Gromovs polynomial growth theorem and approximate groups. E. - - PowerPoint PPT Presentation

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

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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 (i) A contains id, and is symmetric (i.e. a ∈ A ⇐ ⇒ a−1 ∈ 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.

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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|.

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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.

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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 = OK(1) such that A ⊂ P and |P| OK(1)|A|.

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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 = OK(1) such that A ⊂ P and |P| OK(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 r

i=1[−Ni, Ni] ⊂ Zr, and π : Zr → Z is a homomorphism. The

integer r is the rank of P.

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Of course every K-approximate group has doubling at most K.

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Of course every K-approximate group has doubling at most K. Conversely, Proposition (Ruzsa, Tao) If |AAA| K|A|, then A1 := (A ∪ A−1 ∪ {1})2 satisfies: (i) A1 is a O(K O(1))-approximate group, (ii) Moreover A is contained in O(K O(1)) translates of A1.

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Of course every K-approximate group has doubling at most K. Conversely, Proposition (Ruzsa, Tao) If |AAA| K|A|, then A1 := (A ∪ A−1 ∪ {1})2 satisfies: (i) A1 is a O(K O(1))-approximate group, (ii) Moreover A is contained in O(K O(1)) translates of A1. 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.

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Examples of approximate groups

a finite group is a 1-approximate group. a progression of rank r is a 2r-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 Or,s-approximate group. “extensions” of such.

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Examples of approximate groups

a nilprogression of rank r and step s is a Or,s-approximate group. “extensions” of such. What is a nilprogression ?

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Examples of approximate groups

a nilprogression of rank r and step s is a Or,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.

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Examples of approximate groups

a nilprogression of rank r and step s is a Or,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

• n the free nilpotent Lie group.

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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

• n the free nilpotent Lie group.

Example: If N, M ∈ N, set A :=      1 x z 1 y 1   ; |x|, |y| N; |z| M    It is a ”box” if M ≃ N2. It is a nilprogression of step 2 and rank 3 if M N2.

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Main theorem

Theorem (BGST) Let A be a finite K-approximate subgroup of an ambient group G. Then A4 contains an approximate subgroup A′ with (i) A′ is a coset nilprogression of rank and step OK(1), (ii) Moreover A can be covered by OK(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

• f Lindenstrauss and of Helfgott regarding the classification of

approximate groups.

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Main theorem

Theorem (BGST) Let A be a finite K-approximate subgroup of an ambient group G. Then A4 contains an approximate subgroup A′ with (i) A′ is a coset nilprogression of rank and step OK(1), (ii) Moreover A can be covered by OK(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

• f Lindenstrauss and of Helfgott regarding the classification of

approximate groups. (ii) We recover Gromov’s theorem on groups of polynomial growth.

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relation to Gromov’s theorem

(ii) We recover Gromov’s theorem on groups of polynomial growth. How ?

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relation to Gromov’s theorem

(ii) We recover Gromov’s theorem on groups of polynomial growth. How ? Suppose |B(n)| nK for all n ≫ 1. There are arbitrarily large scales r such that |B(3r)| 3K|B(r)|.

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relation to Gromov’s theorem

(ii) We recover Gromov’s theorem on groups of polynomial growth. How ? Suppose |B(n)| nK for all n ≫ 1. There are arbitrarily large scales r such that |B(3r)| 3K|B(r)|. By the theorem B(4r) contains a coset nilprogression HL with L of rank and step OK(1) and s.t. B(r) ⊂ X(HL), with |X| OK(1)

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relation to Gromov’s theorem

(ii) We recover Gromov’s theorem on groups of polynomial growth. How ? Suppose |B(n)| nK for all n ≫ 1. There are arbitrarily large scales r such that |B(3r)| 3K|B(r)|. By the theorem B(4r) contains a coset nilprogression HL with L of rank and step OK(1) and s.t. B(r) ⊂ X(HL), with |X| OK(1) The subgroup HL is virtually nilpotent.

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relation to Gromov’s theorem

(ii) We recover Gromov’s theorem on groups of polynomial growth. How ? Suppose |B(n)| nK for all n ≫ 1. There are arbitrarily large scales r such that |B(3r)| 3K|B(r)|. By the theorem B(4r) contains a coset nilprogression HL with L of rank and step OK(1) and s.t. B(r) ⊂ X(HL), with |X| OK(1) The subgroup HL is virtually nilpotent. If r > |X|, it is also

• f finite index |X|.

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relation to Gromov’s theorem

Suppose |B(n)| nK for all n ≫ 1. There are arbitrarily large scales r such that |B(3r)| 3K|B(r)|. By the theorem applied to A := B(r) we get that B(4r) contains a coset nilprogression HL with L of rank and step OK(1) and s.t. B(r) ⊂ X(HL), with |X| OK(1) The subgroup HL is virtually nilpotent. If r > |X|, it is also

• f finite index |X|.

Remark: We did not need to assume |B(n)| nK for all n ≫ 1.

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relation to Gromov’s theorem

Suppose |B(n)| nK for all n ≫ 1. There are arbitrarily large scales r such that |B(3r)| 3K|B(r)|. By the theorem applied to A := B(r) we get that B(4r) contains a coset nilprogression HL with L of rank and step OK(1) and s.t. B(r) ⊂ X(HL), with |X| OK(1) The subgroup HL is virtually nilpotent. If r > |X|, it is also

• f finite index |X|.

Remark: We did not need to assume |B(n)| nK for all n ≫ 1. In fact, this argument works as soon as |B(n)| nK for some n some function of K only.

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In fact, this argument works as soon as |B(n)| nK for some n some function of K only. → this gives another proof of the Shalom-Tao result (at least qualitatively).

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In fact, this argument works as soon as |B(n)| nK for some n some function of K only. → this gives another proof of the Shalom-Tao result (at least qualitatively). Other by-products include: A new proof of Freiman’s theorem (avoiding harmonic analysis).

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In fact, this argument works as soon as |B(n)| nK for some n some function of K only. → this gives another proof of the Shalom-Tao result (at least qualitatively). Other by-products include: A new proof of Freiman’s theorem (avoiding harmonic analysis). Hrushovski’s improvement of Gromov’s theorem: if a f.g. group G can be exhausted by sets of doubling at most K, then G is virtually nilpotent.

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In fact, this argument works as soon as |B(n)| nK for some n some function of K only. → this gives another proof of the Shalom-Tao result (at least qualitatively). Other by-products include: A new proof of Freiman’s theorem (avoiding harmonic analysis). Hrushovski’s improvement of Gromov’s theorem: if a f.g. group G can be exhausted by sets of doubling at most K, then G is virtually nilpotent. A purely group theoretical proof that Ricci almost non-negatively curved manifolds have a virtually nilpotent π1 (Kapovitch-Petrunin-Tuschmann, Kapovitch-Wilking).

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About the proof: a sketch

Take any sequence of K-approximate groups An and form the ultraproduct A :=

• U

An We will study this infinite object (which is still a K-approximate group and we call it a non-standard finite K-approximate group) to transfer its properties to the original An’s. By a definable set (or internal) we mean a subset Q ⊂ Ak of the form Q =

• U

Qn, where Qn ⊂ Ak

n.

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By a definable set (or internal) we mean a subset Q ⊂ Ak of the form Q =

• U

Qn, where Qn ⊂ Ak

n.

For example if A4 contains a definable set Q such that (i) Q is a subgroup, and (ii) A ⊂ XQ, X finite, Then the same holds for U-almost all An: i.e. An ⊂ XnQn with Qn subgroup ⊂ A4

n, and |Xn| |X|.

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Steps of the proof of the main theorem

Step 1: Define a locally compact topology on A, so A maps onto a compact neighborhood of 1 in a locally compact (local) group. Step 2: Show that there is H U ⊂ A4, where U is a neighborhood of 1, H a definable subgroup normalized by U such that U/H has no subgroups and maps onto a neighborhood of 1 a local Lie group L. Step 3: Show that L is nilpotent and induct on its dimension to build the nilprogression. Remarks: Step 1 relies on the Croot-Sisask-Sanders-Hrushovski lemma. Step 2 uses ideas and techniques behind the proof the Hilbert 5-th problem (Gleason’s lemmas). To perform induction in Step 3, we take advantage of properties of local groups (e.g. the notion of a quotient of two local groups).

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Step 1: build a locally compact topology

This relies on: Lemma (Croot-Sisask-Sanders-Hrushovski) If A is a finite K-approximate group, then for every k 2, there is Sk ⊂ A4 with 1 ∈ Sk = S−1

k

such that Sk

k ⊂ A4 and

|Sk| |A|/Ok,K(1). Can choose Sk nested in such a way that S2

k+1 ⊂ Sk. Do this for

all An, get sets Sk,n and put Sk :=

U Sn,k.

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Then: the Sk form a basis of neighborhoods of 1 in the non-standard approximate group A4. the fact that A4 can be covered by finitely many translates of each Sk easily implies that the topology is locally compact. group operations are continuous → get a locally compact (local) group G. the topology is ”generated” by definable sets in the sense that ∀K ⊂ U with K compact and U open in G, ∃D definable with K ⊂ D ⊂ U. definable sets are closed. A4 is a compact neighorhood of 1. after quotienting by {1} we get a Hausdorff locally compact local group. Note: {1} may not be definable.

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From the structure theorem for locally compact groups (solution to Hilbert 5th problem or its version for local groups due to Goldbring) we can conclude that up to passing to a large A′ ⊂ A4, A′ has a local epimorphism onto a neighborhood of 1 in a connected Lie group L. This already gets you a long way! For instance: Corollary 1 (Hrushovski) If the locally compact local group G is totally disconnected (i.e. L = {1}), then A (hence also the An’s simultaneously) are covered by boundedly many cosets of a subgroup Hn ⊂ A4

n.

This happens for example if all An have bounded exponent, i.e. ∃e ∈ N s.t. ∀n∀x ∈ An, xe = 1.

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This already gets you a long way! For instance: Corollary 1 (Hrushovski) If the locally compact local group G is totally disconnected (i.e. L = {1}), then A (hence also the An’s simultaneously) are covered by boundedly many cosets of a subgroup Hn ⊂ A4

n.

This happens for example if all An have bounded exponent, i.e. ∃e ∈ N s.t. ∀n∀x ∈ An, xe = 1. Corollary 2 (Hrushovski) Any finite K-approximate group A has an A′ ⊂ A4 s.t. [A′, A′] ⊂ A′ and |A′| |A|/OK(1). Indeed: take A′ the pullback of a small neighborhood of {1} in L.

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Hilbert 5-th problem

In 1953, after the combined efforts of several mathematicians including Von Neumann, Montgomerry and Zippin, Iwasawa, Kuranishi, Gleason and Yamabe, the following theorem was established: Theorem (Structure of locally compact groups) Every locally compact group G is a generalized Lie group.

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Hilbert 5-th problem

In 1953, after the combined efforts of several mathematicians including Von Neumann, Montgomerry and Zippin, Iwasawa, Kuranishi, Gleason and Yamabe, the following theorem was established: Theorem (Structure of locally compact groups) Every locally compact group G is a generalized Lie group. ”generalized Lie group” means: G has an open subgroup G ′ such that every neighborhood of the identity in G ′ contains a closed normal subgroup N of G ′ such that G ′/N is a Lie group with finitely many connected components.

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Hilbert 5-th problem

The proof was notoriously technical but roughly went as follows: show that G is a generalized NSS group (i.e. replace “Lie group” by “NSS group” in the above statement) show that NSS groups are Lie groups. NSS is for “No Small Subgroups”.

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Hilbert 5-th problem

The proof was notoriously technical but roughly went as follows: show that G is a generalized NSS group (i.e. replace “Lie group” by “NSS group” in the above statement) show that NSS groups are Lie groups. NSS is for “No Small Subgroups”. A locally compact group G is NSS if there is a neighborhood U of the identity in G containing no non-trivial subgroups.

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Hilbert 5-th problem

The proof was notoriously technical but roughly went as follows: show that G is a generalized NSS group (i.e. replace “Lie group” by “NSS group” in the above statement) show that NSS groups are Lie groups. NSS is for “No Small Subgroups”. A locally compact group G is NSS if there is a neighborhood U of the identity in G containing no non-trivial subgroups. Note: when G is compact, this is the Peter-Weyl/Von Neumann theorem.

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Hilbert 5-th problem

Key to both steps are the so-called Gleason lemmas, whose purpose is to show that the property for an element g ∈ G of being close to the identity together with all its powers up to some large number is closed under products.

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Hilbert 5-th problem

Key to both steps are the so-called Gleason lemmas, whose purpose is to show that the property for an element g ∈ G of being close to the identity together with all its powers up to some large number is closed under products. One way to phrase it is in terms of escape norms: ||g||U := inf{ 1 n + 1; gi ∈ U for all i n}. where U is a compact neighborhood of {1}.

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One way to phrase it is in terms of escape norms: ||g||U := inf{ 1 n + 1; gi ∈ U for all i n}. then, Lemma (Gleason lemma) If G is locally compact, then every neighborhood of the identity contains a smaller neighborhood U such that (i) ||hgh−1||U C||g||U if h ∈ U (ii) ||gh||U C(||g||U + ||h||U) (iii) ||[g, h]||U C||g||U||h||U

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Lemma (Gleason lemma) If G is locally compact, then every neighborhood of the identity contains a smaller neighborhood U such that (i) ||hgh−1||U C||g||U if h ∈ U (ii) ||gh||U C(||g||U + ||h||U) (iii) ||[g, h]||U C||g||U||h||U From (i) and (ii), we get that H := {g ∈ U, ||g||U = 0} is a normal subgroup. And U/H is NSS.

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Lemma (Gleason lemma) If G is locally compact, then every neighborhood of the identity contains a smaller neighborhood U such that (i) ||hgh−1||U C||g||U if h ∈ U (ii) ||gh||U C(||g||U + ||h||U) (iii) ||[g, h]||U C||g||U||h||U From (i) and (ii), we get that H := {g ∈ U, ||g||U = 0} is a normal subgroup. And U/H is NSS. From (ii) and (iii), one can derive that if X(t) and Y (t) are

• ne-parameter subgroups R → G, then

X + Y := t → lim

n (X(1

n)Y (1 n))[nt] is well-defined in G.

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From (ii) and (iii), one can derive that if X(t) and Y (t) are

• ne-parameter subgroups R → G, then

X + Y := t → lim

n (X(1

n)Y (1 n))[nt] is well-defined in G. As a consequence: Proposition Let G be a locally compact group and L(G) be the set of

• ne-parameter subgroups. Then L(G) has naturally the structure
• f a topological vector space.

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From (ii) and (iii), one can derive that if X(t) and Y (t) are

• ne-parameter subgroups R → G, then

X + Y := t → lim

n (X(1

n)Y (1 n))[nt] is well-defined in G. As a consequence: Proposition Let G be a locally compact group and L(G) be the set of

• ne-parameter subgroups. Then L(G) has naturally the structure
• f a topological vector space.

If G has NSS, then one shows easily that L(G) is locally compact hence finite dimensional and exp : X ∈ L(G) → X(1) ∈ G is a local homeomorphism. From there one can use the adjoint representation, Cartan’s embedding theorem and Kuranishi’s theorem that extensions of Lie are Lie to get that G is a Lie group (alternatively work out the C 1,1-Campbell-Baker -Hausdorff formula as in Terry’s blog post).

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Gleason’s lemma is best proved using non-standard analysis → J. Hirschfeld, L. van den Dries, I. Goldbring...

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Back to approximate groups and Step 2 of the main theorem

We prove a version of Gleason’s lemma for non-standard finite approximate groups. Again, the escape norm for any U ⊂ A4 is defined by the non-standard quantity: ||g||U := inf{ 1 n + 1; gi ∈ U for all i n}. Lemma (Gleason for approximate groups) If A is non-standard finite K-approximate group, then A4 contains a compact (definable) neighborhood U of 1 such that (i) ||hgh−1||U C||g||U if h ∈ U (ii) ||gh||U C(||g||U + ||h||U) (iii) ||[g, h]||U C||g||U||h||U

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Lemma (Gleason for approximate groups) If A is non-standard finite K-approximate group, then A4 contains a compact (definable) neighborhood U of 1 such that (i) ||hgh−1||U C||g||U if h ∈ U (ii) ||gh||U C(||g||U + ||h||U) (iii) ||[g, h]||U C||g||U||h||U Consequences: from (i) and (ii) we see that H := {g ∈ U; ||g||U = 0} is a subgroup normalized by U. from (ii) and (iii) we see that the element e with smallest non zero norm ||e||U = inf{||g||U = 0, g ∈ U} is centralized by U modulo H.

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from (i) and (ii) we see that H := {g ∈ U; ||g||U = 0} is a (definable) subgroup normalized by U. from (ii) and (iii) we see that the element e with smallest non zero norm ||e||U = inf{||g||U = 0, g ∈ U} is centralized by U modulo H. Consequences: setting A′ := U (1) H is a definable subgroup and A′/H has no subgroups.

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from (i) and (ii) we see that H := {g ∈ U; ||g||U = 0} is a (definable) subgroup normalized by U. from (ii) and (iii) we see that the element e with smallest non zero norm ||e||U = inf{||g||U = 0, g ∈ U} is centralized by U modulo H. Consequences: setting A′ := U (1) H is a definable subgroup and A′/H has no subgroups. (2) A′′ := A′/H maps onto a neighborhood of the identity of an NSS locally compact (local) group, i.e. a connected (local) Lie group L.

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from (i) and (ii) we see that H := {g ∈ U; ||g||U = 0} is a (definable) subgroup normalized by U. from (ii) and (iii) we see that the element e with smallest non zero norm ||e||U = inf{||g||U = 0, g ∈ U} is centralized by U modulo H. Consequences: setting A′ := U (1) H is a definable subgroup and A′/H has no subgroups. (2) A′′ := A′/H maps onto a neighborhood of the identity of an NSS locally compact (local) group, i.e. a connected (local) Lie group L. (3) the (non-standard) progression P := {en, |n|

1 ||e||U } maps to

a one-parameter central local subgroup of L.

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Step 3: building the nilprogression

Main point: we can quotient out the central progression P without introducing new torsion.

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Step 3: building the nilprogression

Main point: we can quotient out the central progression P without introducing new torsion. → A′′/P will still have no subgroups and will map onto a connected (local) Lie group of dimension dim L − 1.

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Step 3: building the nilprogression

Main point: we can quotient out the central progression P without introducing new torsion. → A′′/P will still have no subgroups and will map onto a connected (local) Lie group of dimension dim L − 1. → this process will thus stop in finite time until dim L = 0.

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Thank you!

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