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A non- embedding result for A non-embedding result for Thompsons Group V Thompsons Group V Nathan Corwin Introduction co CF groups Nathan Corwin Wreath Products Thompsons University of Nebraska Lincoln Group V Dynamics of

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

A non-embedding result for Thompson’s Group V

Nathan Corwin

University of Nebraska – Lincoln

Groups St Andrews 2013 7 August 2013 s-ncorwin1@math.unl.edu

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Overview

Theorem (C. 2013)

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Overview

Theorem (C. 2013)

Z ≀ Z2 does not embed into Thompson’s Group V . Give some motivation.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Overview

Theorem (C. 2013)

Z ≀ Z2 does not embed into Thompson’s Group V . Give some motivation. Define wreath product.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Overview

Theorem (C. 2013)

Z ≀ Z2 does not embed into Thompson’s Group V . Give some motivation. Define wreath product. Define Thompson’s Group V.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Overview

Theorem (C. 2013)

Z ≀ 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.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Overview

Theorem (C. 2013)

Z ≀ 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.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

C Fand coC F

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

C Fand coC F

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

C Fand coC F

In the mid-1980’s Muller and Schupp showed that the class of all of groups that have a context free word problem (denoted C F) is equivalent to the the class of groups that are virtually free. A natural generalization of C F is the class coC F: all groups for which the coword problem is context free.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

C Fand coC F

In the mid-1980’s Muller and Schupp showed that the class of all of groups that have a context free word problem (denoted C F) is equivalent to the the class of groups that are virtually free. A natural generalization of C F is the class coC F: all groups for which the coword problem is context free. This class was first introduced by Holt, Rees, R¨

ver, and

Thomas in 2006.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

C Fand coC F

In the mid-1980’s Muller and Schupp showed that the class of all of groups that have a context free word problem (denoted C F) is equivalent to the the class of groups that are virtually free. A natural generalization of C F is the class coC F: all groups for which the coword problem is context free. This class was first introduced by Holt, Rees, R¨

ver, and

Thomas in 2006. They showed that coC Fhas many closure properties. Closed under:

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

coC F

Two conjectures from that paper:

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

coC F

Two conjectures from that paper:

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

coC F

Two conjectures from that paper:

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

My theorem supports this conjecture.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

coC F

Two conjectures from that paper:

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

My theorem supports this conjecture.

2 coC Fis not closed under free products.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

coC F

Two conjectures from that paper:

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

My theorem supports this conjecture.

2 coC Fis not closed under free products.

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

coC F and V

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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 in coC Fas V contains many copies of Z and Z2 and free products of subgroups are common in V .

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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 in coC Fas V contains many copies of Z and Z2 and free products of subgroups are common in V . In 2009, Bleak and Salazar-D` ıaz showed that Z ∗ Z2 does not embed into V .

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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 in coC Fas V contains many copies of Z and Z2 and free products of subgroups are common in V . In 2009, Bleak and Salazar-D` ıaz showed that Z ∗ Z2 does not embed into V . In that paper, they conjectured that Z ≀ Z2 does not embed into V .

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Other motivation

Structure of the R. Thompson’s groups

Richard Thompson discovered F < T < V in 1965.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Other motivation

Structure of the R. Thompson’s groups

Richard Thompson discovered F < T < V in 1965. In 1999, Guba and Sapir showed that Z ≀ Z embeds into F, and thus embeds into T and V as well.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Other motivation

Structure of the R. Thompson’s groups

Richard Thompson discovered F < T < V in 1965. In 1999, Guba and Sapir showed that Z ≀ Z embeds into F, and thus embeds into T and V as well. In 2008, Bleak showed that Z ≀ Z2 does not embed into F.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Other motivation

Structure of the R. Thompson’s groups

Richard Thompson discovered F < T < V in 1965. In 1999, Guba and Sapir showed that Z ≀ Z embeds into F, and thus embeds into T and V as well. In 2008, Bleak showed that Z ≀ Z2 does not embed into F. In 2009, Bleak, Kassabov, and Matucci showed Z ≀ Z2 does not embed into T.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Some notation

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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).

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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 = x}. Note, this differs slightly from the standard analysis definition.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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 = x}. Note, this differs slightly from the standard analysis definition. Fact: Supp (ab) = Supp (a)b.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Wreath Products

Let A and T be groups.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Wreath Products

Let A and T be groups. Set B = ⊕t∈T A.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Wreath Products

Let A and T be groups. Set B = ⊕t∈T A. Then, the Wreath Product of A and T is A ≀ T = B ⋊ T (where the semi-direct product action of T on B is right multiplication on the index in the direct product). We say T is the top group, A is the bottom group, and B is called the base group.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

First look at R. Thompson’s group V

V is finitely presented.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

First look at R. Thompson’s group V

V is finitely presented. Standard presentation has 4 generators,

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

First look at R. Thompson’s group V

V is finitely presented. Standard presentation has 4 generators, and 13 relations.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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 are A, B, C, π0.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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 are A, B, C, π0. The relations:

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

First look at R. Thompson’s group V

[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;

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Second look at R. Thompson’s group V

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

An element of V

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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 some arbitrary n.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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 some arbitrary n. Let σ ∈ Sn.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Example of an element U in V

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Example of an element U in V

(0101100 . . . )U

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Example of an element U in V

(0101100 . . . )U

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Example of an element U in V

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Not unique representation

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Not unique representation

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Not unique representation

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

A simplifying assumption for this talk

I will assume that every element of V ( I discuss ) has no non-trivial orbits when it acts on the Cantor set.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

A simplifying assumption for this talk

I will assume that every element of V ( I discuss ) has no non-trivial orbits when it acts on the Cantor set. This is not a big assumption (for me ) as for any v ∈ V there is an n ∈ N such that vn has this condition.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

A simplifying assumption for this talk

I will assume that every element of V ( I discuss ) has no non-trivial orbits when it acts on the Cantor set. This is not a big assumption (for me ) as for any v ∈ V there is an n ∈ N such that vn has this condition. This assumption is not needed to understand the dynamics

f V , but makes things simpler to explain.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Revealing Pairs

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Revealing Pairs

Consider the common tree C = D ∩ R.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Revealing Pairs

Consider the common tree C = D ∩ R.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Revealing Pairs and Important points

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Revealing Pairs and Important points

A revealing pair is a particularly tree pair of an element of V . Each element of V has a revealing pair.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Revealing Pairs and Important points

A revealing pair is a particularly tree pair of an element of V . Each element of V has a revealing pair. Each connected component of D \ C has a unique fixed

point. We will call it a repelling fixed point.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Revealing Pairs and Important points

A revealing pair is a particularly tree pair of an element of V . Each element of V has a revealing pair. Each connected component of D \ C has a unique fixed

point. We will call it a repelling fixed point.

Each connected component of R \ C has a unique fixed

point. We will call it an attracting fixed point.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Revealing Pairs and Important points

A revealing pair is a particularly tree pair of an element of V . Each element of V has a revealing pair. Each connected component of D \ C has a unique fixed

point. We will call it a repelling fixed point.

Each connected component of R \ C has a unique fixed

point. We will call it an attracting fixed point.

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Revealing Pairs and Important points

A revealing pair is a particularly tree pair of an element of V . Each element of V has a revealing pair. Each connected component of D \ C has a unique fixed

point. We will call it a repelling fixed point.

Each connected component of R \ C has a unique fixed

point. We will call it an attracting fixed point.

The set of the attracting and repelling fixed points are the set of important points for u. This set is denoted by I(u). Fact: Supp (a) = Supp (a) ∪ I(a).

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Flow graph

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Flow graph

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Flow graph

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Flow graph

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Flow graph

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Components of support

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

A lemma

Lemma (Bleak, Salazar-D` ıaz, 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 h

respectively, 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, there are non-trivial powers m and n such that gm = hn over X.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof (outline) of main result

Recall

Theorem (C. 2013)

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof (outline) of main result

Recall

Theorem (C. 2013)

Z ≀ Z2 does not embed into Thompson’s Group V . Proof: Step 1: Suppose there is an injection φ : Z ≀ Z2 → V

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof (outline) of main result

Recall

Theorem (C. 2013)

Z ≀ Z2 does not embed into Thompson’s Group V . Proof: Step 1: Suppose there is an injection φ : Z ≀ Z2 → V Step 2: Clean up injection

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 2: Clean up injection

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 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 no non-trivial finite orbits.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 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 no non-trivial finite orbits. Fix an element γ′

0 in the bottom group with no non-trivial

finite orbits.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 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 no non-trivial finite orbits. Fix an element γ′

0 in the bottom group with no non-trivial

finite orbits. Repeatedly apply two technical lemmas of Bleak and Salazar-D` ıaz to eventually replace γ′

0 with γ0.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 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 no non-trivial finite orbits. Fix an element γ′

0 in the bottom group with no non-trivial

finite orbits. Repeatedly apply two technical lemmas of Bleak and Salazar-D` ıaz to eventually replace γ′

0 with γ0.

γ0 will:

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 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 no non-trivial finite orbits. Fix an element γ′

0 in the bottom group with no non-trivial

finite orbits. Repeatedly apply two technical lemmas of Bleak and Salazar-D` ıaz to eventually replace γ′

0 with γ0.

γ0 will:

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

We have s, t, γ0 ∼ = Z ≀ Z2.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Proof: Step 1 Suppose there is an injection φ : Z ≀ Z2 → V Step 2: Clean up injection

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Proof: Step 1 Suppose there is an injection φ : Z ≀ Z2 → V Step 2: Clean up injection Step 3: Make sequence of γi’s

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 3: Make sequence of γi’s

Consider the subgroup s, r. It has components of support X1, . . . , Xk.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 3: Make sequence of γi’s

Consider the subgroup s, r. It has components of support X1, . . . , Xk. On X1, s and r commute, so there are integers r, q = 0 such that u = srtq is trivial on X1.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 3: Make sequence of γi’s

Consider the subgroup s, r. It has components of support X1, . . . , Xk. On X1, s and r commute, so there are integers r, q = 0 such that u = srtq is trivial on X1. Thus, there is a power p such that Supp (u) ∩ Supp (γ0) ∩ Supp (γ0)up = ∅. Set w = up.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 3: Make sequence of γi’s

Consider the subgroup s, r. It has components of support X1, . . . , Xk. On X1, s and r commute, so there are integers r, q = 0 such that u = srtq is trivial on X1. Thus, there is a power p such that Supp (u) ∩ Supp (γ0) ∩ Supp (γ0)up = ∅. Set w = up. Define γ1 = [γ0, w]. This element is nontrivial and of infinite order and will have no important points in X1.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 3: Make sequence of γi’s

Consider the subgroup s, r. It has components of support X1, . . . , Xk. On X1, s and r commute, so there are integers r, q = 0 such that u = srtq is trivial on X1. Thus, there is a power p such that Supp (u) ∩ Supp (γ0) ∩ Supp (γ0)up = ∅. Set w = up. Define γ1 = [γ0, w]. This element is nontrivial and of infinite order and will have no important points in X1. One can show that s, t, γ1 ∼ = Z ≀ Z2.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 3: Make sequence of γi’s

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

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 3: Make sequence of γi’s

Recursively repeat this process: given a γi−1, we can make a γi. Each time, we have γi is nontrivial and of infinite order. Further, s, t, γi ∼ = Z ≀ Z2.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 3: Make sequence of γi’s

Recursively repeat this process: given a γi−1, we can make a γi. Each time, we have γi is nontrivial and of infinite order. Further, s, t, γi ∼ = Z ≀ Z2. These elements were made so that γi has no important points in Xi.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 3: Make sequence of γi’s

Recursively repeat this process: given a γi−1, we can make a γi. Each time, we have γi is nontrivial and of infinite order. Further, s, t, γi ∼ = Z ≀ Z2. These elements were made so that γi has no important points in Xi. One can show that γi has no important points in Xj for j < i.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Proof sketch continued

Step 3: Make sequence of γi’s

Recursively repeat this process: given a γi−1, we can make a γi. Each time, we have γi is nontrivial and of infinite order. Further, s, t, γi ∼ = Z ≀ Z2. These elements were made so that γi has no important points in Xi. One can show that γi has no important points in Xj for j < i. In particular, γk has no important points at all, thus it is trivial.

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Finish proof sketch

Proof: Step 1: Suppose there is an injection φ : Z ≀ Z2 → V Step 2: Clean up injection Step 3: Make sequence of γi’s Step 4: Show that γk is a nontrivial element of infinite

rder that is also trivial

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Finish proof sketch

Proof: Step 1: Suppose there is an injection φ : Z ≀ Z2 → V Step 2: Clean up injection Step 3: Make sequence of γi’s Step 4: Show that γk is a nontrivial element of infinite

rder that is also trivial

Step 5: Notice that is a contradiction

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Finish proof sketch

Proof: Step 1: Suppose there is an injection φ : Z ≀ Z2 → V Step 2: Clean up injection Step 3: Make sequence of γi’s Step 4: Show that γk is a nontrivial element of infinite

rder that is also trivial

Step 5: Notice that is a contradiction

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A non- embedding result for Thompson’s Group V Nathan Corwin Introduction coCFgroups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result