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 { u 1 , . . . , u n } of eigenvectors, gu i = λ i u i , such that λ 1 > λ 2 � λ 3 � . . . � λ n − 1 > λ n > 0 . Denote A ( g ) = [ u 1 ] and H ( g ) = [ Span { u 2 , . . . , u n } ]. Then A ( g − 1 ) = [ u n ] and H ( g − 1 ) = [ Span { u 1 , . . . , u n − 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 { u 1 , . . . , u n } , { v 1 , . . . , v n } 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 g N and h N generate a free non-abelian subgroup of 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 g ± 1 , h ± 1 � , ∃ N such that α m with m ≥ N � Step 1: ∀ ε > 0 and ∀ α ∈ maps complementary of N ε ( H ( α )) → B ( A ( α ) , ε ) . Wlog we may assume { u 1 , . . . , u n } is the standard basis because every M ∈ GL ( n , K ) induces a bi-Lipschitz transformation of PK n . the chordal metric on PK n is dist ([ v ] , [ w ]) = � v ∧ w � � v � · � w � . If H = ker f is a hyperplane in K n then | f ( v ) | dist ([ v ] , [ H ]) = � 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 g n ( A ) ⊂ B for every n ∈ Z \ { 0 } , h m ( 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 K n . Then Γ contains a free non-abelian subgroup. Recall that: an algebraic group G contains a radical Rad G := 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 ( g i ) in GL ( n , K ), i.e. the matrix norms � g i � diverge to infinity. If we write Cartan decompositions g i = k i d i h i with k i , h i ∈ O ( n ) and d i diagonal with entries a 1 ( g i ) � a 2 ( g i ) � · · · � a n ( g i ) > 0 then a 1 ( g i ) → ∞ . Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 7 / 13
Proof for unbounded subgroups Contracting sequences A sequence ( g i ) is m -contracting, for m < n , if its elements have Cartan decompositions g i = k i d i h i such that: 1 k i and h i converge to k and h in O ( n ); 2 d i are diagonal matrices with diagonal entries a 1 ( g i ) , . . . , a n ( g i ) such that a 1 ( g i ) � a 2 ( g i ) � . . . � a n ( g i ) , a 1 ( g i ) → ∞ and | a m ( g i ) | lim | a 1 ( g i ) | > 0 . i →∞ 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 = ( g i ) be an m -contracting sequence. A ( g i ) = k i [Span( e 1 , . . . , e m )] and A ( σ ) = k [Span( e 1 , . . . , e m )] . We call A ( σ ) the attracting subspace of the sequence σ . Since k i → k , A ( g i ) converge to A ( σ ) with respect to the Hausdorff metric. Likewise [Span( e m +1 , . . . , e n )] and E ( σ ) = h − 1 [Span( e m +1 , . . . , e n )] . E ( g i ) = h − 1 i We call E ( σ ) the repelling subspace of the sequence σ . Since h i → h , E ( g i ) 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 = ( g i ) be an m-contracting sequence. For each compact K ⊂ ∁ E ( σ ) there exist L and i 0 so that g i is L–Lipschitz on K, for every i � i 0 . Main ingredient in the proof: Lemma Let u be a unit vector and v ∈ K n another vector such that | u i − v i | � ε for all i. Then � v ∧ w � � 2 n ε. 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 a 1 , . . . , a n such that a 1 < ε 2 a 2 √ n , maps the complement of the ε –neighborhood of the hyperplane H = h − 1 [Span( e 2 , . . . , e n )] into the ball with center k [ e 1 ] and radius ε . Proposition If σ = ( g i ) is 1 -contracting with attracting point p = A ( σ ) and repelling hyperplane H ( σ ) , then for every closed ball B ⊆ ∁ H ( σ ) , the maps g i | 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 ( g i ) 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 g i | B converge uniformly to the constant function on B which maps everything to the point p, then ( g i ) contains a 1 -contracting subsequence with attracting point p. 2 If, moreover, there exists a hyperplane H such that for every closed ball B ⊆ H c , g i | B converge uniformly to the constant function on B which maps everything to the point p, then ( g i ) 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 ( g i ) is 1-contracting, and f , h ∈ GL ( n , K ) then the sequence ( fg i h ) contains a 1-contracting subsequence σ = ( g ′ i ) such that E ( σ ) = h − 1 E ( σ ) . A ( σ ) = f ( A ( σ )) , Lemma Given ( g i ) 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 ( g i )) is 1 -contracting in PGL ( W ) . Cornelia Drut ¸u (Oxford) Amenable groups, Alternative Theorem TCC Course 2014, Lecture 7 13 / 13
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