introduced by Holt, Rees, Rover, andThomas in 2006.

They showed that coC Fhas many closure properties.Closed under:

direct products;standard restricted wreath products where the top group isC F ;passing to finitely generated subgroups;passing to finite index over-groups.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

C Fand coC F

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

C Fand coC F

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

C Fand coC F

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

C Fand coC F

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Two conjectures from that paper:

1 If C o T is in coC F , then T must be in C F ;

My theorem supports this conjecture.

2 coC F is not closed under free products.

The leading candidate to show the second conjecture is thegroup Z ∗ Z2.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

coC F and V

In 2007 Lehnert and Schweitzer showed that R.Thompson’s group V is in coC F .

This was a surprising result.

It also put into doubt the belief that Z ∗ Z2 is not incoC Fas V contains many copies of Z and Z2 and freeproducts of subgroups are common in V .

In 2009, Bleak and Salazar-Dıaz showed that Z ∗ Z2 doesnot embed into V .

In that paper, they conjectured that Z o Z2 does notembed into V .

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

coC F and V

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

coC F and V

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

coC F and V

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

coC F and V

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Other motivationStructure of the R. Thompson’s groups

Richard Thompson discovered F < T < V in 1965.

In 1999, Guba and Sapir showed that Z oZ embeds into F ,and thus embeds into T and V as well.

In 2008, Bleak showed that Z o Z2 does not embed into F .

In 2009, Bleak, Kassabov, and Matucci showed Z oZ2 doesnot embed into T .

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Some notation

Suppose that a, b ∈ G and G acts on a set X.

We will use right actions.We write (x)a or just xa instead of a(x).

Conjugation: ab = b−1ab.

Commutator: [a, b] = a−1b−1ab = a−1ab = (b−1)ab.

Support of a function (element):Supp (a) = {x ∈ X|xa 6= x}.Note, this differs slightly from the standard analysisdefinition.

Fact: Supp (ab) = Supp (a)b.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Some notation

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Some notation

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Some notation

Commutator: [a, b] = a−1b−1ab

= a−1ab = (b−1)ab.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Some notation

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Some notation

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Some notation

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Wreath Products

Let A and T be groups.

Set B = ⊕t∈TA.

Then, the Wreath Product of A and T is A o T = B o T(where the semi-direct product action of T on B is rightmultiplication on the index in the direct product).

We say T is the top group, A is the bottom group, and B iscalled the base group.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Wreath Products

Set B = ⊕t∈TA.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Wreath Products

Set B = ⊕t∈TA.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

First look at R. Thompson’s group V

V is finitely presented.

Standard presentation has 4 generators, and 13 relations.

The generators of the standard presentation areA,B,C, π0.

The relations:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Standard presentation has 4 generators,

and 13 relations.

The relations:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

The relations:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

The relations:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

The relations:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

[AB−1, A−1BA] = 1;

[AB−1, A−2BA2] = 1;

C = BA−1CB;

A−1CBA−1BA = BA−2CB2;

CA = (A−1CB)2;

C3 = 1;

((A−1CB)−1π0A−1CB)2 = 1;

[(A−1CB)−1π0A−1CB,A−2(A−1CB)−1π0A

−1CBA2] = 1;

(A−1(A−1CB)−1π0A−1CBA(A−1CB)−1π0A

−1CB)3 = 1;

[A−2BA2, (A−1CB)−1π0A−1CB] = 1;

(A−1CB)−1π0A−1CBA−1BA =

BA−1(A−1CB)−1π0A−1CBA(A−1CB)−1π0A

−1CB;

A−1(A−1CB)−1π0A−1CBAB =

BA−2(A−1CB)−1π0A−1CBA2;

(A−1CB)−1π0A−1CBA−2CB2 =

A−2CB2A−1(A−1CB)−1π0A−1CBA;

((A−1CB)−1π0A−1CBA−1CB)3 = 1〉

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Second look at R. Thompson’s group V

Let T be the infinite, rooted, directed, binary tree.

Note that the limit space is the Cantor set.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Second look at R. Thompson’s group V

Let T be the infinite, rooted, directed, binary tree.

Note that the limit space is the Cantor set.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

An element of V

Let D and R be two finite connected rooted subgraphs of T(with the same root as T ) both with n leaves for somearbitrary n.Let σ ∈ Sn.Then u = (D,R, σ) is a representative of an element of V .(Tree pair representative)

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

An element of V

Let D and R be two finite connected rooted subgraphs of T(with the same root as T ) both with n leaves for somearbitrary n.

Let σ ∈ Sn.Then u = (D,R, σ) is a representative of an element of V .(Tree pair representative)

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

An element of V

Let D and R be two finite connected rooted subgraphs of T(with the same root as T ) both with n leaves for somearbitrary n.Let σ ∈ Sn.

Then u = (D,R, σ) is a representative of an element of V .(Tree pair representative)

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

An element of V

Let D and R be two finite connected rooted subgraphs of T(with the same root as T ) both with n leaves for somearbitrary n.Let σ ∈ Sn.Then u = (D,R, σ) is a representative of an element of V .(Tree pair representative)

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Example of an element U in V

(0101100 . . . )U

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

(0101100 . . . )U

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

(0101100 . . . )U

= 101100 . . .

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

(0101100 . . . )U = 101100 . . .

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Not unique representation

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

A simplifying assumption for this talk

I will assume that every element of V ( I discuss ) has nonon-trivial orbits when it acts on the Cantor set.

This is not a big assumption (for me ) as for any v ∈ Vthere is an n ∈ N such that vn has this condition.

This assumption is not needed to understand the dynamicsof V , but makes things simpler to explain.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Revealing Pairs

Consider the common tree C = D ∩R.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Revealing Pairs

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Revealing Pairs

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Revealing Pairs and Important points

A revealing pair is a particularly tree pair of an element ofV .

Each element of V has a revealing pair.

Each connected component of D \ C has a unique fixedpoint. We will call it a repelling fixed point.

Each connected component of R \ C has a unique fixedpoint. We will call it an attracting fixed point.

The set of the attracting and repelling fixed points are theset of important points for u. This set is denoted by I(u).

Fact: Supp (a) = Supp (a) ∪ I(a).

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Flow graph

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Flow graph

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Flow graph

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Flow graph

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Flow graph

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Components of support

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

A lemma

Lemma (Bleak, Salazar-Dıaz, 2009)

Suppose g, h ∈ V , each with no non-trivial periodic orbits. For(i) and (ii), suppose further that g and h commute. Then:

i. I(g) ∩ I(h) = I(g) ∩ Supp (h) = I(h) ∩ Supp (g);

ii. If X and Y are components of support of g and hrespectively, then X = Y or X ∩ Y = ∅;

iii. Suppose g and h have a common component of support X,and on X the actions of g and h commute. Then, thereare non-trivial powers m and n such that gm = hn over X.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Proof (outline) of main result

Recall

Theorem (C. 2013)

Proof:

Step 1: Suppose there is an injection φ : Z o Z2 → V

Step 2: Clean up injection

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Recall

Theorem (C. 2013)

Proof:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Recall

Theorem (C. 2013)

Proof:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Proof sketch continuedStep 2: Clean up injection

Let s′ and t′ be the images of the generators of the Z2.

Raise s′ and t′ to powers to obtain s and t with nonon-trivial finite orbits.

Fix an element γ′0 in the bottom group with no non-trivialfinite orbits.

Repeatedly apply two technical lemmas of Bleak andSalazar-Dıaz to eventually replace γ′0 with γ0.

γ0 will:

be a non-trivial element of the base with no non-trivialfinite orbits;have support disjoint from a neighborhood of theimportant points of s and t.

We have 〈s, t, γ0〉 ∼= Z o Z2.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

γ0 will:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

γ0 will:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

γ0 will:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

γ0 will:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

γ0 will:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Proof sketch continued

Proof:

Step 1 Suppose there is an injection φ : Z o Z2 → V

Step 3: Make sequence of γi’s

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Proof sketch continued

Proof:

Step 1 Suppose there is an injection φ : Z o Z2 → V

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Proof sketch continuedStep 3: Make sequence of γi’s

Consider the subgroup 〈s, r〉. It has components ofsupport X1, . . . , Xk.

On X1, s and r commute, so there are integers r, q 6= 0such that u = srtq is trivial on X1.

Thus, there is a power p such thatSupp (u) ∩ Supp (γ0) ∩ Supp (γ0)up = ∅. Set w = up.

Define γ1 = [γ0, w]. This element is nontrivial and ofinfinite order and will have no important points in X1.

One can show that 〈s, t, γ1〉 ∼= Z o Z2.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Recursively repeat this process: given a γi−1, we can makea γi.

Each time, we have γi is nontrivial and of infinite order.Further, 〈s, t, γi〉 ∼= Z o Z2.

These elements were made so that γi has no importantpoints in Xi.

One can show that γi has no important points in Xj forj < i.

In particular, γk has no important points at all, thus it istrivial.

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Finish proof sketch

Proof:

Step 4: Show that γk is a nontrivial element of infiniteorder that is also trivial

Step 5: Notice that is a contradiction

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Finish proof sketch

Proof:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Finish proof sketch

Proof:

Thompson’sGroup V

NathanCorwin

Introduction

coCFgroups

WreathProducts

Thompson’sGroup V

Dynamics ofV

Proof of MainResult

Thank You

Thank you for your attention.

A non-embedding result for Thompson's Group V · 2013. 8. 12. · A non-embedding result for...

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Video: Danny Thompson’s Sights Set For Bonneville To …static1.squarespace.com/static/4fe4b72984aefa6aa5a93f72/t/53ce11... · Video: Danny Thompson’s Sights Set For Bonneville

THOMPSON’S GROUP Fpi.math.cornell.edu/~belk/Thesis.pdf · The Group F Thompson’s group F is a certain group of piecewise-linear homeomorphisms of [0;1]. We de ne F in section

Embedding SQL Server Express into Custom Applications - Web viewFilename: #1266 Embedding Express Cust_Apps3. Embedding SQL Server Express into Custom Applications3. Embedding SQL

THOMPSON’S POINT - LoopNet...One Canal Plaza, Ste 500 Portland, ME 04101 207.772.1333 The Boulos Company FOR LEASE CLASS A OFFICE Thompson’s Point Portland, ME Building Feature

Chapter 4.1 An Atomic Model of Matter The Greek Model The Greek Model Dalton’s model Dalton’s model Thompson’s model Thompson’s model Rutherford’s model.

UNIQUE GEODESICS FOR THOMPSON’S METRIC …

Random subgroups of Thompson’s groupandrewr/pubs/randomF-nov07.pdf · Random subgroups of Thompson’s group F Sean Cleary Department of Mathematics, The City College of New York

Thompson’s Gazelle

Conjugacy in Thompson’s Group Jim Belk (joint with Francesco Matucci)

Solar Thompson’s Pies

Thompson’s , the very essence of the Bordeaux region

Thompson’s List

Embedding Environmental Sustainability - Family Repositoryfamilyrepository.lppkn.gov.my/601/1/Embedding Environmental Sustainability.pdf · Embedding Environmental Sustainability

Our Country’s Geography Mr. Thompson’s 5 th Grade Social Studies.