Compact groups with a set of positive Haar measure satisfying a nilpotent law
aa r X i v : . [ m a t h . G R ] J a n COMPACT GROUPS WITH A SET OF POSITIVE HAARMEASURE SATISFYING A NILPOTENT LAW
ALIREZA ABDOLLAHI AND MEISAM SOLEIMANI MALEKAN
Abstract.
The following question is proposed in [4, Question 1.20].Let G be a compact group, and suppose that N k ( G ) = { ( x , . . . , x k +1 ) ∈ G k +1 | [ x , . . . , x k +1 ] = 1 } has positive Haar measure in G k +1 . Does G havean open k -step nilpotent subgroup?The case k = 1 is already known. We positively answer it for k = 2. Introduction and Results
Let G be a (Hausdorff) compact group. Then G has a unique normalized Haarmeasure denoted by m G . The following question is proposed in [4]. Question 1.1. [4, Question 1.20] Let G be a compact group, and suppose that N k ( G ) = { ( x , . . . , x k +1 ) ∈ G k +1 | [ x , . . . , x k +1 ] = 1 } has positive Haar measurein G k +1 . Does G have an open k -step nilpotent subgroup?Positive answer to Question 1.1 is known for k = 1 (see [3, Theorem 1.2]). Itfollows from [4, Theorem 1.19] that Question 1.1 has positive answer for arbitrary k whenever we further assume that G is totally disconnected i.e. G is a profinitegroup. Here we answer positively Question 1.1 for k = 2 (see Theorem 3.2 below).2. A preliminary lemma
We need the following lemma in the proof of our main result.
Lemma 2.1.
Suppose that G is a group and x , x , x , g , g , g ∈ G are such that = [ x , x , x ] = [ x g , x g , x g ] = [ x g , x g , x ] = [ x g , x , x g ] = [ x g , x , x ] = [ x g , x , x g ] = [ x , x , x g ] = [ x , x g , x g ] = [ x , x g , x ] = [ x , x , x g ] = [ x , x g , x g ] = [ x , x , x g ] . Then [ g , g , g ] = 1 .Proof. We will throughout using famous commutator calculus identities.1 = [ x g , x g , g ] = [[ x g , g ][ x g , x ] g , g ] by (2) and (3)= [[ x g , g ][ x g , x ] , g ] by (4) and (5)= [ x g , g , g ] = [[ x , g ] g [ g , g ] , g ] by (5) and (6). (I) Mathematics Subject Classification.
Key words and phrases.
Compact groups, subsets with positive Haar measure, 2-step nilpotentgroups.
On the other hand,1 = [ x , x g , g ] by (8) and (9)= [[ x , g ][ x , x ] g , g ] = [[ x , g ][ x , x ] , g ] by (1) and (10)= [ x , g , g ] by (1) and (7). (II)Also, 1 = [ x , x g , g ] by (9) and (11)= [[ x , g ][ x , x ] g , g ] = [[ x , g ][ x , x ] , g ] by (1) and (10)= [ x , g , g ] by (1) and (12). (III)Now it follows from (I), (II) and (III) that [ g , g , g ] = 1. (cid:3) Remark 2.2.
The “left version” ( g i x j instead of x j g i ) of Lemma 3.2 is not clearto hold. The validity of a similar result to Lemma 3.2 for commutators with lengthmore than 3 is also under question.3. Compact groups with many elements satisfying the -step nilpotentlaw We need the “right version” of [5, Theorem 2.3] as follows.
Theorem 3.1. If A is a measurable subset with positive Haar measure in a com-pact group G , then for any positive integer k there exists an open subset U of G containing such that m G ( A ∩ Au ∩ · · · ∩ Au k ) > for all u , . . . , u k ∈ U .Proof. Since m G ( A ) = m G ( A − ), it follows from Theorem 2.3 of [5] that thereexists an open subset V of G containing 1 such that m G ( A − ∩ v A − ∩ · · · ∩ v k A − ) > v , . . . , v k ∈ V . By [2, Theorem 4.5], there exists an open subset U ⊆ V suchthat 1 ∈ U and U = U − . Thus for all u , . . . , u k ∈ U < m G ( A − ∩ u − A − ∩ · · · ∩ u − k A − )= m G (( A ∩ Au ∩ · · · ∩ Au k ) − )= m G ( A ∩ Au ∩ · · · ∩ Au k )This completes the proof. (cid:3) Now we can prove our main result.
Theorem 3.2.
Let G be a compact group, and suppose that N ( G ) = { ( x , x , x ) ∈ G × G × G | [ x , x , x ] = 1 } has positive Haar measure in G × G × G . Then G hasan open -step nilpotent subgroup.Proof. Let X := N ( G ). It follows from Theorem 3.1 and [2, Theorem 4.5] thatthere exists an open subset U = U − of G containing 1 such that X ∩ X ¯ u ∩ · · · ∩ X ¯ u = ∅ ( ∗ ) OMPACT GROUPS 3 for all ¯ u , . . . , ¯ u ∈ U × U × U . Now take arbitrary elements g , g , g ∈ U andconsider¯ u = ( g − , g − , g − ) , ¯ u = ( g − , g − , , ¯ u = ( g − , , g − )¯ u = ( g − , , , ¯ u = ( g − , , g − ) , ¯ u = (1 , , g − ) , ¯ u = (1 , g − , g − )¯ u = (1 , g − , , ¯ u = (1 , , g − ) , ¯ u = (1 , g − , g − ) , ¯ u = (1 , , g − ) . By ( ∗ ), there exists ( x , x , x ) ∈ X such that all the following 3-tuples are in X .( x g , x g , x g ) , ( x g , x g , x ) , ( x g , x , x g )( x g , x , x ) , ( x g , x , x g ) , ( x , x , x g ) , ( x , x g , x g )( x , x g , x ) , ( x , x , x g ) , ( x , x g , x g ) , ( x , x , x g ) . Now Lemma 2.1 implies that h g , g , g i is nilpotent of class at most 2. Thereforethe subgroup H := h U i generated by U is 2-step nilpotent. Since H = S n ∈ N U n , H is open in G . This completes the proof. (cid:3) References [1] G. B. Folland, A Course in Abstract Harmonic Analysis, Stud. Adv. Math., Taylor & Francis,London, 1994.[2] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis: Structure and Analysis for CompactGroups Analysis on Locally Compact Abelian Groups, Grundlehren Math. Wiss., Springer,Berlin, 2013.[3] K. H. Hofmann and F. G. Russo, The probability that x and y commute in a compact group,Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 3, 557-571.[4] A. Martino, M. C. H. Tointon, M. Valiunas and E. Ventura, Probabilistic Nilpotence in infinitegroups, to appear in Israel J. Math.[5] M. Soleimani Malekan, A. Abdollahi and M. Ebrahimi, Compact groups with many elementsof bounded order, J. Group Theory, 23 (2020) no. 6 991–998. Department of Pure Mathematics, Faculty of Mathematics and Statistics, Universityof Isfahan, Isfahan 81746-73441, Iran.
Email address : [email protected] Department of Mathematics, University of Isfahan, Isfahan 81746-73441, Iran; Insti-tute for Research in Fundamental Sciences, School of Mathematics, Tehran, Iran.
Email address ::