Gromov’s polynomial growth theorem and approximate groups. E. Breuillard, joint with B. Green, T. Sanders and T. Tao Universit´ e Paris-Sud, Orsay, France IHP, Paris, July 4th, 2011 1 / 30
Approximate groups, definition A few years ago T. Tao introduced the following definition: Definition (Approximate subgroups) Let K � 1 be a parameter. A subset A of a group G is said to be a K -approximate subgroup of G if ⇒ a − 1 ∈ A ) ( i ) A contains id, and is symmetric (i.e. a ∈ A ⇐ ( ii ) there is a symmetric subset X ⊂ G with cardinal � K such that AA ⊂ XA Remarks: 1) abusing language for a moment, we will speak about approximate groups, when we mean “approximate subgroups of an ambient group”. We will come back to this point later in the talk. 2) We will be mainly interested in finite approximate groups, although considering infinite ones as well will be crucial to our approach. 2 / 30
Finite approximate groups are closely related to sets of small doubling, a notion much studied in additive combinatorics. A subset A ⊂ G of an ambient group G has doubling at most K if | AA | � K | A | . 3 / 30
Finite approximate groups are closely related to sets of small doubling, a notion much studied in additive combinatorics. A subset A ⊂ G of an ambient group G has doubling at most K if | AA | � K | A | . A central problem in additive combinatorics is to understand the structure of such sets. 3 / 30
Finite approximate groups are closely related to sets of small doubling, a notion much studied in additive combinatorics. A subset A ⊂ G of an ambient group G has doubling at most K if | AA | � K | A | . A central problem in additive combinatorics is to understand the structure of such sets. For example, one has: Theorem (Freiman’s theorem) If A ⊂ Z has | AA | � K | A | , then there is a generalized progression P of rank r = O K (1) such that A ⊂ P and | P | � O K (1) | A | . 3 / 30
Finite approximate groups are closely related to sets of small doubling, a notion much studied in additive combinatorics. A subset A ⊂ G of an ambient group G has doubling at most K if | AA | � K | A | . A central problem in additive combinatorics is to understand the structure of such sets. For example, one has: Theorem (Freiman’s theorem) If A ⊂ Z has | AA | � K | A | , then there is a generalized progression P of rank r = O K (1) such that A ⊂ P and | P | � O K (1) | A | . Remark: A generalized progression (or “GAP”) is a linear image of a box, i.e. a subset P ⊂ Z of the form π ( B ), where B is the box i =1 [ − N i , N i ] ⊂ Z r , and π : Z r → Z is a homomorphism. The � r integer r is the rank of P . 3 / 30
Of course every K -approximate group has doubling at most K . 4 / 30
Of course every K -approximate group has doubling at most K . Conversely, Proposition (Ruzsa, Tao) If | AAA | � K | A | , then A 1 := ( A ∪ A − 1 ∪ { 1 } ) 2 satisfies: ( i ) A 1 is a O ( K O (1) ) -approximate group, ( ii ) Moreover A is contained in � O ( K O (1) ) translates of A 1 . 4 / 30
Of course every K -approximate group has doubling at most K . Conversely, Proposition (Ruzsa, Tao) If | AAA | � K | A | , then A 1 := ( A ∪ A − 1 ∪ { 1 } ) 2 satisfies: ( i ) A 1 is a O ( K O (1) ) -approximate group, ( ii ) Moreover A is contained in � O ( K O (1) ) translates of A 1 . Remarks: 1) Under the weaker assumption that | AA | � K | A | , one has the same conclusion after passing to a large subset A ′ of A . 2) This essentially reduces the study of sets of small doubling to that of finite approximate groups. 4 / 30
Examples of approximate groups a finite group is a 1-approximate group. a progression of rank r is a 2 r -approximate group. a small ball around the identity in a Lie group (not a finite approximate group though!). a nilprogression of rank r and step s is a O r , s -approximate group. “extensions” of such. 5 / 30
Examples of approximate groups a nilprogression of rank r and step s is a O r , s -approximate group. “extensions” of such. What is a nilprogression ? 6 / 30
Examples of approximate groups a nilprogression of rank r and step s is a O r , s -approximate group. “extensions” of such. What is a nilprogression ? “Nilprogression” = a homomorphic image P = π ( B ) of a box B in the free nilpotent group of rank r and step s . 6 / 30
Examples of approximate groups a nilprogression of rank r and step s is a O r , s -approximate group. “extensions” of such. What is a nilprogression ? “Nilprogression” = a homomorphic image P = π ( B ) of a box B in the free nilpotent group of rank r and step s . “Box” means: ball for a left invariant Riemannian (or CC) metric on the free nilpotent Lie group. 6 / 30
Examples of approximate groups What is a nilprogression ? “Nilprogression” = a homomorphic image P = π ( B ) of a box B in the free nilpotent group of rank r and step s . “Box” means: ball for a left invariant Riemannian (or CC) metric on the free nilpotent Lie group. Example: If N , M ∈ N , set 1 x z ; | x | , | y | � N ; | z | � M A := 0 1 y 0 0 1 It is a ”box” if M ≃ N 2 . It is a nilprogression of step 2 and rank � 3 if M � N 2 . 7 / 30
Main theorem Theorem (BGST) Let A be a finite K-approximate subgroup of an ambient group G. Then A 4 contains an approximate subgroup A ′ with ( i ) A ′ is a coset nilprogression of rank and step � O K (1) , ( ii ) Moreover A can be covered by � O K (1) left translates of A ′ . A coset nilprogression is a finite set of the form A ′ = HL , where H is a finite subgroup normalized by the finite set L , in such a way that H \ HL is a nilprogression. Remarks: ( i ) This extends a theorem of Hrushovski and answers conjectures of Lindenstrauss and of Helfgott regarding the classification of approximate groups. 8 / 30
Main theorem Theorem (BGST) Let A be a finite K-approximate subgroup of an ambient group G. Then A 4 contains an approximate subgroup A ′ with ( i ) A ′ is a coset nilprogression of rank and step � O K (1) , ( ii ) Moreover A can be covered by � O K (1) left translates of A ′ . A coset nilprogression is a finite set of the form A ′ = HL , where H is a finite subgroup normalized by the finite set L , in such a way that H \ HL is a nilprogression. Remarks: ( i ) This extends a theorem of Hrushovski and answers conjectures of Lindenstrauss and of Helfgott regarding the classification of approximate groups. ( ii ) We recover Gromov’s theorem on groups of polynomial growth. 8 / 30
relation to Gromov’s theorem ( ii ) We recover Gromov’s theorem on groups of polynomial growth. How ? 9 / 30
relation to Gromov’s theorem ( ii ) We recover Gromov’s theorem on groups of polynomial growth. How ? Suppose | B ( n ) | � n K for all n ≫ 1. There are arbitrarily large scales r such that | B (3 r ) | � 3 K | B ( r ) | . 9 / 30
relation to Gromov’s theorem ( ii ) We recover Gromov’s theorem on groups of polynomial growth. How ? Suppose | B ( n ) | � n K for all n ≫ 1. There are arbitrarily large scales r such that | B (3 r ) | � 3 K | B ( r ) | . By the theorem B (4 r ) contains a coset nilprogression HL with L of rank and step O K (1) and s.t. B ( r ) ⊂ X ( HL ) , with | X | � O K (1) 9 / 30
relation to Gromov’s theorem ( ii ) We recover Gromov’s theorem on groups of polynomial growth. How ? Suppose | B ( n ) | � n K for all n ≫ 1. There are arbitrarily large scales r such that | B (3 r ) | � 3 K | B ( r ) | . By the theorem B (4 r ) contains a coset nilprogression HL with L of rank and step O K (1) and s.t. B ( r ) ⊂ X ( HL ) , with | X | � O K (1) The subgroup � HL � is virtually nilpotent. 9 / 30
relation to Gromov’s theorem ( ii ) We recover Gromov’s theorem on groups of polynomial growth. How ? Suppose | B ( n ) | � n K for all n ≫ 1. There are arbitrarily large scales r such that | B (3 r ) | � 3 K | B ( r ) | . By the theorem B (4 r ) contains a coset nilprogression HL with L of rank and step O K (1) and s.t. B ( r ) ⊂ X ( HL ) , with | X | � O K (1) The subgroup � HL � is virtually nilpotent. If r > | X | , it is also of finite index � | X | . 9 / 30
relation to Gromov’s theorem Suppose | B ( n ) | � n K for all n ≫ 1. There are arbitrarily large scales r such that | B (3 r ) | � 3 K | B ( r ) | . By the theorem applied to A := B ( r ) we get that B (4 r ) contains a coset nilprogression HL with L of rank and step O K (1) and s.t. B ( r ) ⊂ X ( HL ) , with | X | � O K (1) The subgroup � HL � is virtually nilpotent. If r > | X | , it is also of finite index � | X | . Remark: We did not need to assume | B ( n ) | � n K for all n ≫ 1. 10 / 30
relation to Gromov’s theorem Suppose | B ( n ) | � n K for all n ≫ 1. There are arbitrarily large scales r such that | B (3 r ) | � 3 K | B ( r ) | . By the theorem applied to A := B ( r ) we get that B (4 r ) contains a coset nilprogression HL with L of rank and step O K (1) and s.t. B ( r ) ⊂ X ( HL ) , with | X | � O K (1) The subgroup � HL � is virtually nilpotent. If r > | X | , it is also of finite index � | X | . Remark: We did not need to assume | B ( n ) | � n K for all n ≫ 1. In fact, this argument works as soon as | B ( n ) | � n K for some n � some function of K only. 10 / 30
Recommend
More recommend
