The Banach-Tarski Paradox, the von Neumann-Day conjecturepeople.maths.ox.ac.uk › drutu › tcc6...

The Banach-Tarski Paradox, the von Neumann-Dayconjecture

Cornelia Drutu

Oxford

TCC Course 2019, Lecture 4

Cornelia Drutu (Oxford) Banach-Tarski, von Neumann-Day TCC Course 2019, Lecture 4 1 / 23

Assessment

Exercise Sheet 1 is now on the webpage

http://people.maths.ox.ac.uk/drutu/tcc6/

For any type of assessment (broadening courses or others) answers mustbe sent in by email at [email protected] before June 12.


Last lecture

Last lecture I

Gromov’s Question: When are two quasi-isometric spaces actuallybi-Lipschitz equivalent? Especially if the spaces are separated.Key examples: Finitely generated groups, separated nets in Euclideanspaces.

Burago-Kleiner, McMullen: examples of separated nets in R2 notbi-Lipschitz equivalent.T. Dymarz: examples of lamplighter groups (which are amenablegroups) quasi-isometric, not bi-Lipschitz equivalent.

K. Whyte: A quasi-isometry between two non-amenable finitelygenerated groups is at bounded distance from a bi-Lipschitz map.More generally true for two non-amenable graphs of boundedgeometry.


Last lecture

Last lecture II

A group is amenable if there exists a mean m on `∞(G ) invariant byleft multiplication.Equivalently, if there exists a finitely additive probability measure µ onP(G ), the set of all subsets of G , invariant by left multiplication.

Invariant by left multiplication (left-invariance) can be replaced byright-invariance or by bi-invariance.

A group is amenable iff one (every) Cayley graph of G is amenable.A Cayley graph G is amenable: ∃ a Følner sequence (Ωn) ⊂ G .

For every A ⊂ G define

µn(A) =|A ∩ Ωn||Ωn|

.

|µn(A)− µn(Ag)| ≤ 2∂V (Ωn)|Ωn| when g ∈ S .

Let ω be a non-principal ultrafilter on N.Take µ(A) = ω-limµn(A) .


Last lecture

Last lecture III

A finitely generated group is either paradoxical or amenable.

One can define amenability for groups that are not finitely generated,using an invariant mean or probability measure.

The metric definition is no longer equivalent.


Group operations, more examples

Group operations

Proposition

A subgroup of an amenable group is amenable.

Proof: Let H ≤ G and µ be G -invariant finitely additive probabilitymeasure on P(G ).

Axiom of Choice: ∃ subset D of G intersecting each right coset Hg inexactly one point.

Define ν(A) := µ(AD) left-invariant f.a.p. measure on H.

Corollary

Any group containing a free non-abelian subgroup is non-amenable.



1 A finite extension of an amenable group is amenable.

2 Let N be a normal subgroup of a group G . The group G is amenableif and only if both N and G/N are amenable.

3 The direct limit G of a directed system (Hi )i∈I of amenable groupsHi , is amenable.

Corollary

A group G is amenable if and only if all finitely generated subgroups of Gare amenable.

Proof of (2). µG/N(A) = µG (π−1(A)), where π : G → G/N.Let ν be a left-invariant f.a.p. measure on G/N, and λ a left-invariantf.a.p. measure on N.On every left coset gN define a f.a.p. measure λg (A) = λ(g−1A).For every subset B in G define µ(B) =

∫G/N λg (B ∩ gN)dν(gN) .



Solvable groups

Corollary

Every solvable group (not necessarily finitely generated) is amenable.

G ′ = derived subgroup [G ,G ] of the group G .The iterated commutator subgroups G (k) defined inductively by:

G (0) = G ,G (1) = G ′, . . . ,G (k+1) =(G (k)

)′, . . .

The derived series of G is

G D G ′ D . . .D G (k) D G (k+1) D . . .

G is solvable if there exists k such that G (k) = 1.The minimal k such that G (k) = 1 is the derived length of G .

Example: The group of upper triangular n × n matrices in GL(n,K ), Kfield, is solvable.



Solvable groups continued

Milnor-Wolf: A finitely generated solvable group is either virtuallynilpotent or of exponential growth.

A group G is virtually (***) if it has a finite index subgroup with property(***).

Exercise

Suppose G is direct limit of Gi , i ∈ I . Assume that there exist k ,m ∈ N sothat for every i ∈ I , the group Gi contains a solvable subgroup Hi of index6 k and derived length 6 m. Then G contains a subgroup H of index 6 kand derived length 6 m.

Back to

Corollary

Every solvable group is amenable.



Elementary amenable

Definition (M. Day)

The class of elementary amenable groups EA= the smallest classcontaining all finite groups, all abelian groups and closed under finite-indexextensions, direct limits, subgroups, quotients and extensions

1→ G1 → G2 → G3 → 1,

where both G1,G3 are elementary amenable.

There exist groups of intermediate growth (hence amenable) but notelementary amenable: Grigorchuk groups

There exist elementary amenable groups not virtually solvable (moreoverfinitely presented): the Houghton’s groups Hn, satisfying

1→ Σ→ Hn → Zn−1

where Σ is the group of permutations with finite support of Z.See Lee, Sang Rae, Geometry of Houghton’s groups, PhD Dissertation,University of Oklahoma.



An alternative for EA

Theorem (Chou)

A finitely generated elementary amenable group either is virtually nilpotentor it contains a free non-abelian subsemigroup.

Theorem (Chou)

The class of elementary amenable groups is the smallest class containingall finite groups, all abelian groups and closed under direct limits andextensions

1→ G1 → G2 → G3 → 1,

where both G1,G3 are in the class.

Reference: Ching Chou, Elementary amenable groups, Illinois J. Math. 24(1980).


Von Neuman-Day conjecture

We are now able to relate amenable groups to the Banach–Tarski paradox.

Proposition

1 The group of isometries Isom(Rn) with n = 1, 2 is amenable.

2 The group of isometries Isom(Rn) with n > 3 is non-amenable.

Does there exist a purely algebraic definition of amenability for groups?

Conjecture

Does every non-amenable group contain a free non-abelian subgroup?

The question is implicit in von Neumann’s initial paper (1929), formulatedexplicitly by Day in 1957.

Counter-examples:

Al. Olshanskii (1980);

S, Adyan (1982): the free Burnside group B(n,m) with n ≥ 2 andm ≥ 665, m odd.



Positive answers

Theorem (Jacques Tits 1972)

A subgroup G of GL(n,F ), where F is a field of zero characteristic, iseither virtually solvable or it contains a free nonabelian subgroup.

Also true for fields of positive characteristic, if G finitely generated.

Theorem (Mostow–Tits)

A discrete amenable subgroup G of a Lie group L with finitely manycomponents, contains a polycyclic group of index at most η(L).



Other classes of groups for which the von Neumann-Day conjecture has apositive answer (in fact Tits’ theorem is true):

1 subgroups of Gromov hyperbolic groups (Gromov);

2 subgroups of the mapping class group of a surface (Ivanov 1992);

3 subgroups of Out(Fn) (Bestvina-Feighn-Handel 2000,2004,2005);

4 fundamental groups of compact manifolds of nonpositive curvature(Ballmann 1995).



A metric von Neumann-Day

A metric version of the von Neumann-Day conjecture established byBenjamini and Schramm:

A locally finite graph G with positive Cheeger constant contains a treewith positive Cheeger constant.Uniform bound on the valency is not assumed.

Cheeger constant is considered with edge–boundary.

If, moreover, the Cheeger constant of G is at least an integer n > 0,then G contains a spanning subgraph, with each connectedcomponent is a rooted tree with all vertices of valency n, except theroot, of valency n + 1.



Metric von Neumann-Day continued

If X is either a graph or a Riemannian manifold with infinitediameter, bounded geometry and positive Cheeger constant (inparticular, if X is the Cayley graph of a paradoxical group) then Xcontains a bi-Lipschitz embedding of the binary rooted tree.

Related to the above, the following is asked:

Open question

(Benjamini-Schramm 1997) Does every Cayley graph of every finitelygenerated group with exponential growth contain a tree with positiveCheeger constant?

Open case: amenable non-linear groups with exponential growth.



Next, we shall overview briefly quantitative approaches to:

non-amenability: Tarski numbers for groups;

amenability: uniform amenability and Følner functions.


Quantitative non-amenability


One can measure “how paradoxical” a group G is via the Tarski number.

In this discussion, groups are not required to be finitely generated.

Recall that a paradoxical decomposition of a group G is a partition

G = X1 t ... t Xk t Y1 t ... t Ym

for which ∃ g1, ..., gk , h1, ..., hm in G , so that

g1 X1 t ... t gk Xk = G

andh1 Y1 t ... t hm Ym = G .

The Tarski number of the decomposition is k + m.

The Tarski number Tar(G ) of the group = minimum of the Tarskinumbers of paradoxical decompositions.

If G is amenable then set Tar(G ) =∞.Cornelia Drutu (Oxford) Banach-Tarski, von Neumann-Day TCC Course 2019, Lecture 4 18 / 23


Tarski numbers and group operations

Proposition

1 Tar(G ) ≥ 4 for every group G .

2 If H 6 G then Tar(G ) 6 Tar(H).

3 Tar(G ) = 4 if and only if G contains a free non-abelian sub-group.

4 Every paradoxical group G contains a finitely generated subgroup Hwith Tar(G ) generators, such that Tar(G ) = Tar(H).

5 If N is a normal subgroup of G then Tar(G ) 6 Tar(G/N).



Co-embeddable groups

Two groups G1 and G2 are co-embeddable if there exist injective grouphomomorphisms G1 → G2 and G2 → G1.

1 All countable free groups are co-embeddable.

2 Sirvanjan-Adyan: for every odd m > 665, two free Burnside groupsB(n;m) and B(k;m) of exponent m, with n > 2 and k > 2, areco-embeddable.

G1 (non-)amenable iff G2 (non-)amenable. Moreover Tar(G1) = Tar(G2).

Consequence: For every odd m ≥ 665, and n ≥ 2, the Tarski number ofB(n;m) is independent of the number of generators.



Paradoxical decomposition and torsion

Proposition

1 If G admits a paradoxical decomposition

G = X1 t X2 t Y1 t . . . t Ym,

then G contains an element of infinite order.

2 If G is a torsion group then Tar(G ) ≥ 6.

The Tarski numbers help to classify the groups non-amenable and withoutan F2 subgroup (“infinite monsters”).

Ceccherini, Grigorchuk, de la Harpe: The Tarski number of a free Burnsidegroup B(n;m) with n > 2 and m > 665, m odd, is at most 14.



Tarski numbers, final

We proved that a paradoxical group G contains a finitely generatedsubgroup H with Tar(G ) generators, such that Tar(G ) = Tar(H).

Consequence: if G is such that all m generated subgroups are amenablethen Tar(G ) ≥ m + 1.

M. Ershov: certain Golod-Shafarevich groups G

have an infinite quotient with property (T);

for every m large enough, G contains finite index subgroups Hm withthe property that all their m-generated subgroups are finite.

Consequences:

the set of Tarski numbers is unbounded;

Tarski numbers, when large, are not quasi-isometry invariants. Noteven commensurability invariants.

Ershov-Golan-Sapir: D. Osin’s torsion groups have Tarski number 6.Cornelia Drutu (Oxford) Banach-Tarski, von Neumann-Day TCC Course 2019, Lecture 4 22 / 23


Questions

Question

How does the Tarski number of a free Burnside group B(n;m) depend onthe exponent m? What are its possible values?

Question

Is the Tarski number of groups a quasi-isometry invariant, when it takessmall values?

For Tar(G ) = 4 this question is equivalent to a well-known open problem.

A group G is small if it contains no free nonabelian subgroups. Thus, G issmall iff Tar(G ) > 4.

Question

Is smallness invariant under quasi-isometries of finitely generated groups?


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