embeddings into thompson's group v and cocf groups · 2013. 8. 15. · zz2 does not embed into...
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![Page 1: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/1.jpg)
Embeddings into Thompson’s group V and coCFgroups
Francesco Matucci (joint with C. Bleak, M. Neunhoffer)
Groups St. Andrews 2013
St. Andrews
August 7, 2013
![Page 2: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/2.jpg)
Finite state automaton
Mathematical model of computation to design computer programs.
Definition (DFSA)
A (deterministic) finite state automaton is a quintuple(S ,A, µ,Y , s0), where
I S is a finite set, called the state set,
I A is a finite set, called the alphabet,
I µ : A× S → S is a function, called the transition function,
I Y is a (possibly empty) subset of S called the subset ofaccept states,
I s0 ∈ S is called the start state.
If we allow µ : A× S → P(S), we call it a non-deterministic FSA.
RemarkEvery NDFSA is equivalent to a DFSA.
![Page 3: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/3.jpg)
Finite state automaton
Mathematical model of computation to design computer programs.
Definition (DFSA)
A (deterministic) finite state automaton is a quintuple(S ,A, µ,Y , s0), where
I S is a finite set, called the state set,
I A is a finite set, called the alphabet,
I µ : A× S → S is a function, called the transition function,
I Y is a (possibly empty) subset of S called the subset ofaccept states,
I s0 ∈ S is called the start state.
If we allow µ : A× S → P(S), we call it a non-deterministic FSA.
RemarkEvery NDFSA is equivalent to a DFSA.
![Page 4: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/4.jpg)
Finite state automaton
Mathematical model of computation to design computer programs.
Definition (DFSA)
A (deterministic) finite state automaton is a quintuple(S ,A, µ,Y , s0), where
I S is a finite set, called the state set,
I A is a finite set, called the alphabet,
I µ : A× S → S is a function, called the transition function,
I Y is a (possibly empty) subset of S called the subset ofaccept states,
I s0 ∈ S is called the start state.
If we allow µ : A× S → P(S), we call it a non-deterministic FSA.
RemarkEvery NDFSA is equivalent to a DFSA.
![Page 5: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/5.jpg)
Finite state automaton
Mathematical model of computation to design computer programs.
Definition (DFSA)
A (deterministic) finite state automaton is a quintuple(S ,A, µ,Y , s0), where
I S is a finite set, called the state set,
I A is a finite set, called the alphabet,
I µ : A× S → S is a function, called the transition function,
I Y is a (possibly empty) subset of S called the subset ofaccept states,
I s0 ∈ S is called the start state.
If we allow µ : A× S → P(S), we call it a non-deterministic FSA.
RemarkEvery NDFSA is equivalent to a DFSA.
![Page 6: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/6.jpg)
Finite state automaton
Mathematical model of computation to design computer programs.
Definition (DFSA)
A (deterministic) finite state automaton is a quintuple(S ,A, µ,Y , s0), where
I S is a finite set, called the state set,
I A is a finite set, called the alphabet,
I µ : A× S → S is a function, called the transition function,
I Y is a (possibly empty) subset of S called the subset ofaccept states,
I s0 ∈ S is called the start state.
If we allow µ : A× S → P(S), we call it a non-deterministic FSA.
RemarkEvery NDFSA is equivalent to a DFSA.
![Page 7: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/7.jpg)
Finite state automaton: an example
Start0
0
1
1Accept
![Page 8: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/8.jpg)
Finite state automaton: an example
Start0
0
1
1Accept
![Page 9: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/9.jpg)
Regular language and finite groups
DefinitionGiven a FSA a regular language is the language given by all pathsinside the FSA which begin at the start state and end at an acceptstate.
DefinitionGiven a finitely generated group G = 〈X | R〉, the language ofthe word problem is
WP(G ) = {words w in the monoid of X ∪ X−1 such that w ≡G 1}.
Theorem (Anisimov)
A finitely generated group G is finite if and only if WP(G ) is aregular language.
![Page 10: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/10.jpg)
Regular language and finite groups
DefinitionGiven a FSA a regular language is the language given by all pathsinside the FSA which begin at the start state and end at an acceptstate.
DefinitionGiven a finitely generated group G = 〈X | R〉, the language ofthe word problem is
WP(G ) = {words w in the monoid of X ∪ X−1 such that w ≡G 1}.
Theorem (Anisimov)
A finitely generated group G is finite if and only if WP(G ) is aregular language.
![Page 11: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/11.jpg)
Regular language and finite groups
DefinitionGiven a FSA a regular language is the language given by all pathsinside the FSA which begin at the start state and end at an acceptstate.
DefinitionGiven a finitely generated group G = 〈X | R〉, the language ofthe word problem is
WP(G ) = {words w in the monoid of X ∪ X−1 such that w ≡G 1}.
Theorem (Anisimov)
A finitely generated group G is finite if and only if WP(G ) is aregular language.
![Page 12: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/12.jpg)
Regular language and finite groups
DefinitionGiven a FSA a regular language is the language given by all pathsinside the FSA which begin at the start state and end at an acceptstate.
DefinitionGiven a finitely generated group G = 〈X | R〉, the language ofthe word problem is
WP(G ) = {words w in the monoid of X ∪ X−1 such that w ≡G 1}.
Theorem (Anisimov)
A finitely generated group G is finite if and only if WP(G ) is aregular language.
![Page 13: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/13.jpg)
Pushdown automaton
Definition (PDA - Handwaving)
A PDA is like a FSA but it also employs a stack in its transitionfunction. The transition function pops and pushes a symbol at thetop of the stack and uses it to decide which state to reach.
RemarkPDA adds the stack as a parameter for choice. Finite statemachines just look at the input signal and the current state: theyhave no stack to work with.
It is not true that all NDPDA are equivalent to DPDA.
![Page 14: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/14.jpg)
Pushdown automaton
Definition (PDA - Handwaving)
A PDA is like a FSA but it also employs a stack in its transitionfunction. The transition function pops and pushes a symbol at thetop of the stack and uses it to decide which state to reach.
RemarkPDA adds the stack as a parameter for choice. Finite statemachines just look at the input signal and the current state: theyhave no stack to work with.
It is not true that all NDPDA are equivalent to DPDA.
![Page 15: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/15.jpg)
Pushdown automaton
Definition (PDA - Handwaving)
A PDA is like a FSA but it also employs a stack in its transitionfunction. The transition function pops and pushes a symbol at thetop of the stack and uses it to decide which state to reach.
RemarkPDA adds the stack as a parameter for choice. Finite statemachines just look at the input signal and the current state: theyhave no stack to work with.
It is not true that all NDPDA are equivalent to DPDA.
![Page 16: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/16.jpg)
Pushdown automaton
Definition (PDA - Handwaving)
A PDA is like a FSA but it also employs a stack in its transitionfunction. The transition function pops and pushes a symbol at thetop of the stack and uses it to decide which state to reach.
RemarkPDA adds the stack as a parameter for choice. Finite statemachines just look at the input signal and the current state: theyhave no stack to work with.
It is not true that all NDPDA are equivalent to DPDA.
![Page 17: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/17.jpg)
Pushdown automaton: an example
finite state
automaton
![Page 18: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/18.jpg)
Pushdown automaton: an example
finite state
automaton
![Page 19: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/19.jpg)
Context free languages and virtually free groups
DefinitionContext-free languages are those accepted by PDAs.
DefinitionA finitely generated group is a context-free group if WP(G ) is acontext-free language.
Theorem (Muller-Schupp)
A finitely generated group G is virtually free if and only if it iscontext-free.
![Page 20: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/20.jpg)
Context free languages and virtually free groups
DefinitionContext-free languages are those accepted by PDAs.
DefinitionA finitely generated group is a context-free group if WP(G ) is acontext-free language.
Theorem (Muller-Schupp)
A finitely generated group G is virtually free if and only if it iscontext-free.
![Page 21: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/21.jpg)
Context free languages and virtually free groups
DefinitionContext-free languages are those accepted by PDAs.
DefinitionA finitely generated group is a context-free group if WP(G ) is acontext-free language.
Theorem (Muller-Schupp)
A finitely generated group G is virtually free if and only if it iscontext-free.
![Page 22: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/22.jpg)
Context free languages and virtually free groups
DefinitionContext-free languages are those accepted by PDAs.
DefinitionA finitely generated group is a context-free group if WP(G ) is acontext-free language.
Theorem (Muller-Schupp)
A finitely generated group G is virtually free if and only if it iscontext-free.
![Page 23: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/23.jpg)
Co-context-free (Co-CF) groups
DefinitionGiven a finitely generated group G = 〈X | R〉, the language ofthe coword problem is
coWP(G ) = {words w in the monoid of X ∪ X−1 such that w 6≡G 1}.
DefinitionA finitely generated group is a co-context-free group (coCF) ifcoWP(G ) is a context-free language. Let coCF be the class of allcoCF groups.
RemarkEvery CF group is in coCF (the converse is not true).
![Page 24: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/24.jpg)
Co-context-free (Co-CF) groups
DefinitionGiven a finitely generated group G = 〈X | R〉, the language ofthe coword problem is
coWP(G ) = {words w in the monoid of X ∪ X−1 such that w 6≡G 1}.
DefinitionA finitely generated group is a co-context-free group (coCF) ifcoWP(G ) is a context-free language. Let coCF be the class of allcoCF groups.
RemarkEvery CF group is in coCF (the converse is not true).
![Page 25: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/25.jpg)
Co-context-free (Co-CF) groups
DefinitionGiven a finitely generated group G = 〈X | R〉, the language ofthe coword problem is
coWP(G ) = {words w in the monoid of X ∪ X−1 such that w 6≡G 1}.
DefinitionA finitely generated group is a co-context-free group (coCF) ifcoWP(G ) is a context-free language. Let coCF be the class of allcoCF groups.
RemarkEvery CF group is in coCF (the converse is not true).
![Page 26: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/26.jpg)
Co-context-free (Co-CF) groups
DefinitionGiven a finitely generated group G = 〈X | R〉, the language ofthe coword problem is
coWP(G ) = {words w in the monoid of X ∪ X−1 such that w 6≡G 1}.
DefinitionA finitely generated group is a co-context-free group (coCF) ifcoWP(G ) is a context-free language. Let coCF be the class of allcoCF groups.
RemarkEvery CF group is in coCF (the converse is not true).
![Page 27: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/27.jpg)
Closure properties of the class coCF
Theorem (Holt-Rover-Rees-Thomas)
The class coCF is closed under taking
I taking finite direct products,
I taking restricted standard wreath products with context-freetop groups,
I passing to finitely generated subgroups
I passing to finite index overgroups.
Conjecture (Holt-Rover-Rees-Thomas)coCF is not closed for free products. Candidate: Z ∗ Z2.
Theorem (Bleak-Salazar)
Z ∗ Z2 does not embed into Thompson’s group V .
![Page 28: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/28.jpg)
Closure properties of the class coCF
Theorem (Holt-Rover-Rees-Thomas)
The class coCF is closed under taking
I taking finite direct products,
I taking restricted standard wreath products with context-freetop groups,
I passing to finitely generated subgroups
I passing to finite index overgroups.
Conjecture (Holt-Rover-Rees-Thomas)coCF is not closed for free products. Candidate: Z ∗ Z2.
Theorem (Bleak-Salazar)
Z ∗ Z2 does not embed into Thompson’s group V .
![Page 29: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/29.jpg)
Closure properties of the class coCF
Theorem (Holt-Rover-Rees-Thomas)
The class coCF is closed under taking
I taking finite direct products,
I taking restricted standard wreath products with context-freetop groups,
I passing to finitely generated subgroups
I passing to finite index overgroups.
Conjecture (Holt-Rover-Rees-Thomas)coCF is not closed for free products. Candidate: Z ∗ Z2.
Theorem (Bleak-Salazar)
Z ∗ Z2 does not embed into Thompson’s group V .
![Page 30: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/30.jpg)
Closure properties of the class coCF
Theorem (Holt-Rover-Rees-Thomas)
The class coCF is closed under taking
I taking finite direct products,
I taking restricted standard wreath products with context-freetop groups,
I passing to finitely generated subgroups
I passing to finite index overgroups.
Conjecture (Holt-Rover-Rees-Thomas)coCF is not closed for free products. Candidate: Z ∗ Z2.
Theorem (Bleak-Salazar)
Z ∗ Z2 does not embed into Thompson’s group V .
![Page 31: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/31.jpg)
Closure properties of the class coCF
Theorem (Holt-Rover-Rees-Thomas)
The class coCF is closed under taking
I taking finite direct products,
I taking restricted standard wreath products with context-freetop groups,
I passing to finitely generated subgroups
I passing to finite index overgroups.
Conjecture (Holt-Rover-Rees-Thomas)coCF is not closed for free products. Candidate: Z ∗ Z2.
Theorem (Bleak-Salazar)
Z ∗ Z2 does not embed into Thompson’s group V .
![Page 32: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/32.jpg)
Closure properties of the class coCF
Theorem (Holt-Rover-Rees-Thomas)
The class coCF is closed under taking
I taking finite direct products,
I taking restricted standard wreath products with context-freetop groups,
I passing to finitely generated subgroups
I passing to finite index overgroups.
Conjecture (Holt-Rover-Rees-Thomas)coCF is not closed for free products. Candidate: Z ∗ Z2.
Theorem (Bleak-Salazar)
Z ∗ Z2 does not embed into Thompson’s group V .
![Page 33: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/33.jpg)
Closure properties of the class coCF
Theorem (Holt-Rover-Rees-Thomas)
The class coCF is closed under taking
I taking finite direct products,
I taking restricted standard wreath products with context-freetop groups,
I passing to finitely generated subgroups
I passing to finite index overgroups.
Conjecture (Holt-Rover-Rees-Thomas)coCF is not closed for free products. Candidate: Z ∗ Z2.
Theorem (Bleak-Salazar)
Z ∗ Z2 does not embed into Thompson’s group V .
![Page 34: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/34.jpg)
Closure properties of the class coCF
Theorem (Holt-Rover-Rees-Thomas)
The class coCF is closed under taking
I taking finite direct products,
I taking restricted standard wreath products with context-freetop groups,
I passing to finitely generated subgroups
I passing to finite index overgroups.
Conjecture (Holt-Rover-Rees-Thomas)coCF is not closed for free products. Candidate: Z ∗ Z2.
Theorem (Bleak-Salazar)
Z ∗ Z2 does not embed into Thompson’s group V .
![Page 35: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/35.jpg)
Thompson’s group F
Thompson’s group F is the group PL2(I ), with respect tocomposition, of all piecewise-linear homeomorphisms of the unitinterval I = [0, 1] with a finite number of breakpoints, such that
I all slopes are integral powers of 2,
I all breakpoints have dyadic rational coordinates.
![Page 36: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/36.jpg)
Thompson’s group F
Thompson’s group F is the group PL2(I ), with respect tocomposition, of all piecewise-linear homeomorphisms of the unitinterval I = [0, 1] with a finite number of breakpoints, such that
I all slopes are integral powers of 2,
I all breakpoints have dyadic rational coordinates.
![Page 37: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/37.jpg)
Thompson’s group F
Thompson’s group F is the group PL2(I ), with respect tocomposition, of all piecewise-linear homeomorphisms of the unitinterval I = [0, 1] with a finite number of breakpoints, such that
I all slopes are integral powers of 2,
I all breakpoints have dyadic rational coordinates.
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Thompson’s group F
Thompson’s group F is the group PL2(I ), with respect tocomposition, of all piecewise-linear homeomorphisms of the unitinterval I = [0, 1] with a finite number of breakpoints, such that
I all slopes are integral powers of 2,
I all breakpoints have dyadic rational coordinates.
![Page 39: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/39.jpg)
Thompson’s group F
Thompson’s group F is the group PL2(I ), with respect tocomposition, of all piecewise-linear homeomorphisms of the unitinterval I = [0, 1] with a finite number of breakpoints, such that
I all slopes are integral powers of 2,
I all breakpoints have dyadic rational coordinates.
![Page 40: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/40.jpg)
Thompson’s group F
Thompson’s group F is the group PL2(I ), with respect tocomposition, of all piecewise-linear homeomorphisms of the unitinterval I = [0, 1] with a finite number of breakpoints, such that
I all slopes are integral powers of 2,
I all breakpoints have dyadic rational coordinates.
![Page 41: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/41.jpg)
Thompson’s group T
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Thompson’s group T
Similar to F , but defined on the unit circle: it preserves the cyclicorder of the intervals
![Page 43: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/43.jpg)
Thompson’s group T
Similar to F , but defined on the unit circle: it preserves the cyclicorder of the intervals
![Page 44: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/44.jpg)
Thompson’s group T
Similar to F , but defined on the unit circle: it preserves the cyclicorder of the intervals
![Page 45: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/45.jpg)
Thompson’s group V
![Page 46: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/46.jpg)
Thompson’s group V
Similar to F , but not continuous: it permutes the order of theintervals and can be seen as a group of homeomorphisms of theCantor set C to itself:
![Page 47: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/47.jpg)
Thompson’s group V
Similar to F , but not continuous: it permutes the order of theintervals and can be seen as a group of homeomorphisms of theCantor set C to itself:
![Page 48: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/48.jpg)
Multiplication of tree pairs
![Page 49: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/49.jpg)
Multiplication of tree pairs
![Page 50: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/50.jpg)
Multiplication of tree pairs
![Page 51: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/51.jpg)
Multiplication of tree pairs
![Page 52: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/52.jpg)
Multiplication of tree pairs
![Page 53: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/53.jpg)
Multiplication of tree pairs
![Page 54: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/54.jpg)
Multiplication of tree pairs
![Page 55: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/55.jpg)
Multiplication of tree pairs
![Page 56: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/56.jpg)
Multiplication of tree pairs
![Page 57: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/57.jpg)
Multiplication of tree pairs
![Page 58: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/58.jpg)
Multiplication of tree pairs
![Page 59: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/59.jpg)
Multiplication of tree pairs
![Page 60: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/60.jpg)
Multiplication of tree pairs
![Page 61: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/61.jpg)
Multiplication of tree pairs
![Page 62: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/62.jpg)
Multiplication of tree pairs
![Page 63: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/63.jpg)
The group QAut(T2,c)
T2,c is the infinite binary 2-colored tree (left = red, right = blue).
DefinitionQAut(T2,c) is the group of all maps T2,c → T2,c which respect theedge and color relation, except for possibly finitely many vertices.
![Page 64: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/64.jpg)
The group QAut(T2,c)
T2,c is the infinite binary 2-colored tree (left = red, right = blue).
DefinitionQAut(T2,c) is the group of all maps T2,c → T2,c which respect theedge and color relation, except for possibly finitely many vertices.
![Page 65: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/65.jpg)
The group QAut(T2,c)
T2,c is the infinite binary 2-colored tree (left = red, right = blue).
DefinitionQAut(T2,c) is the group of all maps T2,c → T2,c which respect theedge and color relation, except for possibly finitely many vertices.
![Page 66: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/66.jpg)
The group QAut(T2,c)
ε
0
00 01 10 11
1
000 001 010 011 100 101 110 111
ε
0
00 01 10 11
1
000 001 010 011 100 101 110 111
1
0
0000 0001 0010 0011 10 11
ε
00
000 001 01
010 011
01
010 011
![Page 67: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/67.jpg)
The group QAut(T2,c)
ε
0
00 01 10 11
1
000 001 010 011 100 101 110 111
ε
0
00 01 10 11
1
000 001 010 011 100 101 110 111
1
0
0000 0001 0010 0011 10 11
ε
00
000 001 01
010 011
01
010 011
![Page 68: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/68.jpg)
Lehnert’s conjecture
Theorem (Lehnert)
QAut(T2,c) is in coCF .
Conjecture (Lehnert)QAut(T2,c) is a universal coCF group.
![Page 69: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/69.jpg)
Lehnert’s conjecture
Theorem (Lehnert)
QAut(T2,c) is in coCF .
Conjecture (Lehnert)QAut(T2,c) is a universal coCF group.
![Page 70: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/70.jpg)
Lehnert’s conjecture
Theorem (Lehnert)
QAut(T2,c) is in coCF .
Conjecture (Lehnert)QAut(T2,c) is a universal coCF group.
![Page 71: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/71.jpg)
The relation between V and QAut(T2,c)
Theorem (Lehnert)
V ↪→ QAut(T2,c).
Our version of his proposed embedding:
I Given a tree T , regard it as a subtree of T2,c with root 0 (leftchild of the root of T2,c)
I Define a bijection ωT : {leaves of T} → {nodes of T} ∪ {ε}in the left-to-right order so the the rightmost leaf goes to ε.
I Given (D,R, σ) ∈ V define its image this way:
1. σ takes subtrees of T2,c at leaves D to those at leaves of R.
2. If n is a node of D or the root of T2,c , send it to nω−1D σωR .
Corollary (Lehnert-Schweitzer)
Thompson’s group V is in coCF .
![Page 72: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/72.jpg)
The relation between V and QAut(T2,c)
Theorem (Lehnert)
V ↪→ QAut(T2,c).
Our version of his proposed embedding:
I Given a tree T , regard it as a subtree of T2,c with root 0 (leftchild of the root of T2,c)
I Define a bijection ωT : {leaves of T} → {nodes of T} ∪ {ε}in the left-to-right order so the the rightmost leaf goes to ε.
I Given (D,R, σ) ∈ V define its image this way:
1. σ takes subtrees of T2,c at leaves D to those at leaves of R.
2. If n is a node of D or the root of T2,c , send it to nω−1D σωR .
Corollary (Lehnert-Schweitzer)
Thompson’s group V is in coCF .
![Page 73: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/73.jpg)
The relation between V and QAut(T2,c)
Theorem (Lehnert)
V ↪→ QAut(T2,c).
Our version of his proposed embedding:
I Given a tree T , regard it as a subtree of T2,c with root 0 (leftchild of the root of T2,c)
I Define a bijection ωT : {leaves of T} → {nodes of T} ∪ {ε}in the left-to-right order so the the rightmost leaf goes to ε.
I Given (D,R, σ) ∈ V define its image this way:
1. σ takes subtrees of T2,c at leaves D to those at leaves of R.
2. If n is a node of D or the root of T2,c , send it to nω−1D σωR .
Corollary (Lehnert-Schweitzer)
Thompson’s group V is in coCF .
![Page 74: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/74.jpg)
The relation between V and QAut(T2,c)
Theorem (Lehnert)
V ↪→ QAut(T2,c).
Our version of his proposed embedding:
I Given a tree T , regard it as a subtree of T2,c with root 0 (leftchild of the root of T2,c)
I Define a bijection ωT : {leaves of T} → {nodes of T} ∪ {ε}in the left-to-right order so the the rightmost leaf goes to ε.
I Given (D,R, σ) ∈ V define its image this way:
1. σ takes subtrees of T2,c at leaves D to those at leaves of R.
2. If n is a node of D or the root of T2,c , send it to nω−1D σωR .
Corollary (Lehnert-Schweitzer)
Thompson’s group V is in coCF .
![Page 75: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/75.jpg)
The relation between V and QAut(T2,c)
Theorem (Lehnert)
V ↪→ QAut(T2,c).
Our version of his proposed embedding:
I Given a tree T , regard it as a subtree of T2,c with root 0 (leftchild of the root of T2,c)
I Define a bijection ωT : {leaves of T} → {nodes of T} ∪ {ε}in the left-to-right order so the the rightmost leaf goes to ε.
I Given (D,R, σ) ∈ V define its image this way:
1. σ takes subtrees of T2,c at leaves D to those at leaves of R.
2. If n is a node of D or the root of T2,c , send it to nω−1D σωR .
Corollary (Lehnert-Schweitzer)
Thompson’s group V is in coCF .
![Page 76: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/76.jpg)
The relation between V and QAut(T2,c)
Theorem (Lehnert)
V ↪→ QAut(T2,c).
Our version of his proposed embedding:
I Given a tree T , regard it as a subtree of T2,c with root 0 (leftchild of the root of T2,c)
I Define a bijection ωT : {leaves of T} → {nodes of T} ∪ {ε}in the left-to-right order so the the rightmost leaf goes to ε.
I Given (D,R, σ) ∈ V define its image this way:
1. σ takes subtrees of T2,c at leaves D to those at leaves of R.
2. If n is a node of D or the root of T2,c , send it to nω−1D σωR .
Corollary (Lehnert-Schweitzer)
Thompson’s group V is in coCF .
![Page 77: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/77.jpg)
The relation between V and QAut(T2,c)
Theorem (Lehnert)
V ↪→ QAut(T2,c).
Our version of his proposed embedding:
I Given a tree T , regard it as a subtree of T2,c with root 0 (leftchild of the root of T2,c)
I Define a bijection ωT : {leaves of T} → {nodes of T} ∪ {ε}in the left-to-right order so the the rightmost leaf goes to ε.
I Given (D,R, σ) ∈ V define its image this way:
1. σ takes subtrees of T2,c at leaves D to those at leaves of R.
2. If n is a node of D or the root of T2,c , send it to nω−1D σωR .
Corollary (Lehnert-Schweitzer)
Thompson’s group V is in coCF .
![Page 78: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/78.jpg)
The relation between V and QAut(T2,c)
Lemma (Lehnert, Bleak-M-Neunhoffer)
If τ ∈ QAut(T2,c) there is a pair dτ = (vτ , pτ ) representing τ suchthat
I vτ ∈ V acts like τ beneath a suitable level (V -part),
I pτ is a bijection on the nodes above ( bijection part).
We call dτ a disjoint decomposition.
There are many disjoint decompositions, but we can always definea minimal one (in some sense).
![Page 79: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/79.jpg)
The relation between V and QAut(T2,c)
Lemma (Lehnert, Bleak-M-Neunhoffer)
If τ ∈ QAut(T2,c) there is a pair dτ = (vτ , pτ ) representing τ suchthat
I vτ ∈ V acts like τ beneath a suitable level (V -part),
I pτ is a bijection on the nodes above ( bijection part).
We call dτ a disjoint decomposition.
There are many disjoint decompositions, but we can always definea minimal one (in some sense).
![Page 80: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/80.jpg)
The relation between V and QAut(T2,c)
Lemma (Lehnert, Bleak-M-Neunhoffer)
If τ ∈ QAut(T2,c) there is a pair dτ = (vτ , pτ ) representing τ suchthat
I vτ ∈ V acts like τ beneath a suitable level (V -part),
I pτ is a bijection on the nodes above ( bijection part).
We call dτ a disjoint decomposition.
There are many disjoint decompositions, but we can always definea minimal one (in some sense).
![Page 81: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/81.jpg)
The relation between V and QAut(T2,c)
Lemma (Lehnert, Bleak-M-Neunhoffer)
If τ ∈ QAut(T2,c) there is a pair dτ = (vτ , pτ ) representing τ suchthat
I vτ ∈ V acts like τ beneath a suitable level (V -part),
I pτ is a bijection on the nodes above ( bijection part).
We call dτ a disjoint decomposition.
There are many disjoint decompositions, but we can always definea minimal one (in some sense).
![Page 82: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/82.jpg)
The relation between V and QAut(T2,c)
Question (Lehnert-Schweitzer)Does QAut(T2,c) embed into V ?
Theorem (Bleak-M-Neunhoffer)
Yes.
Idea of the embedding: start with τ ∈ QAut(T2,c):
I Build dτ = (vτ , pτ ) with vτ = (Dτ ,Rτ , στ ),
I Build a new tree pair (Dτ , Rτ , στ ) by “expanding vτ” suitably.
![Page 83: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/83.jpg)
The relation between V and QAut(T2,c)
Question (Lehnert-Schweitzer)Does QAut(T2,c) embed into V ?
Theorem (Bleak-M-Neunhoffer)
Yes.
Idea of the embedding: start with τ ∈ QAut(T2,c):
I Build dτ = (vτ , pτ ) with vτ = (Dτ ,Rτ , στ ),
I Build a new tree pair (Dτ , Rτ , στ ) by “expanding vτ” suitably.
![Page 84: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/84.jpg)
The relation between V and QAut(T2,c)
Question (Lehnert-Schweitzer)Does QAut(T2,c) embed into V ?
Theorem (Bleak-M-Neunhoffer)
Yes.
Idea of the embedding: start with τ ∈ QAut(T2,c):
I Build dτ = (vτ , pτ ) with vτ = (Dτ ,Rτ , στ ),
I Build a new tree pair (Dτ , Rτ , στ ) by “expanding vτ” suitably.
![Page 85: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/85.jpg)
The relation between V and QAut(T2,c)
Question (Lehnert-Schweitzer)Does QAut(T2,c) embed into V ?
Theorem (Bleak-M-Neunhoffer)
Yes.
Idea of the embedding: start with τ ∈ QAut(T2,c):
I Build dτ = (vτ , pτ ) with vτ = (Dτ ,Rτ , στ ),
I Build a new tree pair (Dτ , Rτ , στ ) by “expanding vτ” suitably.
![Page 86: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/86.jpg)
The relation between V and QAut(T2,c)
Question (Lehnert-Schweitzer)Does QAut(T2,c) embed into V ?
Theorem (Bleak-M-Neunhoffer)
Yes.
Idea of the embedding: start with τ ∈ QAut(T2,c):
I Build dτ = (vτ , pτ ) with vτ = (Dτ ,Rτ , στ ),
I Build a new tree pair (Dτ , Rτ , στ ) by “expanding vτ” suitably.
![Page 87: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/87.jpg)
The relation between V and QAut(T2,c)
Question (Lehnert-Schweitzer)Does QAut(T2,c) embed into V ?
Theorem (Bleak-M-Neunhoffer)
Yes.
Idea of the embedding: start with τ ∈ QAut(T2,c):
I Build dτ = (vτ , pτ ) with vτ = (Dτ ,Rτ , στ ),
I Build a new tree pair (Dτ , Rτ , στ ) by “expanding vτ” suitably.
![Page 88: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/88.jpg)
The relation between V and QAut(T2,c)
I Replace every node w in Ddτ by a caret (w ,wn,wp),
w
Node with address w... ... becomes a caret in tree for V element.
w
wn wp
(But, not at address w!)
I If eparent, eleft, eright are the edges attached to w , attach eleftand eright to the bottom of wn and eparent to the top of w ,
![Page 89: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/89.jpg)
The relation between V and QAut(T2,c)
I Replace every node w in Ddτ by a caret (w ,wn,wp),
w
Node with address w... ... becomes a caret in tree for V element.
w
wn wp
(But, not at address w!)
I If eparent, eleft, eright are the edges attached to w , attach eleftand eright to the bottom of wn and eparent to the top of w ,
![Page 90: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/90.jpg)
The relation between V and QAut(T2,c)
I Replace every node w in Ddτ by a caret (w ,wn,wp),
w
Node with address w... ... becomes a caret in tree for V element.
w
wn wp
(But, not at address w!)
I If eparent, eleft, eright are the edges attached to w , attach eleftand eright to the bottom of wn and eparent to the top of w ,
![Page 91: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/91.jpg)
The relation between V and QAut(T2,c)
I Replace every node w in Ddτ by a caret (w ,wn,wp),
w
Node with address w... ... becomes a caret in tree for V element.
w
wn wp
(But, not at address w!)
I If eparent, eleft, eright are the edges attached to w , attach eleftand eright to the bottom of wn and eparent to the top of w ,
![Page 92: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/92.jpg)
The relation between V and QAut(T2,c)
![Page 93: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/93.jpg)
The relation between V and QAut(T2,c)
I Apply σdτ to the n-leaves and bdτ to the p-leaves.
![Page 94: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/94.jpg)
The relation between V and QAut(T2,c)
I Apply σdτ to the n-leaves and bdτ to the p-leaves.
ε
0
00 01 10 11
1
000 001 010 011 100 101 110 111
ε
0
00 01 10 11
1
000 001 010 011 100 101 110 111
1
0
0000 0001 0010 0011 10 11
ε
00
000 001 01
010 011
01
010 011
![Page 95: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/95.jpg)
The relation between V and QAut(T2,c)
I Apply σdτ to the n-leaves and bdτ to the p-leaves.
ε
εn εp
0 1
0n 0p 1n 1p
00
n
01
00 p00 n01 p01
ε
εn εp
0 1
0n 0p 1n 1p
10
n
11
10 p10 n11 p11
a b c d
e f g
h
a
c d
h
f g
e
b
![Page 96: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/96.jpg)
The relation between V and QAut(T2,c)
I Apply σdτ to the n-leaves and bdτ to the p-leaves.
ε
εn εp
0 1
0n 0p 1n 1p
00
n
01
00 p00 n01 p01
ε
εn εp
0 1
0n 0p 1n 1p
10
n
11
10 p10 n11 p11
a b c d
e f g
h
a
c d
h
f g
e
b
Lehnert’s conjecture revisitedThompson’s group V is the universal coCF group.
![Page 97: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/97.jpg)
Work in progress on other subgroups of V
We are working on embedding other subgroups into V .
Candidates we are looking at are surface groups:
〈a1, b1, . . . , an, bn | [a1, b1] . . . [an, bn]〉 (orientable)
〈a1, . . . , an | a21 . . . a2n〉 (non-orientable)
Recall:
I finite index subgroups of surface groups are still surfacegroups,
I there exist orientable double covers of non-orientable surfaces.
Theorem (Bleak-Salazar)
Let H ≤ V . Any of its finite index overgroups is a subgroup of V .
![Page 98: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/98.jpg)
Work in progress on other subgroups of V
We are working on embedding other subgroups into V .
Candidates we are looking at are surface groups:
〈a1, b1, . . . , an, bn | [a1, b1] . . . [an, bn]〉 (orientable)
〈a1, . . . , an | a21 . . . a2n〉 (non-orientable)
Recall:
I finite index subgroups of surface groups are still surfacegroups,
I there exist orientable double covers of non-orientable surfaces.
Theorem (Bleak-Salazar)
Let H ≤ V . Any of its finite index overgroups is a subgroup of V .
![Page 99: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/99.jpg)
Work in progress on other subgroups of V
We are working on embedding other subgroups into V .
Candidates we are looking at are surface groups:
〈a1, b1, . . . , an, bn | [a1, b1] . . . [an, bn]〉 (orientable)
〈a1, . . . , an | a21 . . . a2n〉 (non-orientable)
Recall:
I finite index subgroups of surface groups are still surfacegroups,
I there exist orientable double covers of non-orientable surfaces.
Theorem (Bleak-Salazar)
Let H ≤ V . Any of its finite index overgroups is a subgroup of V .
![Page 100: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/100.jpg)
Work in progress on other subgroups of V
We are working on embedding other subgroups into V .
Candidates we are looking at are surface groups:
〈a1, b1, . . . , an, bn | [a1, b1] . . . [an, bn]〉 (orientable)
〈a1, . . . , an | a21 . . . a2n〉 (non-orientable)
Recall:
I finite index subgroups of surface groups are still surfacegroups,
I there exist orientable double covers of non-orientable surfaces.
Theorem (Bleak-Salazar)
Let H ≤ V . Any of its finite index overgroups is a subgroup of V .
![Page 101: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/101.jpg)
Work in progress on other subgroups of V
We are working on embedding other subgroups into V .
Candidates we are looking at are surface groups:
〈a1, b1, . . . , an, bn | [a1, b1] . . . [an, bn]〉 (orientable)
〈a1, . . . , an | a21 . . . a2n〉 (non-orientable)
Recall:
I finite index subgroups of surface groups are still surfacegroups,
I there exist orientable double covers of non-orientable surfaces.
Theorem (Bleak-Salazar)
Let H ≤ V . Any of its finite index overgroups is a subgroup of V .
![Page 102: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/102.jpg)
Work in progress on other subgroups of V
We are attempting to build a surface group inside V .
If one exists, then every other surface group will be in V .
QuestionDo surface groups embed in V ?
Surface groups are special cases of these Fuchsian groups:
〈a1, b1, . . . , an, bn, c1, . . . , ct , | cγ11 , . . . , cγtt ,
c−11 . . . c−1
t [a1, b1] . . . [an, bn]〉, n, s, t ≥ 0
Theorem (Fricke-Klein, Hoare-Karrass-Solitar)
Any finite index group of a Fuchsian group of the type above is aFuchsian group of the same type.
QuestionDo Fuchsian groups embed in V ?
![Page 103: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/103.jpg)
Work in progress on other subgroups of V
We are attempting to build a surface group inside V .
If one exists, then every other surface group will be in V .
QuestionDo surface groups embed in V ?
Surface groups are special cases of these Fuchsian groups:
〈a1, b1, . . . , an, bn, c1, . . . , ct , | cγ11 , . . . , cγtt ,
c−11 . . . c−1
t [a1, b1] . . . [an, bn]〉, n, s, t ≥ 0
Theorem (Fricke-Klein, Hoare-Karrass-Solitar)
Any finite index group of a Fuchsian group of the type above is aFuchsian group of the same type.
QuestionDo Fuchsian groups embed in V ?
![Page 104: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/104.jpg)
Work in progress on other subgroups of V
We are attempting to build a surface group inside V .
If one exists, then every other surface group will be in V .
QuestionDo surface groups embed in V ?
Surface groups are special cases of these Fuchsian groups:
〈a1, b1, . . . , an, bn, c1, . . . , ct , | cγ11 , . . . , cγtt ,
c−11 . . . c−1
t [a1, b1] . . . [an, bn]〉, n, s, t ≥ 0
Theorem (Fricke-Klein, Hoare-Karrass-Solitar)
Any finite index group of a Fuchsian group of the type above is aFuchsian group of the same type.
QuestionDo Fuchsian groups embed in V ?
![Page 105: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/105.jpg)
Work in progress on other subgroups of V
We are attempting to build a surface group inside V .
If one exists, then every other surface group will be in V .
QuestionDo surface groups embed in V ?
Surface groups are special cases of these Fuchsian groups:
〈a1, b1, . . . , an, bn, c1, . . . , ct , | cγ11 , . . . , cγtt ,
c−11 . . . c−1
t [a1, b1] . . . [an, bn]〉, n, s, t ≥ 0
Theorem (Fricke-Klein, Hoare-Karrass-Solitar)
Any finite index group of a Fuchsian group of the type above is aFuchsian group of the same type.
QuestionDo Fuchsian groups embed in V ?
![Page 106: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/106.jpg)
Work in progress on other subgroups of V
We are attempting to build a surface group inside V .
If one exists, then every other surface group will be in V .
QuestionDo surface groups embed in V ?
Surface groups are special cases of these Fuchsian groups:
〈a1, b1, . . . , an, bn, c1, . . . , ct , | cγ11 , . . . , cγtt ,
c−11 . . . c−1
t [a1, b1] . . . [an, bn]〉, n, s, t ≥ 0
Theorem (Fricke-Klein, Hoare-Karrass-Solitar)
Any finite index group of a Fuchsian group of the type above is aFuchsian group of the same type.
QuestionDo Fuchsian groups embed in V ?
![Page 107: Embeddings into Thompson's group V and coCF groups · 2013. 8. 15. · ZZ2 does not embed into Thompson’s group V. Closure properties of the class coCF Theorem (Holt-R over-Rees-Thomas)](https://reader035.vdocuments.net/reader035/viewer/2022071106/5fe08d948a049c689208a190/html5/thumbnails/107.jpg)
Work in progress on other subgroups of V
We are attempting to build a surface group inside V .
If one exists, then every other surface group will be in V .
QuestionDo surface groups embed in V ?
Surface groups are special cases of these Fuchsian groups:
〈a1, b1, . . . , an, bn, c1, . . . , ct , | cγ11 , . . . , cγtt ,
c−11 . . . c−1
t [a1, b1] . . . [an, bn]〉, n, s, t ≥ 0
Theorem (Fricke-Klein, Hoare-Karrass-Solitar)
Any finite index group of a Fuchsian group of the type above is aFuchsian group of the same type.
QuestionDo Fuchsian groups embed in V ?