Minimality in topological groups and Heisenberg type groups
aa r X i v : . [ m a t h . GN ] J un Minimality in topological groups and Heisenberg type groups
Menachem Shlossberg
Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israelemail: [email protected]
June 14, 2007
Abstract
We study relatively minimal subgroups in topological groups. We find, in par-ticular, some natural relatively minimal subgroups in unipotent groups which aredefined over ”good” rings. By ”good” rings we mean archimedean absolute valued(not necessarily associative) division rings. Some of the classical rings which we con-sider besides the field of reals are the ring of quaternions and the ring of octonions.This way we generalize in part a previous result which was obtained by Dikranjanand Megrelishvili [1] and involved the Heisenberg group.
A Hausdorff topological group G is minimal if G does not admit a strictly coarser Haus-dorff group topology or equivalently if every injective continuous group homomorphism G → P into a Hausdorff topological group is a topological embedding. The concept ofminimal topological groups was introduced by Stephenson [9] and Do¨ıchinov [2] in 1971as a natural generalization of compact groups.Heisenberg group and more precisely its generalization, which we present in section2 (see also [4, 7]), provides many examples of minimal groups.Recently Dikranjan and Megrelishvili [1] introduced the concept of co-minimality (seeDefinition 2.5) of subgroups in topological groups after the latter author had introducedthe concept of relative minimality (see Definition 2.3 and also [3]) of subgroups intopological groups and found such subgroups in a generalized Heisenberg group (see[4, 7]).In [1, Proposition 2.4.2] Megrelishvili and Dikranjan proved that the canonical bilin-ear mapping V × V ∗ → R , < v, f > = f ( v ) is strongly minimal (see Definition 2.7) forall normed spaces V. The following result is obtained as a particular case: The inner product map R n × R n → R is strongly minimal. The latter result leads in [1] and [3] to the conclusion that for every n ∈ N the subgroups (cid:26) a I n
00 0 1 (cid:12)(cid:12)(cid:12)(cid:12) a ∈ R n (cid:27) , (cid:26) b (cid:12)(cid:12)(cid:12)(cid:12) b ∈ R n (cid:27) (cid:26) a c I n b (cid:12)(cid:12)(cid:12)(cid:12) a, b ∈ R n , c ∈ R (cid:27) which is known as the classical 2 n + 1-dimensional Heisenberg group (where I n denotesthe identity matrix of size n ). Theorem 3.4 and Corollary 3.6 generalize these resultsand allow us to replace the field of reals by every other archimedean absolute valued (notnecessarily associative) division ring, for example, they can be applied for the ring ofquaternions and the ring of octonions. Theorem 3.9 provides a different generalization.It generalizes the case of the classical real 3-dimensional Heisenberg group. We considerfor every n ∈ N the group of upper unitriangular matrices over an archimedean absolutevalued field of size n + 2 × n + 2 and we find relatively minimal subgroups of this group.This result is a generalization since the classical real 3-dimensional Heisenberg group isa unitriangular group. This theorem is not new when we take n = 1 and consider thefield to reals. However, we obtain a new result even for R when we take n > . Thistheorem can also be applied for the fields Q and C . The group H = (cid:26) x a y (cid:12)(cid:12)(cid:12)(cid:12) x, y, a ∈ R (cid:27) ∼ = ( R × R ) ⋋ R is known as the classical real 3-dimensional Heisenberg Group .We need a far reaching generalization [4, 7, 3], the generalized Heisenberg group ,which is based on biadditive mappings.
Definition 2.1
Let
E, F, A be abelian groups. A map w : E × F → A is said to bebiadditive if the induced mappings w x : F → A, w f : E → A, w x ( f ) := w ( x, f ) =: w f ( x ) are homomorphisms for all x ∈ E and f ∈ F . Definition 2.2 [3, Definition 1.1] Let
E, F and A be Hausdorff abelian topologicalgroups and w : E × F → A be a continuous biadditive mapping. Denote by H ( w ) =( A × E ) ⋋ F the topological semidirect product (say, generalized Heisenberg group in-duced by w ) of F and the group A × E . The group operation is defined as follows: for apair u = ( a , x , f ) , u = ( a , x , f ) we define u u = ( a + a + f ( x ) , x + x , f + f ) where, f ( x ) = w ( x , f ) . Then H ( w ) becomes a Hausdorff topological group. In the caseof a normed space X and a canonical biadditive function w : X × X ∗ → R ( x, f ) f ( x )2 where X ∗ is the Banach space of all continuous functionals from X to R , known as thedual space of X ) we write H ( X ) instead of H ( w ) . Definition 2.3 [1, Definition 1.1.1] Let X be a subset of a Hausdorff topological group ( G, τ ) . We say that X is relatively minimal in G if every coarser Hausdorff group topology σ ⊂ τ of G induces on X the original topology. That is, σ | X = τ | X . Theorem 2.4 [3, Theorem 2.2] The subgroups X and X ∗ are relatively minimal in thegeneralized Heisenberg group H ( X ) = ( R × X ) ⋋ X ∗ for every normed space X . The concept of co-minimality which is presented below played a major role in gener-alizing and strengthen Theorem 2.4. Let H be a subgroup of a topological group ( G, γ ).The quotient topology on the left coset space
G/H := { gH } g ∈ G will be denoted by γ/H. Definition 2.5 [1, Definition 1.1.2] Let X be a topological subgroup of a Hausdorfftopological group ( G, τ ) . We say that X is co-minimal in G if every coarser Hausdorffgroup topology σ ⊂ τ of G induces on the coset space G/X the original topology. Thatis, σ/X = τ /X . Definition 2.6
Let
E, F, A be abelian Hausdorff groups. A biadditive mapping w : E × F → A will be called separated if for every pair ( x , f ) of nonzero elements thereexists a pair ( x, f ) such that f ( x ) = 0 A and f ( x ) = 0 A , where f ( x ) = w ( x, f ) . Definition 2.7 [1, Definition 2.2] Let ( E, σ ) , ( F, τ ) , ( A, ν ) be abelian Hausdorff topolog-ical groups. A continuous separated biadditive mapping w : ( E, σ ) × ( F, τ ) → ( A, ν ) will be called strongly minimal if for every coarser triple ( σ , τ , ν ) of Hausdorff grouptopologies σ ⊂ σ, τ ⊂ τ, ν ⊂ ν such that w : ( E, σ ) × ( F, τ ) → ( A, ν ) is continuous (in such cases we say that the triple ( σ , τ , ν ) is compatible) it followsthat σ = σ, τ = τ . We say that the biadditive mapping is minimal if σ = σ, τ = τ holds for every compatible triple ( σ , τ , ν ) (with ν := ν ). Remark 2.8
The multiplication map A × A → A is minimal for every Hausdorff topolog-ical unital ring A . However note that the multiplication map Z × Z → Z (being minimal)is not strongly minimal. The following theorem which uses the concept of co-minimality and strongly biaddi-tive mappings generalizes Theorem 2.4.
Theorem 2.9 [1, Theorem 4.1] Let w : ( E, σ ) × ( F, τ ) → ( A, ν ) be a strongly minimalbiadditive mapping. Then:1. A, A × E and A × F are co-minimal subgroups of the Heisenberg group H ( w ) .2. E × F is a relatively minimal subset in H ( w ) .3. The subgroups E and F are relatively minimal in H ( w ) . emark 2.10 The mapping w : X × X ∗ → R ( x, f ) f ( x ) is strongly minimal forevery normed space X . Therefore, Theorem 2.9 is indeed a generalization of Theorem2.4. Corollary 2.11 [1, Corollary 4.2] The following conditions are equivalent:1. H ( w ) is a minimal group.2. w is a minimal biadditive mapping and A is a minimal group. Since Z with the p -adic topology τ p is a minimal group for every prime p [6] the followingcorollary is obtained by Remark 2.8: Corollary 2.12 [1, Corollary 4.6.2] The Heisenberg group H ( w ) = ( Z × Z ) ⋋ Z of themapping ( Z , τ p ) × ( Z , τ p ) → ( Z , τ p ) is a minimal two step nilpotent precompact group forevery p -adic topology τ p . In this paper rings are not assumed to be necessarily associative. However, when weconsider division rings we assume they are associative unless otherwise is stated.
Definition 3.1
An absolute value A on a (not necessarily associative) division ring K is archimedean if there exists n ∈ N such that A ( n ) > (where, for any n ∈ N , n := n. · · · + 1 (n terms). From now on we use the following notations for a commutative group G which isdenoted additively: the zero element is denoted by 0 G . If G is also a ring with multi-plicative unit we denote this element by 1 G . In the case of a group G which is a directproduct of groups we shall use slightly different notation and denote the zero element by¯0 G . Lemma 3.2
Let X be a (not necessarily associative) division ring with an archimedeanabsolute value A and denote by τ the ring topology induced by the absolute value. Let σ ⊂ τ be a strictly coarser group topology with respect to the additive structure of X .Then, every σ -neighborhood of X is unbounded with respect to the absolute value. Proof.
Since σ is strictly coarser than τ , there exists an open ball B (0 , r ) with r > σ -neighborhood of 0 X . Then, for every σ -neighborhood U of 0 X thereexists x in U such that A ( x ) ≥ r . Fix a σ -neighborhood V of 0 X . We show that V isunbounded with respect to the absolute value A . Since A is an archimedean absolutevalue there exists n ∈ N such that A ( n ) = c > . Clearly, for every m ∈ N there existsa σ -neighborhood W of 0 X such that W + W + · · · + W | {z } n m ⊂ V.
4y our assumption there exists x ∈ W such that A ( x ) ≥ r . Now for the element n m x := x + x + · · · + x | {z } n m ∈ V we obtain that A ( n m x ) = A ( n ) m A ( x ) ≥ c m r . This clearly means that V is unbounded. (cid:3) Lemma 3.3
Let ( G i ) i ∈ I be a family of topological groups. For each i ∈ I denote by τ i the topology of G i and by p i the projection of G := Q i ∈ I G i to G i . Suppose that σ is agroup topology on G which is strictly coarser than the product topology on G denoted by τ . Then there exist j ∈ I and a group topology σ j on G j which is strictly coarser than τ j , such that B j = p j ( B ) , where B j is the neighborhood filter of G j with respect to σ j and B is the neighborhood filter of ¯0 G with respect to σ . Proof.
Since the topology σ is strictly coarser than τ which is the product topology on G , we get that there exists j ∈ I for which the projection p j : ( G, σ ) → ( G j , τ j ) is notcontinuous at ¯0 G . Hence, there exist a τ j -neighborhood V of 0 G j such that p j ( O ) * V for every O ∈ B . Hence, if p j ( B ) is the neighborhood filter of 0 G j for some grouptopology σ j on G j then this topology is strictly coarser than τ j . We shall prove thatthis formulation defines a group topology σ j . Indeed, consider the normal subgroup H = Q i ∈ I F i of G where F i = (cid:26) G i if i = j { G i } if i = j . It is easy to show that ( G j , τ j ) is topologically isomorphic to the quotient group G/H of (
G, τ ). Let σ j be the finest topology on G j for which the projection p j : ( G, σ ) → G j is continuous. It is exactly the quotient topology on G j = G/H for the topological group(
G, σ ). By our construction σ j is strictly coarser than τ j . Then indeed σ j is the desiredgroup topology on G j and B j = p j ( B ) is the desired neighborhood filter. (cid:3) Theorem 3.4
Let F be a (not necessarily associative) division ring furnished with anarchimedean absolute value A . For each n ∈ N , w n : F n × F n F, w n (¯ x, ¯ y ) = n X x i y i (where (¯ x, ¯ y ) = (( x , . . . x n ) , ( y , . . . y n )) is a strongly minimal biadditive mapping. Proof.
Clearly, for each n ∈ N , w n is a continuous separated biadditive mapping. Denoteby τ the topology of F induced by A and by τ n the product topology on F n . Considerthe max-metric d on F n . Then its topology is exactly τ n . Let ( σ, σ ′ , ν ) be a compatibletriple with respect to w n . We prove that σ = σ ′ = τ n . Assuming the contrary we getthat at least one of the group topologies σ, σ ′ is strictly coarser than τ n . We first assumethat σ is strictly coarser than τ n . Since ν is Hausdorff and ( σ, σ ′ , ν ) is compatible thereexist a ν -neighborhood Y of 0 := 0 F and V, W which are respectively σ, σ ′ -neighborhoodsof ¯0 F n such that V W ⊂ Y and in addition 1 F / ∈ Y . Since W ∈ σ ′ ⊂ τ n , then thereexists ǫ > d -ball B (0 , ǫ ) is a subset of W . Since σ is5trictly coarser than τ n (by Lemmas 3.2 and 3.3) there exists i ∈ I := { , , · · · , n } suchthat p i ( V ) is norm unbounded. Therefore, there exists ¯ x ∈ V such that A ( p i (¯ x )) > ǫ .Hence, A (( p i (¯ x )) − ) < ǫ . Now, let us consider a vector ¯ a ∈ F n such that for every j = i, a j = 0 and a i = ( p i (¯ x )) − . Clearly, ¯ a ∈ B (0 , ǫ ) ⊂ W . We then get that w n (¯ x, ¯ a ) = 1 F ∈ V W ⊂ Y . This contradicts our assumption. Using the same techniquewe can show that σ ′ can’t be strictly coarser than τ n . (cid:3) Example 3.5
1. Let F ∈ { Q , R , C } with the usual absolute value. Then for each n ∈ N the map w n : F n × F n F is strongly minimal. The case of F equals to R follows also from [1, Proposition2.42].2. For each n ∈ N the map w n : H n × H n H is strongly minimal, where H is the quaternions ring equipped with the archimedeanabsolute value defined by: k q k = ( a + b + c + d ) for each q = a + bi + cj + dk ∈ H .
3. Let G be the non-associative ring of octonions.This ring can be defined (see [11]) as pairs of quaternions (this is the Cayley-Dickson construction). Addition is defined pairwise. The product of two pairs ofquaternions ( a, b ) and ( c, d ) is defined by ( a, b )( c, d ) = ( ac − db ∗ , a ∗ d + cb ) where z ∗ = e − f i − gj − hk denotes the conjugate of z = e + f i + gj + hk. We define anorm on G as follows: k ( a + bi + cj + dk, e + f i + gj + hk ) k = ( a + b + c + d + e + f + g + h ) . This norm agrees with the standard Euclidean norm on R . It can be proved thatfor each x , x ∈ G, k x x k = k x k · k x k hence k k is an absolute value andclearly it is archimedean. Again by Theorem 3.4 the map w n : G n × G n G is strongly minimal for each n ∈ N . Corollary 3.6
Under the conditions of Theorem 3.4 we obtain the following results:1. ( F × { ¯0 F n } ) ⋋ { ¯0 F n } , ( F × F n ) ⋋ { ¯0 F n } and ( F × { ¯0 F n } ) ⋋ F n are co-minimalsubgroups of the Heisenberg group H ( w n ) .2. ( { F } × F n ) ⋋ F n is a relatively minimal subset in H ( w n ) .3. The subgroups ( { F }× F n ) ⋋ { ¯0 F n } and ( { F }× { ¯0 F n } ) ⋋ F n are relatively minimalin H ( w n ) . roof. Apply Theorem 2.9 to the strongly minimal biadditive mapping w n . (cid:3) Remark 3.7
We replace H ( w n ) by H ( F n ) for convenience ( w n is the strongly minimalbiadditive mapping from 3.4). In terms of matrices: H ( F n ) is the n + 1 -dimensionalHeisenberg group with coefficients from F which consists of square matrices of size n + 2 : A = F x x . . . x n − x n r F F F F F F y F F . . . . . . . . . ... y ... ... . . . . . . . . . F ...... ... . . . . . . F F y n − F F . . . . . . F F y n F F F . . . . . . F F and by the result (2) of Corollary 3.6 we obtain that the set of matrices B = F x x . . . x n − x n F F F F F F F y F F . . . . . . . . . ... y ... ... . . . . . . . . . F ...... ... . . . . . . F F y n − F F . . . . . . F F y n F F F . . . . . . F F is a relatively minimal subset of H ( F n ) . Lemma 3.8
1. If H is a subgroup of a topological group ( G, τ ) and X is a relativelyminimal subset in H, then X is also relatively minimal in G.
2. Let ( G , τ ) , ( G , τ ) be topological groups and H , H be their subgroups (respec-tively). If H is relatively minimal in G and there exists a topological isomorphism f : ( G , τ ) → ( G , τ ) such that the restriction to H is a topological isomorphismonto H , then H is relatively minimal in G .
3. Let ( G, τ ) be a topological group and let X be a subset of G . If X is relativelyminimal in ( G, τ ) , then every subset of X is also relatively minimal in ( G, τ ) . Proof. (1): Let σ ⊂ τ be a coarser Hausdorff group topology of G, then σ | H ⊂ τ | H is acoarser Hausdorff group topology of H. Since X is a relatively minimal subset in H, weget that σ | X = ( σ | H ) | X = ( τ | H ) | X = τ | X . Hence, X is relatively minimal in G. (2): Observe that if σ ⊂ τ is a coarser Hausdorff group topology of G , then f − ( σ ) = { f − ( U ) | U ∈ σ } ⊂ τ
7s a coarser group topology of G . Since H is relatively minimal in ( G , τ ) we obtainthat τ | H = f − ( σ ) | H . This implies that τ | H = σ | H . This completes our proof.(3): Let Y be a subset of X and σ ⊂ τ a coarser Hausdorff group topology. Then, bythe fact that X is relatively minimal in ( G, τ ) and since Y is a subset of X we obtainthat σ | Y = ( σ | X ) | Y = ( τ | X ) | Y = τ | Y . Hence, Y is relatively minimal in G. (cid:3) The following is new even for the case of F = R (for n > Theorem 3.9
Let F be a field furnished with an archimedean absolute value A . For all n ∈ N denote by U n +2 ( F ) the topological group of all n + 2 × n + 2 upper unitriangularmatrices with entries from F. Then ∀ n ∈ N and for each i, j such that i < j, ( i, j ) =(1 , n + 2) each of the subgroups G n +2 ij ( F ) := ( F F F F F F F F F F ... . . . . . . a ij ... F F F F F F . . . . . . F F ∈ U n +2 ( F ) ) (where a ij is in the ij entry) is relatively minimal in U n +2 ( F ) . Proof.
We prove the assertion for two cases: First case: i = 1 or j = n + 2 (that is theindexes from the first row or from the last column) and the second case: i > , j < n + 2.Let us consider the first case: we know by Remark 3.7 that the set S of square matricesof size n + 2: B = F x x . . . x n − x n F F F F F F F y F F . . . . . . . . . ... y ... ... . . . . . . . . . 0 F ...... ... . . . . . . 1 F F y n − F F . . . . . . 0 F F y n F F F . . . . . . F F is relatively minimal in H ( F n ) . Since H ( F n ) is a subgroup of U n +2 ( F ) we get by Lemma3.8 that S is relatively minimal in U n +2 ( F ) . Now, G n +21 j ( F ) ⊂ S for every 1 < j < n + 2and G n +2 in +2 ( F ) ⊂ S for every 1 < i < n + 2 . By Lemma 3.8 we obtain that G n +2 ij ( F ) isrelatively minimal in U n +2 ( F ) for every pair of indexes ( i, j ) such that i = 1 or j = n + 2(in addition to the demands: i < j and( i, j ) = (1 , n + 2)).Case 2: i > , j < n + 2. Fix n ∈ N and a pair ( i, j ) such that 1 < i < j < n + 2.We shall show that G n +2 ij ( F ) is relatively minimal in U n +2 ( F ) . We define the followingsubgroup of U n +2 ( F ):˜ U n +2 ( F ) := { A ∈ U n +2 ( F ) | a kl = 0 F if l = k < i } (it means that the first i − F at each entry (be-sides the diagonal)). Clearly, this group is isomorphic to the group U ( n +2 − ( i − ( F ) =8 n +3 − i ( F ) . Indeed, for every matrix A ∈ ˜ U n +2 ( F ) if we delete the first i − i − U n +3 − i ( F ) and it also clearthat this way we obtain an isomorphism. Denote this isomorphism by f. Now, G n +2 ij ( F )is a subgroup of ˜ U n +2 ( F ) and f ( G n +2 ij ( F )) = G n +3 − i j +1 − i ( F ) . Since 1 < i < j < n + 2 weobtain that i ≤ n and hence n + 3 − i ≥ . Therefore, we can use the reduction to case(1) to obtain that G n +3 − i j +1 − i ( F ) is relatively minimal in U n +3 − i ( F ) . By applying Lemma3.8 (with G := U n +3 − i ( F ) , G := ˜ U n +2 ( F ) , H := G n +3 − i j +1 − i ( F ) and H := G n +2 ij ( F )) wecan conclude that G n +2 ij ( F ) is relatively minimal in ˜ U n +2 ( F ) and hence also in U n +2 ( F )which contains ˜ U n +2 ( F ) as a subgroup. This completes our proof. (cid:3) Remark 3.10
In the particular case of F = R we obtain by previous results that forevery n ∈ N each of the subgroups G n +2 ij ( R ) is relatively minimal in SL n +2 ( R ). It isderived from the fact that SL m ( R ) is minimal for every m ∈ N (see [8, 1]). Thesegroups are also relatively minimal in GL n +2 ( R ) which contains SL n +2 ( R ) as a subgroup(see Lemma 3.8). Nevertheless, the fact that these groups are relatively minimal in U n +2 ( R ) is not derived from the minimality of SL n +2 ( R ) since U n +2 ( R ) is contained in SL n +2 ( R ) and not the opposite (that is SL n +2 ( R ) is not a subset of U n +2 ( R ) ). Definition 3.11
Let K be a Hausdorff topological division ring. A topological K -vectorspace E is straight if E is Hausdorff and for every nonzero c ∈ E, λ → λc is a homeo-morphism from K to the one-dimensional subspace Kc of E . The Hausdorff topologicaldivision ring is straight if every Hausdorff K -vector space is straight. Theorem 3.12 [10, Theorem 13.8] A nondiscrete locally retrobounded division ring isstraight. In particular, a division ring topologized by a proper absolute value is straight.
Lemma 3.13
Let ( F, τ ) be a unital Hausdorff topological ring. Consider the followingcases:1. ( F, τ ) is a minimal topological group.2. The multiplication map w : ( F, τ ) × ( F, τ ) → ( F, τ ) is strongly minimal.3. ( F, τ ) is minimal as a topological module over ( F, τ ) (i.e. there is no strictly coarserHausdorff topology σ on F for which ( F, σ ) is a topological module over ( F, τ ) ).4. ( F, τ ) is minimal as a topological ring (i.e. there is no strictly coarser Hausdorffring topology on F ).Then: (1) ⇒ (2) ⇒ (3) ⇒ (4) . Proof. (1) ⇒ (2): If F is a unital topological ring then w is minimal. Indeed, let( σ , τ , ν ) be a compatible triple then the identity maps ( F, σ ) → ( F, τ ) and (
F, τ ) → ( F, τ ) are continuous since the multiplication map w : ( F, σ ) × ( F, τ ) → ( F, τ ) iscontinuous at ( λ, F ) , (1 F , λ ) for every λ ∈ F and from the fact that ∀ λ ∈ F w ( λ, F ) = w (1 F , λ ) = λ. Clearly, in the case of a minimal topological Hausdorff group the definition of a minimalbiadditive mapping and a strongly minimal biadditive mapping coincide. The rest of theimplications are trivial. (cid:3) emark 3.14 Although (1) ⇒ (2) , the converse implication in general is not true. Forinstance, the multiplication map w : R × R → R is strongly minimal but R is not minimalas a topological Hausdorff group. Lemma 3.15
Let ( R, τ ) be a straight division ring. Let τ be a strictly coarser Hausdorfftopology on τ . Then ( R, τ ) is not a topological vector space over ( R, τ ) . Proof.
Let τ ⊂ τ . We shall show that if ( R, τ ) is a topological vector space then τ = τ .In the definition of straight division ring let K = ( R, τ ) and E = ( R, τ ) also let c = 1.Then it is clear that the identity mapping ( R, τ ) → ( R, τ ) is a homeomorphism. Hence, τ = τ . (cid:3) Remark 3.16
By our new results we get that in the case of archimedean absolute value,conditions (2)-(4) of Lemma 3.13 hold. Since a proper non-archimedean absolute valueddivision ring is a straight division ring we get by Lemma 3.15 that the conditions (3)-(4)in Lemma 3.13 hold in this situation. The question that remains open is whether themultiplication map w : ( F, τ ) × ( F, τ ) → ( F, τ ) is strongly minimal where F is a division ring and the topology τ is induced by a propernon-archimedean absolute value. We ask even more concretely: is the multiplication map w : Q × Q → Q strongly minimal when Q is equipped with the p -adic topology? I would like to thank D. Dikranjan and M. Megrelishvili for their suggestions and remarks.
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