algorithmic and asymptotic properties of groups
TRANSCRIPT
![Page 1: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/1.jpg)
Algorithmic and asymptotic properties of groups
Mark Sapir
![Page 2: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/2.jpg)
Group presentations
The object of our study - finitely generated groups given bypresentations 〈a1, ..., an | r1, r2, ...〉,
![Page 3: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/3.jpg)
Group presentations
The object of our study - finitely generated groups given bypresentations 〈a1, ..., an | r1, r2, ...〉, where ri is a word in a1, ..., an.
![Page 4: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/4.jpg)
Group presentations
The object of our study - finitely generated groups given bypresentations 〈a1, ..., an | r1, r2, ...〉, where ri is a word in a1, ..., an.That is groups generated by a1, ..., an
![Page 5: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/5.jpg)
Group presentations
The object of our study - finitely generated groups given bypresentations 〈a1, ..., an | r1, r2, ...〉, where ri is a word in a1, ..., an.That is groups generated by a1, ..., an with relationsr1 = 1, r2 = 1, ... imposed.
![Page 6: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/6.jpg)
Group presentations
The object of our study - finitely generated groups given bypresentations 〈a1, ..., an | r1, r2, ...〉, where ri is a word in a1, ..., an.That is groups generated by a1, ..., an with relationsr1 = 1, r2 = 1, ... imposed.
For example, the free Burnside group of exponent n with twogenerators is given by the presentation
![Page 7: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/7.jpg)
Group presentations
The object of our study - finitely generated groups given bypresentations 〈a1, ..., an | r1, r2, ...〉, where ri is a word in a1, ..., an.That is groups generated by a1, ..., an with relationsr1 = 1, r2 = 1, ... imposed.
For example, the free Burnside group of exponent n with twogenerators is given by the presentation
〈a, b | un = 1〉
for all words u in the alphabet a, b.
![Page 8: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/8.jpg)
Group presentations
The object of our study - finitely generated groups given bypresentations 〈a1, ..., an | r1, r2, ...〉, where ri is a word in a1, ..., an.That is groups generated by a1, ..., an with relationsr1 = 1, r2 = 1, ... imposed.
For example, the free Burnside group of exponent n with twogenerators is given by the presentation
〈a, b | un = 1〉
for all words u in the alphabet a, b.
The fundamental group of the orientable surface of genus n isgiven by the presentation
![Page 9: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/9.jpg)
Group presentations
The object of our study - finitely generated groups given bypresentations 〈a1, ..., an | r1, r2, ...〉, where ri is a word in a1, ..., an.That is groups generated by a1, ..., an with relationsr1 = 1, r2 = 1, ... imposed.
For example, the free Burnside group of exponent n with twogenerators is given by the presentation
〈a, b | un = 1〉
for all words u in the alphabet a, b.
The fundamental group of the orientable surface of genus n isgiven by the presentation
〈a1, b1, ..., an, bn | [a1, b1]...[an, bn] = 1〉.
![Page 10: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/10.jpg)
Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem)
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Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
![Page 12: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/12.jpg)
Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
Theorem. (Higman)
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Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has r.e. word problem iff
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Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has r.e. word problem iff it is asubgroup of a f.p. group.
![Page 15: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/15.jpg)
Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has r.e. word problem iff it is asubgroup of a f.p. group.
Theorem. (Adian-Novikov’s solution of Burnside problem)
![Page 16: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/16.jpg)
Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has r.e. word problem iff it is asubgroup of a f.p. group.
Theorem. (Adian-Novikov’s solution of Burnside problem) Thefree Burnside group of exponent n with at least two generators
![Page 17: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/17.jpg)
Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has r.e. word problem iff it is asubgroup of a f.p. group.
Theorem. (Adian-Novikov’s solution of Burnside problem) Thefree Burnside group of exponent n with at least two generators isinfinite for large enough odd n.
![Page 18: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/18.jpg)
Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has r.e. word problem iff it is asubgroup of a f.p. group.
Theorem. (Adian-Novikov’s solution of Burnside problem) Thefree Burnside group of exponent n with at least two generators isinfinite for large enough odd n.
Theorem. (Olshanskii’s solution of Tarski’s and von Neumann’sproblems)
![Page 19: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/19.jpg)
Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has r.e. word problem iff it is asubgroup of a f.p. group.
Theorem. (Adian-Novikov’s solution of Burnside problem) Thefree Burnside group of exponent n with at least two generators isinfinite for large enough odd n.
Theorem. (Olshanskii’s solution of Tarski’s and von Neumann’sproblems) There exists a non-Amenable group
![Page 20: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/20.jpg)
Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has r.e. word problem iff it is asubgroup of a f.p. group.
Theorem. (Adian-Novikov’s solution of Burnside problem) Thefree Burnside group of exponent n with at least two generators isinfinite for large enough odd n.
Theorem. (Olshanskii’s solution of Tarski’s and von Neumann’sproblems) There exists a non-Amenable group with all propersubgroups cyclic of the same prime order.
![Page 21: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/21.jpg)
Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has r.e. word problem iff it is asubgroup of a f.p. group.
Theorem. (Adian-Novikov’s solution of Burnside problem) Thefree Burnside group of exponent n with at least two generators isinfinite for large enough odd n.
Theorem. (Olshanskii’s solution of Tarski’s and von Neumann’sproblems) There exists a non-Amenable group with all propersubgroups cyclic of the same prime order.
Theorem. (Gromov’s solution of Milnor’s problem)
![Page 22: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/22.jpg)
Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has r.e. word problem iff it is asubgroup of a f.p. group.
Theorem. (Adian-Novikov’s solution of Burnside problem) Thefree Burnside group of exponent n with at least two generators isinfinite for large enough odd n.
Theorem. (Olshanskii’s solution of Tarski’s and von Neumann’sproblems) There exists a non-Amenable group with all propersubgroups cyclic of the same prime order.
Theorem. (Gromov’s solution of Milnor’s problem) Any group ofpolynomial growth
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Classical results
Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.
Theorem. (Higman) A group has r.e. word problem iff it is asubgroup of a f.p. group.
Theorem. (Adian-Novikov’s solution of Burnside problem) Thefree Burnside group of exponent n with at least two generators isinfinite for large enough odd n.
Theorem. (Olshanskii’s solution of Tarski’s and von Neumann’sproblems) There exists a non-Amenable group with all propersubgroups cyclic of the same prime order.
Theorem. (Gromov’s solution of Milnor’s problem) Any group ofpolynomial growth has a nilpotent subgroup of finite index.
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Groups turning into machines
Groups −→
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Groups turning into machines
Groups −→Machines −→
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Groups turning into machines
Groups −→Machines −→Groups
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Miller machineLet G = 〈X | R〉 be an f.p. group.
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Miller machineLet G = 〈X | R〉 be an f.p. group.
Consider a new group MG .
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Miller machineLet G = 〈X | R〉 be an f.p. group.
Consider a new group MG . The generators: tape letters X , stateletter q, command letters θx , θr , x ∈ X , r ∈ R.
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Miller machineLet G = 〈X | R〉 be an f.p. group.
Consider a new group MG . The generators: tape letters X , stateletter q, command letters θx , θr , x ∈ X , r ∈ R.
The defining relations:
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Miller machineLet G = 〈X | R〉 be an f.p. group.
Consider a new group MG . The generators: tape letters X , stateletter q, command letters θx , θr , x ∈ X , r ∈ R.
The defining relations:
qxθx = θxxq, qθr = θrqr , xθ = θx
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Miller machineLet G = 〈X | R〉 be an f.p. group.
Consider a new group MG . The generators: tape letters X , stateletter q, command letters θx , θr , x ∈ X , r ∈ R.
The defining relations:
qxθx = θxxq, qθr = θrqr , xθ = θx
We can draw these relations as follows.
![Page 33: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/33.jpg)
Miller machineLet G = 〈X | R〉 be an f.p. group.
Consider a new group MG . The generators: tape letters X , stateletter q, command letters θx , θr , x ∈ X , r ∈ R.
The defining relations:
qxθx = θxxq, qθr = θrqr , xθ = θx
We can draw these relations as follows.
6
- -
- -6
s s s
rss -µ6
-
r r
r r r
6
r p
p r
x
x q
θxθx θr θr
q
q r
6
x
x
θ θ
q
-
-
![Page 34: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/34.jpg)
Why is it a machine?Let us show that this is a machine
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Why is it a machine?Let us show that this is a machine checking whether a word isequal to 1 in G
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Why is it a machine?Let us show that this is a machine checking whether a word isequal to 1 in G
Start with any word uqv = y ...xq...z
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Why is it a machine?Let us show that this is a machine checking whether a word isequal to 1 in G
Start with any word uqv = y ...xq...z where uv = 1 in G
![Page 38: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/38.jpg)
Why is it a machine?Let us show that this is a machine checking whether a word isequal to 1 in G
Start with any word uqv = y ...xq...z where uv = 1 in G
. . . . . .xqy z
Figure: Deduction uqv → ... → q if uv = 1 in G . Here u = y ...x ,v = ...z .
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Why is it a machine?Let us show that this is a machine checking whether a word isequal to 1 in G
Start with any word uqv = y ...xq...z where uv = 1 in G
. . . . . .xq
qx
y zθx
θx-
-? ?
Figure: Deduction uqv → ... → q if uv = 1 in G . Here u = y ...x ,v = ...z .
![Page 40: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/40.jpg)
Why is it a machine?Let us show that this is a machine checking whether a word isequal to 1 in G
Start with any word uqv = y ...xq...z where uv = 1 in G
. . . . . .xq
qx
y
y z
zθx
θx-
-? ?
Figure: Deduction uqv → ... → q if uv = 1 in G . Here u = y ...x ,v = ...z .
![Page 41: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/41.jpg)
Why is it a machine?Let us show that this is a machine checking whether a word isequal to 1 in G
Start with any word uqv = y ...xq...z where uv = 1 in G
. . . . . .xq
qx
y
y z
zθx
θx-
-? ?
Àθrθr
xqr
Figure: Deduction uqv → ... → q if uv = 1 in G . Here u = y ...x ,v = ...z .
![Page 42: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/42.jpg)
Why is it a machine?Let us show that this is a machine checking whether a word isequal to 1 in G
Start with any word uqv = y ...xq...z where uv = 1 in G
. . . . . .xq
qx
y
y z
zθx
θx-
-? ?
Àθrθr
y x zqr
Figure: Deduction uqv → ... → q if uv = 1 in G . Here u = y ...x ,v = ...z .
![Page 43: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/43.jpg)
Why is it a machine?Let us show that this is a machine checking whether a word isequal to 1 in G
Start with any word uqv = y ...xq...z where uv = 1 in G
. . . . . .xq
qx
y
y z
zθx
θx-
-? ?
Àθrθr
y x zqr
. . .
. . .
-q
3 k
Figure: Deduction uqv → ... → q if uv = 1 in G . Here u = y ...x ,v = ...z .
![Page 44: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/44.jpg)
Why is it a machine?Let us show that this is a machine checking whether a word isequal to 1 in G
Start with any word uqv = y ...xq...z where uv = 1 in G
. . . . . .xq
qx
y
y z
zθx
θx-
-? ?
Àθrθr
y x zqr
. . .
. . .
-q
3 k
Figure: Deduction uqv → ... → q if uv = 1 in G . Here u = y ...x ,v = ...z .
Theorem.(Miller)
![Page 45: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/45.jpg)
Why is it a machine?Let us show that this is a machine checking whether a word isequal to 1 in G
Start with any word uqv = y ...xq...z where uv = 1 in G
. . . . . .xq
qx
y
y z
zθx
θx-
-? ?
Àθrθr
y x zqr
. . .
. . .
-q
3 k
Figure: Deduction uqv → ... → q if uv = 1 in G . Here u = y ...x ,v = ...z .
Theorem.(Miller) The group MG has solvable conjugacy problemiff
![Page 46: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/46.jpg)
Why is it a machine?Let us show that this is a machine checking whether a word isequal to 1 in G
Start with any word uqv = y ...xq...z where uv = 1 in G
. . . . . .xq
qx
y
y z
zθx
θx-
-? ?
Àθrθr
y x zqr
. . .
. . .
-q
3 k
Figure: Deduction uqv → ... → q if uv = 1 in G . Here u = y ...x ,v = ...z .
Theorem.(Miller) The group MG has solvable conjugacy problemiff G has solvable word problem.
![Page 47: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/47.jpg)
S-machines
Now we define machines that are groups.
![Page 48: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/48.jpg)
S-machines
Now we define machines that are groups. Here is a definition of asimplest (1-tape, no semigroup part) version of S-machines.
![Page 49: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/49.jpg)
S-machines
Now we define machines that are groups. Here is a definition of asimplest (1-tape, no semigroup part) version of S-machines.
Definition. An S-machine is an HNN extension of a free group.
![Page 50: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/50.jpg)
S-machines
Now we define machines that are groups. Here is a definition of asimplest (1-tape, no semigroup part) version of S-machines.
Definition. An S-machine is an HNN extension of a free group.Generators: tape letters X ,
![Page 51: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/51.jpg)
S-machines
Now we define machines that are groups. Here is a definition of asimplest (1-tape, no semigroup part) version of S-machines.
Definition. An S-machine is an HNN extension of a free group.Generators: tape letters X , state letters Q,
![Page 52: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/52.jpg)
S-machines
Now we define machines that are groups. Here is a definition of asimplest (1-tape, no semigroup part) version of S-machines.
Definition. An S-machine is an HNN extension of a free group.Generators: tape letters X , state letters Q, command letters Θ.
![Page 53: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/53.jpg)
S-machines
Now we define machines that are groups. Here is a definition of asimplest (1-tape, no semigroup part) version of S-machines.
Definition. An S-machine is an HNN extension of a free group.Generators: tape letters X , state letters Q, command letters Θ.
Relations are of one of the forms:
![Page 54: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/54.jpg)
S-machines
Now we define machines that are groups. Here is a definition of asimplest (1-tape, no semigroup part) version of S-machines.
Definition. An S-machine is an HNN extension of a free group.Generators: tape letters X , state letters Q, command letters Θ.
Relations are of one of the forms:
p p p
p p
- - -
-
x q′
6 6
-
-
p p p
p p
θ θ
x
θ θI µ
q x
y
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S-machines
Now we define machines that are groups. Here is a definition of asimplest (1-tape, no semigroup part) version of S-machines.
Definition. An S-machine is an HNN extension of a free group.Generators: tape letters X , state letters Q, command letters Θ.
Relations are of one of the forms:
p p p
p p
- - -
-
x q′
6 6
-
-
p p p
p p
θ θ
x
θ θI µ
q x
y
The main idea: S-machines are much easier to use as buildingblocks of groups than Turing machines.
![Page 56: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/56.jpg)
Applications: Dehn functions of groups
Definition. (Madlener-Otto, Gersten, Gromov)
![Page 57: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/57.jpg)
Applications: Dehn functions of groups
Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group,
![Page 58: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/58.jpg)
Applications: Dehn functions of groups
Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group, w be a word in X , w = 1 in G .
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Applications: Dehn functions of groups
Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group, w be a word in X , w = 1 in G . Thearea
![Page 60: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/60.jpg)
Applications: Dehn functions of groups
Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group, w be a word in X , w = 1 in G . Thearea of w is the minimal number of cells in a van Kampen diagramwith boundary label w .
![Page 61: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/61.jpg)
Applications: Dehn functions of groups
Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group, w be a word in X , w = 1 in G . Thearea of w is the minimal number of cells in a van Kampen diagramwith boundary label w . That is how long it takes to deduce theequality w = 1 from the defining relations.
![Page 62: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/62.jpg)
Applications: Dehn functions of groups
Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group, w be a word in X , w = 1 in G . Thearea of w is the minimal number of cells in a van Kampen diagramwith boundary label w . That is how long it takes to deduce theequality w = 1 from the defining relations.
Definition. (Dehn function)
![Page 63: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/63.jpg)
Applications: Dehn functions of groups
Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group, w be a word in X , w = 1 in G . Thearea of w is the minimal number of cells in a van Kampen diagramwith boundary label w . That is how long it takes to deduce theequality w = 1 from the defining relations.
Definition. (Dehn function) For any n ≥ 1 let d(n) be thelargest area of a word w of length at most n.
![Page 64: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/64.jpg)
Applications: Dehn functions of groups
Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group, w be a word in X , w = 1 in G . Thearea of w is the minimal number of cells in a van Kampen diagramwith boundary label w . That is how long it takes to deduce theequality w = 1 from the defining relations.
Definition. (Dehn function) For any n ≥ 1 let d(n) be thelargest area of a word w of length at most n.
Example. Surface group The Dehn function is linear.
![Page 65: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/65.jpg)
Applications: Dehn functions of groups
Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group, w be a word in X , w = 1 in G . Thearea of w is the minimal number of cells in a van Kampen diagramwith boundary label w . That is how long it takes to deduce theequality w = 1 from the defining relations.
Definition. (Dehn function) For any n ≥ 1 let d(n) be thelargest area of a word w of length at most n.
Example. Surface group The Dehn function is linear.
6
ºN
j
)
zπ
A typical diagram over the surface group presentation (genus > 1)
![Page 66: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/66.jpg)
Applications: Dehn functions of groups
Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group, w be a word in X , w = 1 in G . Thearea of w is the minimal number of cells in a van Kampen diagramwith boundary label w . That is how long it takes to deduce theequality w = 1 from the defining relations.
Definition. (Dehn function) For any n ≥ 1 let d(n) be thelargest area of a word w of length at most n.
Example. Surface group The Dehn function is linear.
6
ºN
j
)
zπ
A typical diagram over the surface group presentation (genus > 1)Definition.
![Page 67: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/67.jpg)
Applications: Dehn functions of groups
Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group, w be a word in X , w = 1 in G . Thearea of w is the minimal number of cells in a van Kampen diagramwith boundary label w . That is how long it takes to deduce theequality w = 1 from the defining relations.
Definition. (Dehn function) For any n ≥ 1 let d(n) be thelargest area of a word w of length at most n.
Example. Surface group The Dehn function is linear.
6
ºN
j
)
zπ
A typical diagram over the surface group presentation (genus > 1)Definition. A group is hyperbolic if its Dehn function is linear.
![Page 68: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/68.jpg)
Dehn functions
Example.
![Page 69: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/69.jpg)
Dehn functions
Example. The Dehn function of the Abelian group〈a, b | ab = ba〉
![Page 70: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/70.jpg)
Dehn functions
Example. The Dehn function of the Abelian group〈a, b | ab = ba〉 is quadratic:
![Page 71: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/71.jpg)
Dehn functions
Example. The Dehn function of the Abelian group〈a, b | ab = ba〉 is quadratic:
6
z
?
¾
b b b b
b b
b ba
a
a
a
a
a
a
a
b-band
a-band
![Page 72: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/72.jpg)
Dehn functions of S-machines
Observation. (Rips)
![Page 73: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/73.jpg)
Dehn functions of S-machines
Observation. (Rips) The Dehn function of any S-machine is atmost cubic.
![Page 74: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/74.jpg)
Dehn functions of S-machines
Observation. (Rips) The Dehn function of any S-machine is atmost cubic.
hi
story
histor
y
input
q
Area ∼ h2
r q
q q
. . .h
h
![Page 75: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/75.jpg)
Dehn functions of S-machines
Observation. (Rips) The Dehn function of any S-machine is atmost cubic.
hi
story
histor
y
input
q
Area ∼ h2
r q
q q
. . .h
h
Theorem. (Olshanskii, S.)
![Page 76: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/76.jpg)
Dehn functions of S-machines
Observation. (Rips) The Dehn function of any S-machine is atmost cubic.
hi
story
histor
y
input
q
Area ∼ h2
r q
q q
. . .h
h
Theorem. (Olshanskii, S.) There exists a f.p. group with Dehnfunction n2 log n
![Page 77: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/77.jpg)
Dehn functions of S-machines
Observation. (Rips) The Dehn function of any S-machine is atmost cubic.
hi
story
histor
y
input
q
Area ∼ h2
r q
q q
. . .h
h
Theorem. (Olshanskii, S.) There exists a f.p. group with Dehnfunction n2 log n and undecidable conjugacy problem.
![Page 78: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/78.jpg)
Dehn functions of S-machines
Observation. (Rips) The Dehn function of any S-machine is atmost cubic.
hi
story
histor
y
input
q
Area ∼ h2
r q
q q
. . .h
h
Theorem. (Olshanskii, S.) There exists a f.p. group with Dehnfunction n2 log n and undecidable conjugacy problem.
We use some (Vassiliev-type) invariants of chord diagrams
![Page 79: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/79.jpg)
Dehn functions of S-machines
Observation. (Rips) The Dehn function of any S-machine is atmost cubic.
hi
story
histor
y
input
q
Area ∼ h2
r q
q q
. . .h
h
Theorem. (Olshanskii, S.) There exists a f.p. group with Dehnfunction n2 log n and undecidable conjugacy problem.
We use some (Vassiliev-type) invariants of chord diagrams toobtain the upper bound.
![Page 80: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/80.jpg)
Corollaries about conjugacy problem
Conjecture.
![Page 81: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/81.jpg)
Corollaries about conjugacy problem
Conjecture. Every group with Dehn function <e n2 log n
![Page 82: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/82.jpg)
Corollaries about conjugacy problem
Conjecture. Every group with Dehn function <e n2 log n hassolvable conjugacy problem.
![Page 83: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/83.jpg)
Corollaries about conjugacy problem
Conjecture. Every group with Dehn function <e n2 log n hassolvable conjugacy problem.
O+S: proved for HNN extensions of free groups.
![Page 84: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/84.jpg)
Corollaries about conjugacy problem
Conjecture. Every group with Dehn function <e n2 log n hassolvable conjugacy problem.
O+S: proved for HNN extensions of free groups.
Together with a result of Bridson+Groves
![Page 85: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/85.jpg)
Corollaries about conjugacy problem
Conjecture. Every group with Dehn function <e n2 log n hassolvable conjugacy problem.
O+S: proved for HNN extensions of free groups.
Together with a result of Bridson+Groves gives a proof ofsolvability of conjugacy problem
![Page 86: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/86.jpg)
Corollaries about conjugacy problem
Conjecture. Every group with Dehn function <e n2 log n hassolvable conjugacy problem.
O+S: proved for HNN extensions of free groups.
Together with a result of Bridson+Groves gives a proof ofsolvability of conjugacy problem for cyclic extensions of free groups.
![Page 87: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/87.jpg)
Corollaries about conjugacy problem
Conjecture. Every group with Dehn function <e n2 log n hassolvable conjugacy problem.
O+S: proved for HNN extensions of free groups.
Together with a result of Bridson+Groves gives a proof ofsolvability of conjugacy problem for cyclic extensions of free groups.
An earlier proof: Bogopolski, Martino, Maslakova, Ventura.
![Page 88: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/88.jpg)
Bigger Dehn functions
Theorem. (Birget-S.)
![Page 89: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/89.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group
![Page 90: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/90.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
![Page 91: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/91.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
Theorem. (S., Birget, Rips, Ann. of Math., 2002)
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Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
Theorem. (S., Birget, Rips, Ann. of Math., 2002) If a timefunction is superadditive and ≻ n4
![Page 93: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/93.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
Theorem. (S., Birget, Rips, Ann. of Math., 2002) If a timefunction is superadditive and ≻ n4 then it is equivalent to theDehn function of a f.p. group.
![Page 94: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/94.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
Theorem. (S., Birget, Rips, Ann. of Math., 2002) If a timefunction is superadditive and ≻ n4 then it is equivalent to theDehn function of a f.p. group.
Theorem. (S.)
![Page 95: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/95.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
Theorem. (S., Birget, Rips, Ann. of Math., 2002) If a timefunction is superadditive and ≻ n4 then it is equivalent to theDehn function of a f.p. group.
Theorem. (S.) Every Turing machine is polynomially equivalentto an S-machine
![Page 96: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/96.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
Theorem. (S., Birget, Rips, Ann. of Math., 2002) If a timefunction is superadditive and ≻ n4 then it is equivalent to theDehn function of a f.p. group.
Theorem. (S.) Every Turing machine is polynomially equivalentto an S-machine with one tape and one state
![Page 97: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/97.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
Theorem. (S., Birget, Rips, Ann. of Math., 2002) If a timefunction is superadditive and ≻ n4 then it is equivalent to theDehn function of a f.p. group.
Theorem. (S.) Every Turing machine is polynomially equivalentto an S-machine with one tape and one state (Miller machine).
![Page 98: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/98.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
Theorem. (S., Birget, Rips, Ann. of Math., 2002) If a timefunction is superadditive and ≻ n4 then it is equivalent to theDehn function of a f.p. group.
Theorem. (S.) Every Turing machine is polynomially equivalentto an S-machine with one tape and one state (Miller machine).
Corollary. (S.)
![Page 99: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/99.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
Theorem. (S., Birget, Rips, Ann. of Math., 2002) If a timefunction is superadditive and ≻ n4 then it is equivalent to theDehn function of a f.p. group.
Theorem. (S.) Every Turing machine is polynomially equivalentto an S-machine with one tape and one state (Miller machine).
Corollary. (S.) For every number α ≥ 4 that is computable intime ≤ 22
n
,
![Page 100: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/100.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
Theorem. (S., Birget, Rips, Ann. of Math., 2002) If a timefunction is superadditive and ≻ n4 then it is equivalent to theDehn function of a f.p. group.
Theorem. (S.) Every Turing machine is polynomially equivalentto an S-machine with one tape and one state (Miller machine).
Corollary. (S.) For every number α ≥ 4 that is computable intime ≤ 22
n
, there exists a f.p. group with Dehn function nα.
![Page 101: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/101.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
Theorem. (S., Birget, Rips, Ann. of Math., 2002) If a timefunction is superadditive and ≻ n4 then it is equivalent to theDehn function of a f.p. group.
Theorem. (S.) Every Turing machine is polynomially equivalentto an S-machine with one tape and one state (Miller machine).
Corollary. (S.) For every number α ≥ 4 that is computable intime ≤ 22
n
, there exists a f.p. group with Dehn function nα.Conversely, every number in the isoperimetric spectrum
![Page 102: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/102.jpg)
Bigger Dehn functions
Theorem. (Birget-S.) Every Dehn function of a f.p. group isequivalent to the time function of a non-deterministic Turingmachine.
Theorem. (S., Birget, Rips, Ann. of Math., 2002) If a timefunction is superadditive and ≻ n4 then it is equivalent to theDehn function of a f.p. group.
Theorem. (S.) Every Turing machine is polynomially equivalentto an S-machine with one tape and one state (Miller machine).
Corollary. (S.) For every number α ≥ 4 that is computable intime ≤ 22
n
, there exists a f.p. group with Dehn function nα.Conversely, every number in the isoperimetric spectrum is
computable in time ≤ 222n
.
![Page 103: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/103.jpg)
Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie)
![Page 104: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/104.jpg)
Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞
![Page 105: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/105.jpg)
Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
![Page 106: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/106.jpg)
Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov)
![Page 107: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/107.jpg)
Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connected
![Page 108: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/108.jpg)
Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
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Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
Papasoglu: The converse statement is true if the Dehn function isquadratic.
![Page 110: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/110.jpg)
Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
Papasoglu: The converse statement is true if the Dehn function isquadratic.
Theorem. (O+S, solving a problem of Drutu)
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Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
Papasoglu: The converse statement is true if the Dehn function isquadratic.
Theorem. (O+S, solving a problem of Drutu) There are f.p.groups with non-simply connected asymptotic cones
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Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
Papasoglu: The converse statement is true if the Dehn function isquadratic.
Theorem. (O+S, solving a problem of Drutu) There are f.p.groups with non-simply connected asymptotic cones but Dehnfunctions arbitrary close to n2
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Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
Papasoglu: The converse statement is true if the Dehn function isquadratic.
Theorem. (O+S, solving a problem of Drutu) There are f.p.groups with non-simply connected asymptotic cones but Dehnfunctions arbitrary close to n2 and linear isodiametric function.
![Page 114: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/114.jpg)
Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
Papasoglu: The converse statement is true if the Dehn function isquadratic.
Theorem. (O+S, solving a problem of Drutu) There are f.p.groups with non-simply connected asymptotic cones but Dehnfunctions arbitrary close to n2 and linear isodiametric function.
Theorem (O+S)
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Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
Papasoglu: The converse statement is true if the Dehn function isquadratic.
Theorem. (O+S, solving a problem of Drutu) There are f.p.groups with non-simply connected asymptotic cones but Dehnfunctions arbitrary close to n2 and linear isodiametric function.
Theorem (O+S) There are f.p. groups with undecidable wordproblem
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Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
Papasoglu: The converse statement is true if the Dehn function isquadratic.
Theorem. (O+S, solving a problem of Drutu) There are f.p.groups with non-simply connected asymptotic cones but Dehnfunctions arbitrary close to n2 and linear isodiametric function.
Theorem (O+S) There are f.p. groups with undecidable wordproblem but Dehn function f (n) bounded by Cn2 for infinitelymany n’s.
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Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
Papasoglu: The converse statement is true if the Dehn function isquadratic.
Theorem. (O+S, solving a problem of Drutu) There are f.p.groups with non-simply connected asymptotic cones but Dehnfunctions arbitrary close to n2 and linear isodiametric function.
Theorem (O+S) There are f.p. groups with undecidable wordproblem but Dehn function f (n) bounded by Cn2 for infinitelymany n’s. Such a group has at least two non-homeomorphic a. c.:
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Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
Papasoglu: The converse statement is true if the Dehn function isquadratic.
Theorem. (O+S, solving a problem of Drutu) There are f.p.groups with non-simply connected asymptotic cones but Dehnfunctions arbitrary close to n2 and linear isodiametric function.
Theorem (O+S) There are f.p. groups with undecidable wordproblem but Dehn function f (n) bounded by Cn2 for infinitelymany n’s. Such a group has at least two non-homeomorphic a. c.:one simply connected and one non-simply connected.
![Page 119: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/119.jpg)
Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
Papasoglu: The converse statement is true if the Dehn function isquadratic.
Theorem. (O+S, solving a problem of Drutu) There are f.p.groups with non-simply connected asymptotic cones but Dehnfunctions arbitrary close to n2 and linear isodiametric function.
Theorem (O+S) There are f.p. groups with undecidable wordproblem but Dehn function f (n) bounded by Cn2 for infinitelymany n’s. Such a group has at least two non-homeomorphic a. c.:one simply connected and one non-simply connected.
Kramer, Shelah, Tent, Thomas:
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Asymptotic cones
Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞ is a Gromov-Hausdorff limit of the spaces G/dn.
Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.
Papasoglu: The converse statement is true if the Dehn function isquadratic.
Theorem. (O+S, solving a problem of Drutu) There are f.p.groups with non-simply connected asymptotic cones but Dehnfunctions arbitrary close to n2 and linear isodiametric function.
Theorem (O+S) There are f.p. groups with undecidable wordproblem but Dehn function f (n) bounded by Cn2 for infinitelymany n’s. Such a group has at least two non-homeomorphic a. c.:one simply connected and one non-simply connected.
Kramer, Shelah, Tent, Thomas: assuming CH is not true.
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Embeddings
Theorem. (Birget, Rips, Olshanskii, S., Ann. of Math.,2002)
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Embeddings
Theorem. (Birget, Rips, Olshanskii, S., Ann. of Math.,2002) A finitely generated group has word problem in NP
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Embeddings
Theorem. (Birget, Rips, Olshanskii, S., Ann. of Math.,2002) A finitely generated group has word problem in NP iff it isinside a finitely presented group with polynomial Dehn function.
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Embeddings
Theorem. (Birget, Rips, Olshanskii, S., Ann. of Math.,2002) A finitely generated group has word problem in NP iff it isinside a finitely presented group with polynomial Dehn function.
Corollary.
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Embeddings
Theorem. (Birget, Rips, Olshanskii, S., Ann. of Math.,2002) A finitely generated group has word problem in NP iff it isinside a finitely presented group with polynomial Dehn function.
Corollary. There exists an NP-complete f.p. group.
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Embeddings
Theorem. (Birget, Rips, Olshanskii, S., Ann. of Math.,2002) A finitely generated group has word problem in NP iff it isinside a finitely presented group with polynomial Dehn function.
Corollary. There exists an NP-complete f.p. group.
Corollary.
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Embeddings
Theorem. (Birget, Rips, Olshanskii, S., Ann. of Math.,2002) A finitely generated group has word problem in NP iff it isinside a finitely presented group with polynomial Dehn function.
Corollary. There exists an NP-complete f.p. group.
Corollary. If a word problem in a f.g. group can be solved inNP-time by a smart algorithm,
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Embeddings
Theorem. (Birget, Rips, Olshanskii, S., Ann. of Math.,2002) A finitely generated group has word problem in NP iff it isinside a finitely presented group with polynomial Dehn function.
Corollary. There exists an NP-complete f.p. group.
Corollary. If a word problem in a f.g. group can be solved inNP-time by a smart algorithm, it can be solved in NP-time by theobvious algorithm
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Embeddings
Theorem. (Birget, Rips, Olshanskii, S., Ann. of Math.,2002) A finitely generated group has word problem in NP iff it isinside a finitely presented group with polynomial Dehn function.
Corollary. There exists an NP-complete f.p. group.
Corollary. If a word problem in a f.g. group can be solved inNP-time by a smart algorithm, it can be solved in NP-time by theobvious algorithm involving relations of a bigger group.
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Idea of the proof
k1
k2 k2
k3 k3
qu
q qq′ q′
q′u′ q′u′
disc 1 disc 2
hub hub
qu → q iff u =G 1 qu → q iff u =G 1
k1q
u =G 1 iff u =G ′ 1
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Collins’ problem
The conjugacy problem is much harder to preserve underembeddings.
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Collins’ problem
The conjugacy problem is much harder to preserve underembeddings.
Collins-Miller and Gorjaga-Kirkinskiı: even subgroups of index 2
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Collins’ problem
The conjugacy problem is much harder to preserve underembeddings.
Collins-Miller and Gorjaga-Kirkinskiı: even subgroups of index 2 offinitely presented groups do not inherit solvability or unsolvabilityof the conjugacy problem.
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Collins’ problem
The conjugacy problem is much harder to preserve underembeddings.
Collins-Miller and Gorjaga-Kirkinskiı: even subgroups of index 2 offinitely presented groups do not inherit solvability or unsolvabilityof the conjugacy problem.
D. Collins (1976)
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Collins’ problem
The conjugacy problem is much harder to preserve underembeddings.
Collins-Miller and Gorjaga-Kirkinskiı: even subgroups of index 2 offinitely presented groups do not inherit solvability or unsolvabilityof the conjugacy problem.
D. Collins (1976) Does there exist a version of the Higman
embedding theorem in which the degree of unsolvability of the
conjugacy problem is preserved?
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Collins’ problem
The conjugacy problem is much harder to preserve underembeddings.
Collins-Miller and Gorjaga-Kirkinskiı: even subgroups of index 2 offinitely presented groups do not inherit solvability or unsolvabilityof the conjugacy problem.
D. Collins (1976) Does there exist a version of the Higman
embedding theorem in which the degree of unsolvability of the
conjugacy problem is preserved?
Theorem (O+S, Memoirs of AMS, 2004)
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Collins’ problem
The conjugacy problem is much harder to preserve underembeddings.
Collins-Miller and Gorjaga-Kirkinskiı: even subgroups of index 2 offinitely presented groups do not inherit solvability or unsolvabilityof the conjugacy problem.
D. Collins (1976) Does there exist a version of the Higman
embedding theorem in which the degree of unsolvability of the
conjugacy problem is preserved?
Theorem (O+S, Memoirs of AMS, 2004) A finitely generatedgroup H has solvable conjugacy problem
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Collins’ problem
The conjugacy problem is much harder to preserve underembeddings.
Collins-Miller and Gorjaga-Kirkinskiı: even subgroups of index 2 offinitely presented groups do not inherit solvability or unsolvabilityof the conjugacy problem.
D. Collins (1976) Does there exist a version of the Higman
embedding theorem in which the degree of unsolvability of the
conjugacy problem is preserved?
Theorem (O+S, Memoirs of AMS, 2004) A finitely generatedgroup H has solvable conjugacy problem if and only if it is Frattiniembedded into a finitely presented group G with solvableconjugacy problem.
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Collins problem: the construction
◮ Embed H into a finitely presented group H1.
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Collins problem: the construction
◮ Embed H into a finitely presented group H1.
◮ Use the Miller S-machine M(H1) to solve the word problem inH.
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Collins problem: the construction
◮ Embed H into a finitely presented group H1.
◮ Use the Miller S-machine M(H1) to solve the word problem inH.
◮ Use Boone-Novikov to make a part of M(H1) act as TM.
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Collins problem: the construction
◮ Embed H into a finitely presented group H1.
◮ Use the Miller S-machine M(H1) to solve the word problem inH.
◮ Use Boone-Novikov to make a part of M(H1) act as TM.
◮ Embed H into a f.p. group G using the new machine.
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Collins problem: the construction
◮ Embed H into a finitely presented group H1.
◮ Use the Miller S-machine M(H1) to solve the word problem inH.
◮ Use Boone-Novikov to make a part of M(H1) act as TM.
◮ Embed H into a f.p. group G using the new machine.
◮ Use Makanin-Razborov to analyze conjugacy problem fortrapezia.
![Page 144: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/144.jpg)
Collins problem: the construction
◮ Embed H into a finitely presented group H1.
◮ Use the Miller S-machine M(H1) to solve the word problem inH.
◮ Use Boone-Novikov to make a part of M(H1) act as TM.
◮ Embed H into a f.p. group G using the new machine.
◮ Use Makanin-Razborov to analyze conjugacy problem fortrapezia.
◮ Analyze annular diagrams to solve conjugacy problem.
![Page 145: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/145.jpg)
Collins problem: the construction
◮ Embed H into a finitely presented group H1.
◮ Use the Miller S-machine M(H1) to solve the word problem inH.
◮ Use Boone-Novikov to make a part of M(H1) act as TM.
◮ Embed H into a f.p. group G using the new machine.
◮ Use Makanin-Razborov to analyze conjugacy problem fortrapezia.
◮ Analyze annular diagrams to solve conjugacy problem.
Problem.
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Collins problem: the construction
◮ Embed H into a finitely presented group H1.
◮ Use the Miller S-machine M(H1) to solve the word problem inH.
◮ Use Boone-Novikov to make a part of M(H1) act as TM.
◮ Embed H into a f.p. group G using the new machine.
◮ Use Makanin-Razborov to analyze conjugacy problem fortrapezia.
◮ Analyze annular diagrams to solve conjugacy problem.
Problem. Is there a version of Higman embedding preserving thecomplexity of conjugacy problem?
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von Neumann’s problem: history
Hausdorff, Banach, Tarski:
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von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand
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von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.
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von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann:
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von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann: The reason is that the group of isometries of theball
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von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann: The reason is that the group of isometries of theball is not amenable,
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von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann: The reason is that the group of isometries of theball is not amenable, i.e. there is no finitely additive left invariantprobability measure on the set of all subsets of the group.
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von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann: The reason is that the group of isometries of theball is not amenable, i.e. there is no finitely additive left invariantprobability measure on the set of all subsets of the group.
The reason for non-amenability:
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von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann: The reason is that the group of isometries of theball is not amenable, i.e. there is no finitely additive left invariantprobability measure on the set of all subsets of the group.
The reason for non-amenability: existence of non-cyclic freesubgroups.
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von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann: The reason is that the group of isometries of theball is not amenable, i.e. there is no finitely additive left invariantprobability measure on the set of all subsets of the group.
The reason for non-amenability: existence of non-cyclic freesubgroups.
Another definition. (Gromov):
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von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann: The reason is that the group of isometries of theball is not amenable, i.e. there is no finitely additive left invariantprobability measure on the set of all subsets of the group.
The reason for non-amenability: existence of non-cyclic freesubgroups.
Another definition. (Gromov):r
r
r
r
r
r
rr
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von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann: The reason is that the group of isometries of theball is not amenable, i.e. there is no finitely additive left invariantprobability measure on the set of all subsets of the group.
The reason for non-amenability: existence of non-cyclic freesubgroups.
Another definition. (Gromov):r
r
r
r
r
r
rr
Question. (Beatles)
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von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann: The reason is that the group of isometries of theball is not amenable, i.e. there is no finitely additive left invariantprobability measure on the set of all subsets of the group.
The reason for non-amenability: existence of non-cyclic freesubgroups.
Another definition. (Gromov):r
r
r
r
r
r
rr
Question. (Beatles) Why can’t we do it in the road?
![Page 160: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/160.jpg)
von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann: The reason is that the group of isometries of theball is not amenable, i.e. there is no finitely additive left invariantprobability measure on the set of all subsets of the group.
The reason for non-amenability: existence of non-cyclic freesubgroups.
Another definition. (Gromov):r
r
r
r
r
r
rr
Question. (Beatles) Why can’t we do it in the road?Answer:
![Page 161: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/161.jpg)
von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann: The reason is that the group of isometries of theball is not amenable, i.e. there is no finitely additive left invariantprobability measure on the set of all subsets of the group.
The reason for non-amenability: existence of non-cyclic freesubgroups.
Another definition. (Gromov):r
r
r
r
r
r
rr
Question. (Beatles) Why can’t we do it in the road?Answer: Because R is amenable.
![Page 162: Algorithmic and asymptotic properties of groups](https://reader035.vdocuments.mx/reader035/viewer/2022070605/62c296ee87cee954090a1e68/html5/thumbnails/162.jpg)
von Neumann’s problem: history
Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann: The reason is that the group of isometries of theball is not amenable, i.e. there is no finitely additive left invariantprobability measure on the set of all subsets of the group.
The reason for non-amenability: existence of non-cyclic freesubgroups.
Another definition. (Gromov):r
r
r
r
r
r
rr
Question. (Beatles) Why can’t we do it in the road?Answer: Because R is amenable.
s r
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von Neumann’s problem
Problem (von Neumann-Day, 50s)
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von Neumann’s problem
Problem (von Neumann-Day, 50s) Is there a non-amenablegroup without non-cyclic free subgroups?
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von Neumann’s problem
Problem (von Neumann-Day, 50s) Is there a non-amenablegroup without non-cyclic free subgroups?
Solved in the 80s:
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von Neumann’s problem
Problem (von Neumann-Day, 50s) Is there a non-amenablegroup without non-cyclic free subgroups?
Solved in the 80s: Olshanskii (Tarski monster),
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von Neumann’s problem
Problem (von Neumann-Day, 50s) Is there a non-amenablegroup without non-cyclic free subgroups?
Solved in the 80s: Olshanskii (Tarski monster), Adian (the freeBurnside groups of large enough exponent).
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Grigorchuk-Cohen problem
Problem.
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Grigorchuk-Cohen problem
Problem. Is there a finitely presented counterexample to vonNeumann’s problem?
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Grigorchuk-Cohen problem
Problem. Is there a finitely presented counterexample to vonNeumann’s problem?
Theorem. (Olshanskii, S., Publ. IHES, 2002)
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Grigorchuk-Cohen problem
Problem. Is there a finitely presented counterexample to vonNeumann’s problem?
Theorem. (Olshanskii, S., Publ. IHES, 2002) For everysufficiently large odd n,
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Grigorchuk-Cohen problem
Problem. Is there a finitely presented counterexample to vonNeumann’s problem?
Theorem. (Olshanskii, S., Publ. IHES, 2002) For everysufficiently large odd n, there exists a finitely presented group G:
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Grigorchuk-Cohen problem
Problem. Is there a finitely presented counterexample to vonNeumann’s problem?
Theorem. (Olshanskii, S., Publ. IHES, 2002) For everysufficiently large odd n, there exists a finitely presented group G:
1. G is an ascending HNN extension of a finitely generatedinfinite group H of exponent n.
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Grigorchuk-Cohen problem
Problem. Is there a finitely presented counterexample to vonNeumann’s problem?
Theorem. (Olshanskii, S., Publ. IHES, 2002) For everysufficiently large odd n, there exists a finitely presented group G:
1. G is an ascending HNN extension of a finitely generatedinfinite group H of exponent n.
2. H is an “almost” finitely presented bounded torsion group.
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Grigorchuk-Cohen problem
Problem. Is there a finitely presented counterexample to vonNeumann’s problem?
Theorem. (Olshanskii, S., Publ. IHES, 2002) For everysufficiently large odd n, there exists a finitely presented group G:
1. G is an ascending HNN extension of a finitely generatedinfinite group H of exponent n.
2. H is an “almost” finitely presented bounded torsion group.
3. G contains a subgroup isomorphic to a free Burnside group ofexponent n with 2 generators.
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Grigorchuk-Cohen problem
Problem. Is there a finitely presented counterexample to vonNeumann’s problem?
Theorem. (Olshanskii, S., Publ. IHES, 2002) For everysufficiently large odd n, there exists a finitely presented group G:
1. G is an ascending HNN extension of a finitely generatedinfinite group H of exponent n.
2. H is an “almost” finitely presented bounded torsion group.
3. G contains a subgroup isomorphic to a free Burnside group ofexponent n with 2 generators.
4. G is a non-amenable finitely presented group without freenon-cyclic subgroups.
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Grigorchuk-Cohen problem
Problem. Is there a finitely presented counterexample to vonNeumann’s problem?
Theorem. (Olshanskii, S., Publ. IHES, 2002) For everysufficiently large odd n, there exists a finitely presented group G:
1. G is an ascending HNN extension of a finitely generatedinfinite group H of exponent n.
2. H is an “almost” finitely presented bounded torsion group.
3. G contains a subgroup isomorphic to a free Burnside group ofexponent n with 2 generators.
4. G is a non-amenable finitely presented group without freenon-cyclic subgroups.
The proof uses all the ideas mentioned above
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Grigorchuk-Cohen problem
Problem. Is there a finitely presented counterexample to vonNeumann’s problem?
Theorem. (Olshanskii, S., Publ. IHES, 2002) For everysufficiently large odd n, there exists a finitely presented group G:
1. G is an ascending HNN extension of a finitely generatedinfinite group H of exponent n.
2. H is an “almost” finitely presented bounded torsion group.
3. G contains a subgroup isomorphic to a free Burnside group ofexponent n with 2 generators.
4. G is a non-amenable finitely presented group without freenon-cyclic subgroups.
The proof uses all the ideas mentioned above plus the Olshanskiitheory of graded diagrams.
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Grigorchuk-Cohen problem
Problem. Is there a finitely presented counterexample to vonNeumann’s problem?
Theorem. (Olshanskii, S., Publ. IHES, 2002) For everysufficiently large odd n, there exists a finitely presented group G:
1. G is an ascending HNN extension of a finitely generatedinfinite group H of exponent n.
2. H is an “almost” finitely presented bounded torsion group.
3. G contains a subgroup isomorphic to a free Burnside group ofexponent n with 2 generators.
4. G is a non-amenable finitely presented group without freenon-cyclic subgroups.
The proof uses all the ideas mentioned above plus the Olshanskiitheory of graded diagrams.
Problem.
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Grigorchuk-Cohen problem
Problem. Is there a finitely presented counterexample to vonNeumann’s problem?
Theorem. (Olshanskii, S., Publ. IHES, 2002) For everysufficiently large odd n, there exists a finitely presented group G:
1. G is an ascending HNN extension of a finitely generatedinfinite group H of exponent n.
2. H is an “almost” finitely presented bounded torsion group.
3. G contains a subgroup isomorphic to a free Burnside group ofexponent n with 2 generators.
4. G is a non-amenable finitely presented group without freenon-cyclic subgroups.
The proof uses all the ideas mentioned above plus the Olshanskiitheory of graded diagrams.
Problem. Is there a finitely presented torsion group?