Amenable Groups; Alternative Theorem

25 Pages·2015·0.2 MB·English

by  C. Drutu

Amenable Groups; Alternative Theorem
C. Dru¸tu
Oxford
TCC 2015, Lecture 1
C.Dru¸tu (Oxford) AmenableGroups;AlternativeTheorem TCC2015,Lecture1 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.Dru¸tu (Oxford) AmenableGroups;AlternativeTheorem TCC2015,Lecture1 2/25
Von Neumann-Day
In the beginning of the XX-th century measure theory emerged (Lebesgue
1901, 1902). Among the questions asked in those days was the following.
Question
Does there exist a measure on Rn that is
finitely additive;
invariant by isometries;
defined on every subset ?
The Banach–Tarski Paradox answered this question.
C.Dru¸tu (Oxford) AmenableGroups;AlternativeTheorem TCC2015,Lecture1 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 = A (cid:116)A (cid:116)···(cid:116)A ,B = B (cid:116)B (cid:116)···(cid:116)B
1 2 n 1 2 n
and A ,B congruent for every i.
i i
C.Dru¸tu (Oxford) AmenableGroups;AlternativeTheorem TCC2015,Lecture1 4/25
Banach–Tarski Paradox II
Theorem (Banach-Tarski)
Every two subsets A.B in Rn,n ≥ 3, A,B with non-empty interiors are
piecewise congruent.
Proof uses the Axiom of Choice.
Examples:
A and λA, for λ > 0;
B(0,1) and B(0,1)(cid:116)B(c,1).
C.Dru¸tu (Oxford) AmenableGroups;AlternativeTheorem TCC2015,Lecture1 5/25
Banach-Tarski Paradox III
In view of the Banach-Tarski Paradox, several conclusions are possible:
Conclusion 1: The answer to the question on measures on Rn is negative.
In particular there exist subsets in Rn “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.Dru¸tu (Oxford) AmenableGroups;AlternativeTheorem TCC2015,Lecture1 6/25
Amenable groups
Conclusion 3: Why the hypothesis “dimension at least 3” in
Banach-Tarski ?
John von Neumann (1929) defined amenable groups to explain this.
Let G be locally compact, second countable.
G amenable = there exists a functional m : (cid:96)∞(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.Dru¸tu (Oxford) AmenableGroups;AlternativeTheorem TCC2015,Lecture1 7/25
Amenable groups II
Equivalently, for every compact K ⊂ G and every (cid:15) > 0 there exists
Ω ⊆ G Haar measurable, of positive measure ν(Ω) > 0 such that
ν(KΩ(cid:52)Ω) ≤ (cid:15)ν(Ω).
Remark
In all the above, left multiplication can be replaced by right multiplication.
Examples
G finite.
G = Rn or Zn.
For G = Z, for every K finite and (cid:15) > 0, Ω = [−n,n]∩Z with n large
enough works.
C.Dru¸tu (Oxford) AmenableGroups;AlternativeTheorem TCC2015,Lecture1 8/25
Amenable Groups III
Proposition (immediate properties)
A subgroup of an amenable group 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.Dru¸tu (Oxford) AmenableGroups;AlternativeTheorem TCC2015,Lecture1 9/25
“Abelian-like” groups
Construction by induction: C1(G) := G,
C2(G) = [G,C1(G)],...,Ci+1(G) = [G,Ci(G)].
Thus, a decreasing series of characteristic subgroups
C1(G)(cid:68)C2(G)(cid:68)···(cid:68)Ci(G)(cid:68)···
called lower central series.
G nilpotent if ∃k s.t. Ck(G) = {1}.
G(cid:48) = [G,G] = the derived group, G(i+1) = (cid:0)G(i)(cid:1)(cid:48).
Another decreasing series of characteristic subgroups
G (cid:68)G(cid:48)(cid:68)···(cid:68)G(i)(cid:68)···
called derived series.
G solvable if ∃k s.t. G(k) = {1}.
C.Dru¸tu (Oxford) AmenableGroups;AlternativeTheorem TCC2015,Lecture1 10/25