Introduction to Amenable Groups

The Margulis Normal Subgroups Theorem Lecture 1: Introduction to Amenable Groups

Dave Witte Morris

February 18, 2008

Definition. G amen: Example. G acts on a compact metric space X (i.e., G Homeo(X)) → G-invariant probability measure µ on X Z is amenable ⇒ ∃ • (µ(X) = 1)

Example. G/N, N amenable G amenable • ⇒ finite groups are amenable • compact groups are amenable abelian groups are amenable • • free group F2 is not amenable • solvable groups are amenable SL(2, R) is not amenable • • ∗ ∗ ∗ e.g., SL(3, R) For additional examples, it is helpful to have a more  ∗ ∗ ⊂ flexible characterization of amenability. ∗   Note. Prob(X) = prob meas on X C(X) { } ⊂ ∗ (weak topology — Frechet space) (closed subgroups of amenable groups are amenable) ∗ • nonempty • Note. discrete Γ free group Γ not amenable compact ⊃ ⇒ • converse not true [Olshanskii, 1980] convex • G-invariant Proposition. G connected Lie group: • G amen closed, solvable N !G s.t. G/N compact Proposition. G amenable G has a fixed point ⇐⇒ ∃ ⇐⇒ in every nonempty, compact, convex G-space C (in a Frechet space). Proof. ( ) amen extension of amen groups is amen. ⇐ Proof. ( ) Prob(X) is a nonempty, compact, convex ( ) structure theory: solvable N ! G (“radical”), ⇐ ⇒ . ∃ G-space. s.t. G/N = finite product of simple groups ( ) G-inv’t prob meas µ on C. (“semisimple”) ⇒ ∃ Center of mass ( C c dµ(c)) is G-invariant. Noncpct simple group SL(2, R) [Jacobson-Morosov] (It is in C, because C is closed and convex.) ⊃ R

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Key Lemma. References

Γ = SL(3, Z), Frederick P. Greenleaf: Invariant Means on Topologi- • cal Groups and Their Applications. Van Nostrand, New G = SL(3, R), • York, 1969. MR0251549 (40 #4776)

∗ ∗ ∗ Jean-Paul Pier: Amenable Locally Compact Groups. G •  ∗ ∗ ⊂ Pure and Applied Mathematics (New York). Wiley- ∗ Interscience Publication. Wiley, New York, 1984. ISBN:   Γ acts on compact metric X 0-471-89390-0 MR0767264 (86a:43001) (essentially) Γ-equivariant measurable ⇒ ∃ ψ : G/P Prob(X) Stan Wagon: The Banach-Tarski Paradox. Cam- → bridge U. Press, Cambridge, 1993. ISBN: 0-521-45704-1 Proof. Let C = Prob(X). MR1251963 (94g:04005) (G; C) is: L∞ Robert J. Zimmer: Ergodic Theory and Semisimple compact in a weak topology • ∗ Groups. Monographs in Mathematics, 81. Birkh¨auser, (since C closed, bounded) [Banach-Alaoglu] Basel, 1984. ISBN: 3-7643-3184-4 MR0776417 (86j:22014) convex (see Section 4.1) • (G; C) = Γ-equivariant ψ ψ(γx) = γ ψ(x) a.e. Draft chapter on “Amenable groups” from my book- LΓ∞ { } · in-progress Introduction to Arithmetic Groups will be closed (so compact) • posted (temporarily?) at convex http://people.uleth.ca/ dave.morris/talks/intro-amen.pdf • ∼ nonempty (def ψ to be constant on a fund dom) • G acts by right translation, so subgroup P also acts. P amen fixed pt ψ (i.e., ψ is P -inv’t) ⇒ ψ(xp) = ψ(x) ⇒ ψ factors thru to well-defined ψ : G/P C ⇒ → Other characterizations of amenability Theorem. TFAE

G is amenable • inv’t prob meas on any compact metric G-space • ∃ fixed pt in every nonempty, cpct, convex G-space • ∃ left-invariant mean on (G) • ∃ L∞ left-invariant finitely additive probability mea- • ∃ sure µ defined on all measurable subsets of G (s.t. µ(null set) = 0)

almost-invariant vectors in 2(G) (or 1(G)) • ∃ L L Følner sequences in G • ∃