aa r X i v : . [ m a t h . G R ] F e b ON THE COKERNEL OF THE BAUMSLAG RATIONALIZATION
SERGEI O. IVANOV
Abstract.
We prove that for the free group of rank two F the cokernel of thehomomorphism to its Baumslag rationalization F → Bau ( F ) is not abelian.Moreover, this cokernel contains a free subgroup of any finite rank. Thisanswers a question of Emmanuel Farjoun. Introduction
A localization on a category C is a couple L = ( L, η ) , where L ∶ C → C is a functorand η ∶ Id → L is a natural transformation such that the natural transformations ηL, Lη ∶ L → L are equal and they are isomorphisms. Equivalently one can definea localization as a monad whose multiplication map is an isomorphism, or as areflection with respect to some reflective subcategory. In recent years there hasbeen an increased interest in localizations on the category of groups (see [1], [5],[9], [4], [11]). In personal communication, Emmanuel Dror Farjoun formulated thefollowing conjectures. Conjecture 1.
Any localization on the category of groups sends a nilpotent groupto a nilpotent group.
Conjecture 2.
For any localization L on the category of groups and for any finite p -group G the map G → LG is surjective. Conjecture 3.
For any localization L on the category of groups and any group G the cokernel of the map G → LG is abelian. Generally Conjectures 1 and 2 are open but there are some particular positiveresults in this direction. Conjecture 1 was proven for nilpotent groups of class ≤ G iscalled uniquely divisible (or a Q -group), if for any n ∈ Z the map G → G, g ↦ g n is bijective. The Baumslag rationalization of a group G is the universal map to auniquely divisible group η G ∶ G ⟶ Bau ( G ) . Baumslag rationalization exists for any group and defines a localization ( Bau , η ) onthe category of groups. The work is supported by: (1) Ministry of Science and Higher Education of the Russian Feder-ation, agreement 075-15-2019-1619; (2) the grant of the Government of the Russian Federation forthe state support of scientific research carried out under the supervision of leading scientists, agree-ment 14.W03.31.0030 dated 15.02.2018; (3) RFBR according to the research project 20-01-00030;(4) Russian Federation President Grant for Support of Young Scientists MK-681.2020.1.
A homomorphism f ∶ H → G is called a Nikolov-Segal map if it satisfies [ G, G ] = [ G, Im ( f )] . Note that the cokernel of a Nikolov-Segal map is abelian. The term ’Nikolov-Segalmap’ is motivated by a theorem of Nikolov and Segal [10, Th. 1.7] which impliesthat for a finitely generated group G the homomorphism to its profinite completion G → ̂ G is a Nikolov-Segal map. It is also known [1, Prop. 8.1] that if G is nilpotent,then the map η G ∶ G → Bau ( G ) is a Nikolov-Segal map.For any group G we consider the cokernel of the map to its Baumslag rational-ization CBau ( G ) ∶ = Coker ( η G ∶ G → Bau ( G )) . In other words,
CBau ( G ) is the quotient of Bau ( G ) by the normal closure of Im ( η G ) . Since η G ∶ G → Bau ( G ) is a Nikolov-Segal map for a nilpotent group G , CBau ( G ) isabelian in this case. So Conjecture 3 holds for the case of Baumslag rationalizationand a nilpotent group G. The aim of this paper is to prove that Conjecture 3 failsfor the Baumslag rationalization and the free group of rank 2. Moreover, we provethe following theorem.
Theorem.
Let F be the free group of rank 2. Then CBau ( F ) contains a freesubgroup of any finite rank. In particular, Conjecture 3 fails. Since
CBau ( F ) contains a free subgroup of any finite rank, it is not abelian andnot even solvable. Moreover, it is not in any proper variety of groups.It seems that right exact localizations on the category of groups that were studiedin [1] have much nicer properties than arbitrary ones. For example, as we noticedabove, Conjectures 1 and 2 hold for right exact localizations. However, Conjecture3 fails even in this case because the Baumslag rationalization is a right exact lo-calization [1, Th 4.1]. But we still have some reasons to believe that Conjecture 3holds for right exact localizations and nilpotent groups. Moreover, we believe thatin this case the map G → LG is a Nikolov-Segal map. We leave it here in the formof new conjecture. Conjecture 4.
For any right exact localization L on the category of groups andany nilpotent group G the map G → LG is a Nikolov-Segal map. Acknowledgments
I am grateful to Emmanuel Farjoun, Danil Akhtiamov and Fedor Pavutnitskiyfor helpful discussions.
Proof of the Theorem
A group G is called divisible (resp. uniquely divisible), if for any n ∈ Z the map G → G, g ↦ g n is surjective (resp. bijective). A group is uniquely divisible if andonly if it is local with respect to the homomorphism Z ↪ Q . The existence of theBaumslag localization for any group follows from [3, Cor. 1.7].The free uniquely divisible group (or the free Q -group ) generated by a set X isdefined as F Q ( X ) = Bau ( F ( X )) , where F ( X ) is the free group generated by X . Recently Jaikin-Zapirain proveda conjecture of Baumslag that F Q ( X ) is residually torsion-free nilpotent [7]. It is N THE COKERNEL OF THE BAUMSLAG RATIONALIZATION 3 easy to see that any map from the set X to a uniquely divisible group ϕ ∶ X → G can be uniquely extended to a homomorphism Φ ∶ F Q ( X ) → G. Gilbert Baumslagproved the following version of this statement for all divisible groups.
Theorem 1 (Baumslag, [2, Th. 39.6]) . Any map from a set to a (possibly non-uniquely) divisible group ϕ ∶ X → G can be extended to a (possibly non-unique)homomorphism Φ ∶ F Q ( X ) → G. Lemma 2.
Let ϕ ∶ X → G be a map from a set X to a divisible group G. Assumethat all elements of ϕ ( X ) are torsion elements. Then the subgroup ⟨ ϕ ( X )⟩ is asubquotient of CBau ( F ( X )) . Proof.
By Theorem 1 we have a homomorphism Φ ∶ F Q ( X ) → G such that Φ ( x ) = ϕ ( x ) for x ∈ X. Since ϕ ( X ) consists of torsion elements, for any x ∈ X there existsa number n x ≥ ϕ ( x ) n x = . By the universal property of F Q ( X ) thereexists a unique homomorphism Ψ ∶ F Q ( X ) → F Q ( X ) such that Ψ ( x ) = x n x forany x ∈ X. Similarly there exists a unique homomorphism Ψ ′ ∶ F Q ( X ) → F Q ( x ) such that Ψ ′ ( x ) = x nx for any x ∈ X. It is easy to see that Ψ − = Ψ ′ , and hence Ψis an automorphism. The compositionΦΨ ∶ F Q ( X ) ⟶ G satisfies ΦΨ ( x ) = ϕ ( x ) n x = x ∈ X. Thus the homomorphism ΦΨ η F ( X ) istrivial, and hence, the homomorphism ΦΨ induces a homomorphismΘ ∶ CBau ( F ( X )) → G. Since Ψ is an automorphism, we have ⟨ ϕ ( X )⟩ ⊆ Im ( Φ ) = Im ( ΦΨ ) = Im ( Θ ) . The assertion follows. (cid:3)
In order to prove the theorem we need to add two additional well known ingre-dients: some information about subgroups of the special orthogonal group SO ( ) , which is an interesting example of a divisible but not uniquely divisible group; andsome information about the kernel of the map from the free product to the directproduct for any groups G ∗ H → G × H. Consider the special orthogonal group SO ( ) , the group of rotations of R . Fol-lowing Radin and Sadun [12], for each positive integer p we denote by R π / px therotation around the axis x on the angle 2 π / p, and for any positive integer q we de-note by R π / qz the rotation around the axis z on on the angle 2 π / q. We also denoteby G ( p, q ) the subgroup of SO ( ) generated by R π / px and R π / qz G ( p, q ) = ⟨ R π / px , R π / qz ⟩ ⊆ SO ( ) . Theorem 3 (Radin, Sadun, [12, Th.2]) . If p and q are odd, then G ( p, q ) ≅ Z / p ∗ Z / q. For any groups
G, H we set G □ H ∶ = Ker ( G ∗ H → G × H ) . The following statement is well-known and the standard reference is [8] (see also[6]).
SERGEI O. IVANOV
Theorem 4 (Levi [8]) . For any groups
G, H the group G □ H is a free group freelygenerated by the commutators [ g, h ] for g ∈ G \ { } , h ∈ H \ { } G □ H ≅ F (( G \ { }) × ( H \ { })) . Now we are ready to prove our theorem.
Proof of Theorem.
Note that the group SO ( ) is divisible. Indeed, any its elementis a rotation around some line on some angle α , and the rotations around the sameline on the angles α / n are roots of this element. Consider two odd positive integers p, q ≥ ϕ ∶ { x, y } → SO ( ) given by ϕ ( x ) = R π / px and ϕ ( y ) = R π / qy . Then by Lemma 2 we obtain that G ( p, q ) is a subquotient of CBau ( F ) , where F = F ( x, y ) is the free group of rank 2. Using Theorem 3, we obtain that Z / p ∗ Z / q is a subquotient of CBau ( F ) . Hence Z / p □ Z / q is also a subquotient of CBau ( F ) . By Theorem 4 the group Z / p □ Z / q is a free group with ( p − )( q − ) generators. Itfollows that a free group with ( p − )( q − ) generators is a subquotient of CBau ( F ) . Since any epimorphism to a free group splits, a free group with ( p − )( q − ) generators is a subgroup of CBau ( F ) . (cid:3) References [1] Danil Akhtiamov, Sergei O Ivanov, and Fedor Pavutnitskiy. “Right exactlocalizations of groups”. In: arXiv preprint arXiv:1905.07612 (2019).[2] Gilbert Baumslag. “Some aspects of groups with unique roots”. In:
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London Math. Soc. Lecture Notes Ser.,Cambridge Univ. Press, Cambridge (1992).[4] Carles Casacuberta. “On structures preserved by idempotent transformationsof groups and homotopy types”. In:
Contemporary Mathematics
262 (2000),pp. 39–68.[5] Ram´on Flores and Jos´e L Rodr´ıguez. “On localizations of quasi-simple groupswith given countable center”. In: arXiv preprint arXiv:1810.11400 (2018).[6] ND Gilbert and Philip J Higgins. “The non-abelian tensor product of groupsand related constructions”. In:
Glasgow Mathematical Journal Q -groups are residually torsion-free nilpotent”.In: (2020).[8] Friedrich Levi. The commutator group of a free product . Verlag nicht ermit-telbar, 1941.[9] Assaf Libman. “Cardinality and nilpotency of localizations of groups andG-modules”. In:
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