Amenable Groups; Alternative Theorem

C. Drut¸u

Oxford

TCC 2015, Lecture 1

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 1 / 25 Overview

The course is centred around two conjectures that attempt to draw a demarcation line between “abelian-like” groups and “free-like” groups. These are the Von Neumann-Day conjecture; the Milnor conjecture.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 2 / 25 Von Neumann-Day

In the beginning of the XX-th century theory emerged (Lebesgue 1901, 1902). Among the questions asked in those days was the following. Question n Does there exist a measure on R that is finitely additive; by isometries; defined on every subset ?

The Banach–Tarski Paradox answered this question.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 3 / 25 Banach-Tarski Paradox

First some terminology: congruent subsets in a metric space = subsets A, B such that Φ(A) = B for some isometry Φ. piecewise congruent subsets in a metric space = subsets A, B such that

A = A1 t A2 t · · · t An, B = B1 t B2 t · · · t Bn

and Ai , Bi congruent for every i.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 4 / 25 Banach–Tarski Paradox II

Theorem (Banach-Tarski) n Every two subsets A.B in R , n ≥ 3,A, B with non-empty interiors are piecewise congruent.

Proof uses the . Examples:

A and λA, for λ > 0; B(0, 1) and B(0, 1) t B(c, 1).

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 5 / 25 Banach-Tarski Paradox III

In view of the Banach-Tarski Paradox, several conclusions are possible: n Conclusion 1: The answer to the question on measures on R is negative. n In particular there exist subsets in R “without volume”. Conclusion 2 (the opposite): Hard to believe that sets “without volume” exist, therefore the Axiom of Choice should not be admitted. Note that Banach-Tarski is not provable nor disprovable only with Zermelo-Frankel; The Axiom of Choice can be replaced with the Hahn-Banach Extension Theorem (a weaker hypothesis).

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 6 / 25 Amenable groups

Conclusion 3: Why the hypothesis “dimension at least 3” in Banach-Tarski ? (1929) defined amenable groups to explain this. Let G be locally compact, second countable. ∞ G amenable = there exists a functional m : ` (G) → R (mean) such that m has norm 1; if f ≥ 0 then m(f ) ≥ 0 (m positive definite); m invariant by G-left translations.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 7 / 25 Amenable groups II

Equivalently, for every compact K ⊂ G and every  > 0 there exists Ω ⊆ G Haar measurable, of positive measure ν(Ω) > 0 such that

ν(KΩ4Ω) ≤ ν(Ω).

Remark In all the above, left multiplication can be replaced by right multiplication.

Examples G finite. n n G = R or Z . For G = Z, for every K finite and  > 0, Ω = [−n, n] ∩ Z with n large enough works.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 8 / 25 Amenable Groups III

Proposition (immediate properties)

A subgroup of an amenable is amenable. Given a short exact sequence

1 → N → G → Q → 1

G amenable iff N and Q amenable. A direct limit of amenable groups is amenable.

Exercise: Prove the above.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 9 / 25 “Abelian-like” groups

Construction by induction: C 1(G) := G, C 2(G) = [G, C 1(G)],..., C i+1(G) = [G, C i (G)]. Thus, a decreasing series of characteristic subgroups

1 2 i C (G) D C (G) D ··· D C (G) D ··· called lower central series. G nilpotent if ∃k s.t. C k (G) = {1}. 0 G 0 = [G, G] = the derived group, G (i+1) = G (i) . Another decreasing series of characteristic subgroups

0 (i) G D G D ··· D G D ··· called derived series. G solvable if ∃k s.t. G (k) = {1}.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 10 / 25 Solvable, paradoxical

Corollary Every is amenable.

Prove the Corollary using Proposition 4. G is paradoxical if

∃ a decomposition G = X1 t · · · t Xk t Y1 t · · · t Ym; ∃ g1,..., gk and h1,..., hm in G s.t.

G = g1X1 t · · · t gk Xk = h1Y1 t · · · t hmYm.

Theorem G is non-amenable if and only if G is paradoxical.

Exercise: Prove that G paradoxical ⇒ G non-amenable. C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 11 / 25 The

Free group of rank 2, F2 = ha, bi= set of words in the alphabet a, b, a−1, b−1 , reduced (i.e. no xx−1). Binary operation is concatenation and reduction.  −1 −1 For x ∈ a, b, a , b , Wx = set of words beginning in x. 0 n Define Wb = Wb \{b | n ≥ 1}, 0 n Wb−1 = Wb−1 t {b | n ≥ 0}.

0 0 0 0 F2 = Wa t Wa−1 t Wb t Wb−1 = Wa t aWa−1 = Wb t bWb−1 .

Corollary

If F2 ≤ G then G is non-amenable.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 12 / 25 von Neumann-Day

Back to the question on dimension for Banach–Tarski: n n The group of isometries of R is O(n) o R . For n = 2, O(2) is virtually abelian.

For n ≥ 3, O(n) contains F2.

G is virtually (*)= ∃ H ≤ G of finite index, H with property (*).

Question Can one find an algebraic definition for amenability/non-amenability ?

Conjecture of von Neumann-Day: Every finitely generated group is either amenable or it contains (a copy of) F2.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 13 / 25 von Neumann-Day

Theorem ( 1972) Let L be a Lie group with finitely many connected components. If G ≤ L, G finitely generated then either F2 ≤ G or G is virtually solvable.

Remark “Virtually solvable” cannot be replaced with “solvable”: take G = F × S, where F is finite in GL(n, R), S solvable in GL(m, R), embed G in GL(n + m, R) using diagonal blocks. Exercise: Prove that virtually solvable groups are also amenable.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 14 / 25 von Neumann-Day

Counter-examples to general von Neumann-Day: Olshanskii (1980), using small cancellation, a technique to construct “infinite monsters”. Adyan: Burnside groups are counter-examples:

m B(n, m) = hx1,..., xn | w = 1i for n ≥ 2, m ≥ 665, m odd.

Olshanskii, Sapir 2003: finitely presented counter-examples. Monod 2012: The group of piecewise projective homeomorphisms of R is non-amenable, does not contain F2. Lodha-Moore 2014: A finitely presented subgroup of it too.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 15 / 25 Milnor’s Conjecture

M Riemannian manifold. Fix p ∈ M. Consider the function r 7→ Vol(B(p, r)). Questions: Dependence on p ? If N compact manifold and M = Ne, can part of the behaviour of the function be detected by looking at π1(N)?

Let f , g : A → R, A ⊆ R. f . g ⇔ ∃a, b, c, d, e positive s.t. f (x) ≤ ag(bx + c) + dx + e. f ≈ g ⇔ f . g, g . f f , g as above are called asymptotically equal.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 16 / 25 Milnor’s Conjecture II

Dependence on p is up to ≈. For G finitely generated define r 7→ ]B(1, r), with B(1, r) ball centred in 1 for some metric on G: fix S finite set generating G, 1 6∈ S, S−1 = S. −1 word metric: distS (g, h)= shortest word in S representing h g.

Growth function on Ne ≈ growth function of G = π1(N) (Milnor; Efremovich- Schwartz).

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 17 / 25 Examples of growth functions

Examples: d d Z has growth function ≈ x ; G nilpotent,

1 2 k C (G) D C (G) D ··· D C (G) = {1} .

i i+1 m Then C (G)/C (G) w Z i × . P m Growth of G ≈ x i i (Bass-Guivarch). x the free group F2 has growth ≈ e .

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 18 / 25 Milnor-Wolf

Theorem (Milnor-Wolf) If G is finitely generated and solvable then G is either virtually nilpotent or of exponential growth.

2 Exercise: Prove that for Z oM Z, where M 2 × 2 matrix with entries x either M has an eigenvalue of absolute value 6= 1, and growth ≈ e ; 2 or both eigenvalues have absolute value 1 and Z oM Z virtually nilpotent.

Milnor’s Conjecture: Every f.g. group has either polynomial or exponential growth.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 19 / 25 Growth and amenability

Proposition

If G has sub-exponential growth then G amenable.

pn Sub-exponential growth= limn→∞ ]B(1, n) = 1. Exercise: prove Proposition. Hint: try second definition of amenability, with Ω = B(1, n).

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 20 / 25 Milnor’s conjecture

Theorem (M. Gromov’s Polynomial Growth Theorem) If a f.g. group has polynomial growth then it is virtually nilpotent.

Milnor’s conjecture false in general; counter-examples√ of Grigorchuk, of n groups with growth function sub-exponential and & e .

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 21 / 25 Gromov’s theorem

Strategy of proof for Gromov’s theorem: d Let G f.g., growth / x . Embed G in a Lie group with finitely many connected components.

Use Alternative Theorem: G either solvable or F2 ≤ G. The latter d impossible, as growth / x . Apply Milnor-Wolf Theorem.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 22 / 25 Main Tool for Gromov’s Thm.

Tool used to embed G in a Lie Groups: Theorem (Montgomery-Zippin) Input: X metric space that is complete; connected and locally connected; proper (i.e. compact balls); of finite Hausdorff dimension. Output: If H = the group of isometries of X acts transitively then H has finitely many conn. comp.;

there exists a homomorphism ϕ : He → GL(n, C) with kernel in the centre Z(He ).

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 23 / 25 First part of the course

Goal: Make G act on a space X . G already acts on (G, word metric ); but this space is not connected;

every Cayley graph Cay(G, S) with the simplicial metric; but for these the group of isometries does not act trasitively.

2 Example: for G = Z and S = {±(1, 0), ±(0, 1)}, the Cayley graph Cay(G, S) is the planar grid with mesh 1. 1 1 If we rescale its metric by ⇒ planar grid with mesh . λ λ 2 The limit of the rescaled grids as λ → ∞ is R , has all the properties required by Montgomery-Zippin Theorem. Gromov’s theorem and its proof will occupy the first part of the course.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 24 / 25 Second part

Second part of the course: Look at actions of G on Hilbert spaces, by affine isometries, i.e. preserving the metric, not necessarily linear. Masur-Ulam: Every affine isometry of a Hilbert space H is of the form v 7→ Av + b, where A unitary transformation, b ∈ H. Assume G locally compact, second countable. Kazhdan’s Property (T): Every continuous action by affine isometries on a Hilbert space has a global fixed point. a-T-menable (): There exists a continuous action by affine isometries on a Hilbert space that is proper: when g leaves every compact, kgvk → ∞, for v arbitrary fixed. Amenable groups are a-T-menable. Groups with Kazhdan’s Property (T) are paradoxical.

C. Drut¸u (Oxford) Amenable Groups; Alternative Theorem TCC 2015, Lecture 1 25 / 25