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Amenable groups, Jacques Tits’ Alternative Theorem

Cornelia Drut ¸u

Oxford

TCC Course 2014, Lecture 7

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 1 / 13

Ping-pong on the projective space

Projective ping-pong

Let g ∈ GL(n, K) be an ordered basis {u1, . . . , un} of eigenvectors, gui = λiui, such that λ1 > λ2 λ3 . . . λn−1 > λn > 0 . Denote A(g) = [u1] and H(g) = [Span {u2, . . . , un}]. Then A(g−1) = [un] and H(g−1) = [Span {u1, . . . , un−1}]. Obviously, A(g) ∈ H(g−1) and A(g−1) ∈ H(g). Proposition (projective ping-pong) Assume that g and h are two elements in GL(n, K) diagonal with respect to bases {u1, . . . , un}, {v1, . . . , vn} respectively. Assume that A(g±1) is not in H(h) ∪ H(h−1), and A(h±1) is not in H(g) ∪ H(g−1). There exists N such that gN and hN generate a free non-abelian subgroup

• f GL(n, K).

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 2 / 13

Ping-pong on the projective space

Proof of Proposition

Step 1: ∀ε > 0 and ∀α ∈

• g±1 , h±1

, ∃N such that αm with m ≥ N maps complementary of Nε(H(α)) → B (A(α) , ε) . Wlog we may assume {u1, . . . , un} is the standard basis because every M ∈ GL(n, K) induces a bi-Lipschitz transformation of PKn. the chordal metric on PKn is dist([v], [w]) = v ∧ w v · w. If H = ker f is a hyperplane in Kn then dist([v], [H]) = |f (v)| v f .

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 3 / 13

Ping-pong on the projective space

General ping-pong

The proof is finished using Lemma (Ping–pong, or Table–tennis, lemma) Let X be a set, g : X → X and h : X → X two bijections. If A, B non-empty subsets of X, such that A ⊂ B and gn(A) ⊂ B for every n ∈ Z \ {0} , hm(B) ⊂ A for every m ∈ Z \ {0} , then g, h is a free non-abelian subgroup. Notation: For every subset A in a fixed space X we denote by ∁A the complementary of A, i.e. X \ A.

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 4 / 13

Outline of the proof of Tits’ Theorem

Outline of the proof of Tits’ Theorem

Two cases: Case 1: Γ GL(n, K) is unbounded (for the norm in M(n, K)). Solved with geometric methods: construction of a ping-pong situation approximating the Projective ping-pong. Case 2: Γ GL(n, K) is relatively compact. Solved with Number Theory methods: a trick reducing it to Case 1.

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 5 / 13

Proof for unbounded subgroups

Case 1 of unbounded subgroups

The first step in the unbounded case is Theorem Let Γ GL(n, K) be unbounded, with Zariski closure semisimple Zariski-irreducible, acting irreducibly on Kn. Then Γ contains a free non-abelian subgroup. Recall that: an algebraic group G contains a radical RadG := the largest irreducible solvable normal algebraic subgroup of G. A group with trivial radical is called semisimple.

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 6 / 13

Proof for unbounded subgroups

Diverging sequences

Γ unbounded ⇒ Γ contains a diverging sequence of elements (gi) in GL(n, K), i.e. the matrix norms gi diverge to infinity. If we write Cartan decompositions gi = kidihi with ki, hi ∈ O(n) and di diagonal with entries a1(gi) a2(gi) · · · an(gi) > 0 then a1(gi) → ∞ .

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 7 / 13

Proof for unbounded subgroups

Contracting sequences

A sequence (gi) is m-contracting, for m < n , if its elements have Cartan decompositions gi = kidihi such that:

1 ki and hi converge to k and h in O(n); 2 di are diagonal matrices with diagonal entries a1(gi), . . . , an(gi) such

that a1(gi) a2(gi) . . . an(gi), a1(gi) → ∞ and lim

i→∞

|am(gi)| |a1(gi)| > 0 .

3 The number m is maximal with the above properties.

Lemma Every diverging sequence has an m–contracting subsequence, for some m < n. Hence every unbounded Γ contains an m–contracting subsequence.

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 8 / 13

Proof for unbounded subgroups

Terminology

Notation Let σ = (gi) be an m-contracting sequence. A(gi) = ki [Span(e1, . . . , em)] and A(σ) = k [Span(e1, . . . , em)] . We call A(σ) the attracting subspace of the sequence σ. Since ki → k, A(gi) converge to A(σ) with respect to the Hausdorff metric. Likewise E(gi) = h−1

i

[Span(em+1, . . . , en)] and E(σ) = h−1 [Span(em+1, . . . , en)] . We call E(σ) the repelling subspace of the sequence σ. Since hi → h, E(gi) converge to E(σ) in the Hausdorff metric.

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 9 / 13

Proof for unbounded subgroups

Uniform Lipschitz

Uniform version for sequences: Proposition Let σ = (gi) be an m-contracting sequence. For each compact K ⊂ ∁ E(σ) there exist L and i0 so that gi is L–Lipschitz

• n K, for every i i0.

Main ingredient in the proof: Lemma Let u be a unit vector and v ∈ Kn another vector such that |ui − vi| ε for all i. Then v ∧ w 2nε.

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 10 / 13

Proof for unbounded subgroups

Dynamics of 1–contracting sequences

Lemma An element g in GL(V ) with Cartan decomposition g = kdh, where d diagonal matrix with positive decreasing entries a1, . . . , an such that

a2 a1 < ε2 √n, maps the complement of the ε–neighborhood of the hyperplane

H = h−1 [Span(e2, . . . , en)] into the ball with center k[e1] and radius ε . Proposition If σ = (gi) is 1-contracting with attracting point p = A(σ) and repelling hyperplane H(σ), then for every closed ball B ⊆ ∁ H(σ), the maps gi|B converge uniformly to the constant function on B which maps everything to p.

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 11 / 13

Proof for unbounded subgroups

Dynamics of 1–contracting sequences II

Proposition Let (gi) be a diverging sequence of elements in GL(V ).

1 If there exists a closed ball B with non-empty interior and a point p

such that gi|B converge uniformly to the constant function on B which maps everything to the point p, then (gi) contains a 1-contracting subsequence with attracting point p.

2 If, moreover, there exists a hyperplane H such that for every closed

ball B ⊆ Hc, gi|B converge uniformly to the constant function on B which maps everything to the point p, then (gi) contains a 1-contracting subsequence with the attracting point p and the repelling hyperplane H.

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 12 / 13

Proof for unbounded subgroups

Constructing 1-contracting sequences

Consequence: If (gi) is 1-contracting, and f , h ∈ GL(n, K) then the sequence (fgih) contains a 1-contracting subsequence σ = (g′

i ) such that

A(σ) = f (A(σ)), E(σ) = h−1E(σ). Lemma Given (gi) m-contracting in PGL(V ), there exists a vector space W and an embedding ρ : GL(V ) ֒ → GL(W ) so that a subsequence in (ρ(gi)) is 1-contracting in PGL(W ).

Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 13 / 13