Algorithmic and asymptotic properties of groups

Algorithmic and asymptotic properties of groups

Mark Sapir

Group presentations

The object of our study - finitely generated groups given bypresentations 〈a1, ..., an | r1, r2, ...〉,

Group presentations

The object of our study - finitely generated groups given bypresentations 〈a1, ..., an | r1, r2, ...〉, where ri is a word in a1, ..., an.

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

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.

Group presentations


For example, the free Burnside group of exponent n with twogenerators is given by the presentation

Group presentations



〈a, b | un = 1〉

for all words u in the alphabet a, b.

Group presentations



〈a, b | un = 1〉


The fundamental group of the orientable surface of genus n isgiven by the presentation

Group presentations



〈a, b | un = 1〉


The fundamental group of the orientable surface of genus n isgiven by the presentation

〈a1, b1, ..., an, bn | [a1, b1]...[an, bn] = 1〉.

Classical results

Theorem. (Boone-Novikov’s solution of Dehn’s problem)

Classical results

Theorem. (Boone-Novikov’s solution of Dehn’s problem) Thereexists a finitely presented group with undecidable word problem.

Classical results


Theorem. (Higman)

Classical results


Theorem. (Higman) A group has r.e. word problem iff

Classical results


Theorem. (Higman) A group has r.e. word problem iff it is asubgroup of a f.p. group.

Classical results



Theorem. (Adian-Novikov’s solution of Burnside problem)

Classical results



Theorem. (Adian-Novikov’s solution of Burnside problem) Thefree Burnside group of exponent n with at least two generators

Classical results



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.

Classical results




Theorem. (Olshanskii’s solution of Tarski’s and von Neumann’sproblems)

Classical results




Theorem. (Olshanskii’s solution of Tarski’s and von Neumann’sproblems) There exists a non-Amenable group

Classical results




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.

Classical results





Theorem. (Gromov’s solution of Milnor’s problem)

Classical results





Theorem. (Gromov’s solution of Milnor’s problem) Any group ofpolynomial growth

Classical results





Theorem. (Gromov’s solution of Milnor’s problem) Any group ofpolynomial growth has a nilpotent subgroup of finite index.

Groups turning into machines

Groups −→


Groups −→Machines −→


Groups −→Machines −→Groups

Miller machineLet G = 〈X | R〉 be an f.p. group.


Consider a new group MG .


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.





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

-

-

Why is it a machine?Let us show that this is a machine

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


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 .



. . . . . .xq

qx

y zθx

θx-

-? ?




. . . . . .xq

qx

y

y z

zθx

θx-

-? ?




. . . . . .xq

qx

y

y z

zθx

θx-

-? ?

Àθrθr

xqr




. . . . . .xq

qx

y

y z

zθx

θx-

-? ?

Àθrθr

y x zqr




. . . . . .xq

qx

y

y z

zθx

θx-

-? ?

Àθrθr

y x zqr

. . .

. . .

-q

3 k




. . . . . .xq

qx

y

y z

zθx

θx-

-? ?

Àθrθr

y x zqr

. . .

. . .

-q

3 k


Theorem.(Miller)



. . . . . .xq

qx

y

y z

zθx

θx-

-? ?

Àθrθr

y x zqr

. . .

. . .

-q

3 k


Theorem.(Miller) The group MG has solvable conjugacy problemiff



. . . . . .xq

qx

y

y z

zθx

θx-

-? ?

Àθrθr

y x zqr

. . .

. . .

-q

3 k


Theorem.(Miller) The group MG has solvable conjugacy problemiff G has solvable word problem.

S-machines

Now we define machines that are groups.

S-machines

Now we define machines that are groups. Here is a definition of asimplest (1-tape, no semigroup part) version of S-machines.

S-machines


Definition. An S-machine is an HNN extension of a free group.

S-machines


Definition. An S-machine is an HNN extension of a free group.Generators: tape letters X ,

S-machines


Definition. An S-machine is an HNN extension of a free group.Generators: tape letters X , state letters Q,

S-machines


Definition. An S-machine is an HNN extension of a free group.Generators: tape letters X , state letters Q, command letters Θ.

S-machines



Relations are of one of the forms:

S-machines




p p p

p p

- - -

-

x q′

6 6

-

-

p p p

p p

θ θ

x

θ θI µ

q x

y

S-machines




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.

Applications: Dehn functions of groups

Definition. (Madlener-Otto, Gersten, Gromov)


Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group,


Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group, w be a word in X , w = 1 in G .


Definition. (Madlener-Otto, Gersten, Gromov) LetG = 〈X | R〉 be a f.p. group, w be a word in X , w = 1 in G . Thearea


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 .


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)



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)





6

ºN

j

)

zπ

A typical diagram over the surface group presentation (genus > 1)Definition.





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.

Dehn functions

Example.

Dehn functions

Example. The Dehn function of the Abelian group〈a, b | ab = ba〉

Dehn functions

Example. The Dehn function of the Abelian group〈a, b | ab = ba〉 is quadratic:

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

Dehn functions of S-machines

Observation. (Rips)


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



hi

story

histor

y

input

q

Area ∼ h2

r q

q q

. . .h

h

Theorem. (Olshanskii, S.)



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



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.



hi

story

histor

y

input

q

Area ∼ h2

r q

q q

. . .h

h


We use some (Vassiliev-type) invariants of chord diagrams



hi

story

histor

y

input

q

Area ∼ h2

r q

q q

. . .h

h


We use some (Vassiliev-type) invariants of chord diagrams toobtain the upper bound.

Corollaries about conjugacy problem

Conjecture.


Conjecture. Every group with Dehn function <e n2 log n


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




Together with a result of Bridson+Groves gives a proof ofsolvability of conjugacy problem




Together with a result of Bridson+Groves gives a proof ofsolvability of conjugacy problem for cyclic 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.

Bigger Dehn functions

Theorem. (Birget-S.)


Theorem. (Birget-S.) Every Dehn function of a f.p. group


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)



Theorem. (S., Birget, Rips, Ann. of Math., 2002) If a timefunction is superadditive and ≻ n4



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.)




Theorem. (S.) Every Turing machine is polynomially equivalentto an S-machine




Theorem. (S.) Every Turing machine is polynomially equivalentto an S-machine with one tape and one state




Theorem. (S.) Every Turing machine is polynomially equivalentto an S-machine with one tape and one state (Miller machine).





Corollary. (S.)





Corollary. (S.) For every number α ≥ 4 that is computable intime ≤ 22

n

,






n

, there exists a f.p. group with Dehn function nα.






n

, there exists a f.p. group with Dehn function nα.Conversely, every number in the isoperimetric spectrum






n

, there exists a f.p. group with Dehn function nα.Conversely, every number in the isoperimetric spectrum is

computable in time ≤ 222n

.

Asymptotic cones

Definition. (Gromov, van den Dries - Wilkie)

Asymptotic cones

Definition. (Gromov, van den Dries - Wilkie) An asymptoticcone of a group G corresponding to a sequences of scalarsdn → ∞

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.

Asymptotic cones


Theorem. (Gromov)

Asymptotic cones


Theorem. (Gromov) If all a. c. of a group are simply connected

Asymptotic cones


Theorem. (Gromov) If all a. c. of a group are simply connectedthen the group has polynomial Dehn function and linearisodiametric function.

Asymptotic cones



Papasoglu: The converse statement is true if the Dehn function isquadratic.

Asymptotic cones




Theorem. (O+S, solving a problem of Drutu)

Asymptotic cones




Theorem. (O+S, solving a problem of Drutu) There are f.p.groups with non-simply connected asymptotic cones

Asymptotic cones




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

Asymptotic cones




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.

Asymptotic cones





Theorem (O+S)

Asymptotic cones





Theorem (O+S) There are f.p. groups with undecidable wordproblem

Asymptotic cones





Theorem (O+S) There are f.p. groups with undecidable wordproblem but Dehn function f (n) bounded by Cn2 for infinitelymany n’s.

Asymptotic cones





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.:

Asymptotic cones





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.

Asymptotic cones






Kramer, Shelah, Tent, Thomas:

Asymptotic cones






Kramer, Shelah, Tent, Thomas: assuming CH is not true.

Embeddings

Theorem. (Birget, Rips, Olshanskii, S., Ann. of Math.,2002)

Embeddings

Theorem. (Birget, Rips, Olshanskii, S., Ann. of Math.,2002) A finitely generated group has word problem in NP

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.

Embeddings


Corollary.

Embeddings


Corollary. There exists an NP-complete f.p. group.

Embeddings



Corollary.

Embeddings



Corollary. If a word problem in a f.g. group can be solved inNP-time by a smart algorithm,

Embeddings



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

Embeddings



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.

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

Collins’ problem

The conjugacy problem is much harder to preserve underembeddings.

Collins’ problem


Collins-Miller and Gorjaga-Kirkinskiı: even subgroups of index 2

Collins’ problem


Collins-Miller and Gorjaga-Kirkinskiı: even subgroups of index 2 offinitely presented groups do not inherit solvability or unsolvabilityof the conjugacy problem.

Collins’ problem



D. Collins (1976)

Collins’ 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?

Collins’ problem






Theorem (O+S, Memoirs of AMS, 2004)

Collins’ problem






Theorem (O+S, Memoirs of AMS, 2004) A finitely generatedgroup H has solvable conjugacy problem

Collins’ problem






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.

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.








Problem. Is there a version of Higman embedding preserving thecomplexity of conjugacy problem?

von Neumann’s problem: history

Hausdorff, Banach, Tarski:


Hausdorff, Banach, Tarski: One can cut a ball into several piecesand


Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.


Hausdorff, Banach, Tarski: One can cut a ball into several piecesand move them to assemble two balls of the same size.von Neumann:


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


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,


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:



The reason for non-amenability: existence of non-cyclic freesubgroups.




Another definition. (Gromov):




Another definition. (Gromov):r

r

r

r

r

r

rr





r

r

r

r

r

rr

Question. (Beatles)





r

r

r

r

r

rr

Question. (Beatles) Why can’t we do it in the road?





r

r

r

r

r

rr

Question. (Beatles) Why can’t we do it in the road?Answer:





r

r

r

r

r

rr

Question. (Beatles) Why can’t we do it in the road?Answer: Because R is amenable.





r

r

r

r

r

rr

Question. (Beatles) Why can’t we do it in the road?Answer: Because R is amenable.

s r

von Neumann’s problem

Problem (von Neumann-Day, 50s)


Problem (von Neumann-Day, 50s) Is there a non-amenablegroup without non-cyclic free subgroups?



Solved in the 80s:



Solved in the 80s: Olshanskii (Tarski monster),



Solved in the 80s: Olshanskii (Tarski monster), Adian (the freeBurnside groups of large enough exponent).

Grigorchuk-Cohen problem

Problem.


Problem. Is there a finitely presented counterexample to vonNeumann’s problem?



Theorem. (Olshanskii, S., Publ. IHES, 2002)



Theorem. (Olshanskii, S., Publ. IHES, 2002) For everysufficiently large odd n,



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








The proof uses all the ideas mentioned above plus the Olshanskiitheory of graded diagrams.









Problem.









Problem. Is there a finitely presented torsion group?

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Algorithmic and asymptotic properties of groups