Subgroups of the Group of Formal Power Series with the Big Powers Condition
aa r X i v : . [ m a t h . G R ] A ug SUBGROUPS OF THE GROUP OF FORMAL POWER SERIES WITHTHE BIG POWERS CONDITION
ALEXANDER BRUDNYI
Abstract.
We study the structure of discrete subgroups of the group G [[ r ]] of complexformal power series under the operation of composition of series. In particular, we provethat every finitely generated fully residually free group is embeddable to G [[ r ]]. Main Result
Let G [[ r ]] be the prounipotent group of formal power series of the form r + P ∞ i =1 c i r i +1 , c i ∈ C , i ∈ N , under the operation ◦ of composition of series. In the paper we study theproblem on the structure of discrete subgroups of G [[ r ]]. The problem is of importance, inparticular, in connection with the classification of local analytic foliations and the holonomyof local differential equations (see, e.g., [C], [CL], [EV], [IP], [L], [NY] and referencestherein). The deep results of [EV] show that in contrast to free prounipotent groups(see [LM, Cor. 4.7]) the group G [[ r ]] contains two-generator discrete subgroups which areneither abelian nor free (see also [NY] for further results in this direction). In turn, in [Br,Problem 4.15] we asked with regard to the center problem for families of Abel differentialequations whether the fundamental groups of orientable compact Riemann surfaces areembeddable to G [[ r ]]. In this paper we answer this question affirmatively. Our approachis purely group-theoretical and can be applied to a wide class of prounipotent groups.To formulate the main result of the paper we introduce several definitions.Let G be a group and u = ( u , . . . , u k ), k ∈ N , be a tuple of non-trivial elements of G . Wesay that u is commutation-free if [ u i , u i +1 ] := u i u j u − i u − j = 1 for all 1 ≤ i ≤ k −
1. In turn, u is called independent if there exists an integer n = n ( u ) ∈ N such that u α · · · u α k k = 1 forany integers α , . . . , α k ≥ n . Definition 1.1.
Group G satisfies the big powers condition if every commutation-free tuplein G is independent. The groups subject to the definition are referred to as BP -groups . The class of BP -groups contains torsion-free abelian groups, free groups and torsion-free hyperbolic groups.Also, subgroups and direct and inverse limits of BP -groups are BP as well. On the otherhand, e.g., nonabelian torsion-free nilpotent groups are not BP (see [KMS, Thm. 1]). We Mathematics Subject Classification.
Primary 20E06. Secondary 20F38.
Key words and phrases.
Group of formal power series, the big powers condition, fully residually freegroup, free product of groups.Research is supported in part by NSERC. recommend the paper [KMS] for the corresponding references and other examples andproperties of BP -groups and their applications in group theory.Let δ be an ordinal of cardinality ≤ c and(1.1) G ≤ G ≤ · · · ≤ G α ≤ G α +1 ≤ · · · ≤ G δ be a chain of subgroups such that for each limit ordinal λG λ := [ α<λ G α . Suppose that for each successor ordinal α + 1 ≤ δ one of the following holds:(i) G α +1 = G α ∗ C α F α , where F α is a nontrivial subgroup of G α , and either C α = { } or C α = C G α ( u ) = C F α ( u ) for some nontrivial u ∈ F α ;(ii) G α +1 is an extension of a centralizer of G α .Recall that an extension of a centralizer of a group G is the group h G, t | [ c, t ] = 1 , c ∈ C G ( u ) i for some nontrivial u ∈ G . Theorem 1.2. G δ is a BP -group embeddable to G [[ r ]] if and only if G is. Example 1.3. (1) Let G ( ∼ = C ) be a one-parametric subgroup of G [[ r ]] and G α +1 = G α ∗ G for all successor ordinals α + 1 ≤ δ , where δ is of the cardinality of the continuum c . Then G δ is isomorphic to the free product of c copies of C and due to Theorem 1.2 it isa BP -group embeddable to G [[ r ]].(2) A group G is called fully residually free if for any finite subset X of G there exists ahomomorphism from G to a free group that is injective on X . The notion was introducedin [B2] and since then extensively studied in connection with important problems of grouptheory and logic. Deep results of [MR] and [KM] assert that a finitely generated fullyresidually free group is embeddable to a finite sequence of extensions of centralizers ofthe free group of rank two. Hence, due Theorem 1.2(b) and part (1) of the example afinitely generated fully residually free group is a BP -group embeddable to G [[ r ]]. Since allnon-exceptional fundamental groups of compact Riemann surfaces (i.e., distinct from thefundamental groups of non-orientable surfaces of Euler characteristic 1 , −
1) are fullyresidually free (see [B1]), they are embeddable to G [[ r ]]. This answers [Br, Problem 4.15].(3) Let G Z [ t ] be the Lyndon’s completion of a finitely generated fully residually free group G . The notion was introduced in [L] in order to describe the solutions of equations in asingle variable with coefficients in a free group. The recent result of [MR] asserts that G Z [ t ] is the direct limit of a countable chain of extensions of centralizers G ≤ G ≤ G ≤ · · · .Hence, Theorem 1.2(b) and part (2) imply that G Z [ t ] is a BP -group embeddable to G [[ r ]]. Remark 1.4. (1) Let F ⊂ C be a subfield and G F [[ r ]] < G [[ r ]] be the subgroup of serieswith coefficients in F . A minor modification of the proof of Theorem 1.2 (see Section 4)leads to the following result. C G ( u ) ≤ G stands for the centralizer of an element u of a group G . UBGROUPS OF THE GROUP OF FORMAL POWER SERIES 3
Theorem 1.5.
Suppose the cardinality of G is less than c . Then G δ is a BP -groupembeddable to G R [[ r ]] if and only if G is. In particular, the Lyndon’s completion G Z [ t ] , where G is a finitely generated fully residuallyfree group, is embeddable to G R [[ r ]].(2) In view of our main result the following questions seem plausible. Problem. (a) Is G [[ r ]] a BP -group? (b) Suppose groups G , G are embeddable to G [[ r ]] . Is G ∗ G embeddable to G [[ r ]] ? (c) Let ¯ Q be the algebraic closure of the field of rational numbers Q . Is a finitely generatedfully residually free group embeddable to G ¯ Q [[ r ]] ? (Note that the proof of Theorem 1.2 uses the fact that the transcendence degree of C is c .)In a forthcoming paper we present some applications of Theorems 1.2 and 1.5 to thecenter problem for ordinary differential equations.2. Auxiliary Results BP condition.We say that a group G satisfies the separation condition if for any positive integer k andany tuples u = ( u , ..., u k ) and g = ( g , ..., g k +1 ) of elements from G such that[ g − i +1 u i g i +1 , u i +1 ] = 1 for i = 1 , . . . , k − , there exists an integer n = n ( u, g ) such that g u α g u α · · · g k u α k k g k +1 = 1for any integers α , . . . , α k ≥ n .It was proved in [KMS, Prop. 1] that a group G satisfies the big powers condition if andonly if it satisfies the separation condition.2.2. We also use some known facts about the prounipotent group G [[ r ]].The Lie algebra g of G [[ r ]] consists of formal vector fields of the form P ∞ j =1 c j e j , c j ∈ C ,where e j := − x j +1 ddx . Here the Lie bracket satisfies the identities [ e i , e j ] = ( i − j ) e i + j forall i, j ∈ N . Moreover, if v r is the formal solution of the initial value problem dvdx = ∞ X j =1 c j v j +1 , v (0) = r, then the exponential map exp : g → G [[ r ]] sends the element P ∞ j =1 c j e j to v r (1), where(2.1) v r (1) = r + ∞ X i =1 X i + ··· + i k = i ( i + 1)( i + i + 1) · · · ( i − i k + 1) c i · · · c i k k ! r i +1 . UBGROUPS OF THE GROUP OF FORMAL POWER SERIES 4
The map exp is bijective. We denote its inverse by log : G [[ r ]] → g . Then for h = r + P ∞ i =1 h i r i +1 , h i ∈ C ,(2.2) log h = ∞ X i =1 P i ( h , . . . , h i ) e i , where P i ∈ Q [ x , . . . , x i ], i ∈ N .In turn, let w ( X , . . . , X n ) be a word in the free group with generators X , . . . , X n . Forsome a , . . . , a n ∈ g we set ˜ w ( a , . . . , a n ) := w (exp( a ) , . . . , exp( a n )). Then the formula forthe composition of series and (2.1) imply that(2.3) ˜ w ( a , . . . , a n ) = r + ∞ X i =1 Q i ( a , . . . , a n ) r i +1 , where Q i is a polynomial with rational coefficients of degree i in the first i coefficients ofthe series expansions of a , . . . , a n .We also use the following fact. Lemma 2.1.
Elements exp( a ) , exp( a ) ∈ G [[ r ]] with nonzero a , a ∈ g commute iff a = λa for some λ ∈ C .Proof. If exp( − a ) exp( a ) exp( a ) = exp( a ), then passing to the logarithm we getad(exp( a ))( a ) = a , where ad is the differential at 1 of the map Ad(exp( a ))( g ) := exp( − a ) g exp( a ), g ∈ G [[ r ]].Multiplying both parts of the previous equation by t ∈ C and taking the exponents weobtain that exp( − a ) exp( ta ) exp( a ) = exp( ta ) for all x ∈ C . This implies[ a , a ] := lim t → t (cid:0) ad(exp( ta ))( a ) − a (cid:1) = 0 . Further, if a k = P ∞ j = j k c jk e j , where c j k k = 0, k = 1 ,
2, then0 = [ a , a ] = ∞ X n =1 X i + j = n c i c j [ e i , e j ] = ∞ X n =1 X i + j = n c i c j ( j − i ) e n . Thus,(2.4) X i + j = n c i c j ( j − i ) = 0 for all n ≥ . In particular, c j c j ( j − j ) = 0, i.e., j = j and there exists a nonzero λ ∈ C such that c j = λc j . UBGROUPS OF THE GROUP OF FORMAL POWER SERIES 5
Assume now that we have proved that c j = λc j for all j ≤ j < n . Let us prove that c n = λc n as well. Indeed, due to (2.4) and our hypothesis we obtain0 = X i + j = n + j c i c j ( j − i ) = c n c j ( j − n ) + λc j c n ( n − j ) + X i + j = n + j , i>j λc i c j ( j − i )= c n c j ( j − n ) + λc j c n ( n − j ) . This gives the required. Hence, we obtain by induction that a = λa .The converse statement is obvious. (cid:3) A subgroup H of a group G is called malnormal if H ∩ g − Hg = { } , g ∈ G implies g ∈ H . A group is called CSA if every maximal abelian subgroup is malnormal.As a corollary of Lemma 2.1 we obtain:
Proposition 2.2.
Any subgroup of G [[ r ]] is CSA .Proof.
Let H ⊂ G [[ r ]] and A ⊂ H be a maximal abelian subgroup of H . Without loss ofgenerality we may assume that H is nontrivial. Then A contains a centralizer C H ( h ) of anontrivial element h ∈ H . Due to Lemma 2.1, each g ∈ H such that [ g, h ] = 1 is of theform exp( λ log( h )) for some nonzero λ ∈ C . Then A = h exp( λ log( h )) : λ ∈ C i ∩ H = C G [[ r ]] ( h ) ∩ H := C H ( h ).Further, suppose ( g − Ag ) ∩ A = { } for some nontrivial g ∈ H . Let us show that g ∈ A .We have g − hg = exp( µ log( h )) for some µ ∈ C . Let h = r + P ∞ j = p h p r p +1 with h p = 0.Let G p +1 < G [[ r ]] be the normal subgroup of series of the form r + P ∞ j = p +1 c j r j +1 , c j ∈ N ,and ϕ p +1 : G [[ r ]] → G [[ r ]] /G p +1 be the quotient homomorphism. Then ϕ p +1 (cid:0) C G [[ r ]] ( h ) (cid:1) belongs to the central subgroup and is isomorphic to C , where the isomorphism sends ϕ p +1 (exp( λ log( h ))) to λh p , λ ∈ C . Hence, ϕ p +1 ( g − hg ) = ϕ p +1 ( h ) = ϕ p +1 (exp( µ log( h )))which implies that µ = 1. Thus [ g, h ] = 1 and by Lemma 2.1 g ∈ C G [[ r ]] ( h ) ∩ H := C H ( h ).This completes the proof of the proposition. (cid:3) Proof of Theorem 1.2 δ of cardinality 2,i.e., the following result. Theorem 3.1. (a)
Let H and H be nontrivial subgroups of a BP -group H ⊂ G [[ r ]] .Then the group H ∗ C H , where either C = { } or H ∩ H = { } and there is a nontrivial u ∈ H ∩ H such that C = C H ( u ) = C H ( u ) , is a BP -group embeddable to G [[ r ]] . (b) An extension of a centralizer of a BP -subgroup of G [[ r ]] is a BP -group embeddable to G [[ r ]] .Proof. (a) Let S ⊂ R be the transcendence basis of C over Q . It is known that S is of thecardinality of the continuum. We write S = S ⊔ S c , where S and S c are of the cardinalityof the continuum, and choose some s, t ∈ S c . Then a bijection S → S extends to an UBGROUPS OF THE GROUP OF FORMAL POWER SERIES 6 embedding σ : C ֒ → C such that s and t are algebraically independent over σ ( C ). Theisomorphism C ∼ = σ ( C ) induces an isomorphism G [[ r ]] ∼ = G σ ( C ) [[ r ]], where the latter is thesubgroup of G [[ r ]] of series with coefficients in σ ( C ). Thus without loss of generality wemay assume that H ≤ G σ ( C ) [[ r ]].Let C ≤ H ∩ H be as in the statement of the theorem. First, we consider the case C = { } . Then C ≤ C G [[ r ]] ( c ) := h c α : α ∈ C i for a fixed c ∈ C \ { } ; here we set forbrevity c α := exp( α log( c )). Lemma 3.2.
The group ¯ H := c − s H c s satisfies ¯ H ∩ H = C .Proof. Since C ≤ C G [[ r ]] ( c ), C ≤ ¯ H ∩ H . Suppose that there exists some u ∈ ( ¯ H ∩ H ) \ C .Then u = c − s vc s for some v ∈ H \ C . Since s is algebraically independent over σ ( C ) andthe coefficients of the series expansion of u belong to σ ( C ), the latter identity implies that u = c − α vc α for all α ∈ C (see (2.2),(2.3)). Thus for α = 0 we have u = v and from here for α = 1 we obtain that [ u, c ] = 1. Then Lemma 2.1 implies that v = u ∈ C G [[ r ]] ( c ) ∩ H = C ,a contradiction that proves the lemma. (cid:3) Let e H ≤ G [[ r ]] be a subgroup generated by ¯ H and H . Consider the epimorphism ϕ : H ∗ H → e H such that f ( h ) := h , h ∈ H , and f ( h ) := c − s h c s ∈ ¯ H , h ∈ H .Since c − s C c s = C , ϕ descends to an epimorphism ˜ ϕ : H ∗ C H → e H . Lemma 3.3. ˜ ϕ is an isomorphism.Proof. Let h ∈ H ∗ C H be such that ˜ ϕ ( h ) = 1. Then there exist h , . . . , h k , where h i − ∈ H , h i ∈ H , 1 ≤ i ≤ k , such that h = h ∗ · · · ∗ h k (here ∗ stands for the producton H ∗ C H ). Thus we have˜ ϕ ( h ) = h c − s h c s · · · h k − c − s h k c s = 1 . Since s is algebraically independent over σ ( C ) the latter implies a similar identity with anarbitrary α ∈ C instead of s (see (2.2),(2.3)). In particular, for all n ∈ Z ,(3.1) h c − n h c n · · · h k − c − n h k c n h k +1 = 1 , h k +1 := 1 . Since the element on the right belongs to the BP -group H , by the separation condition(see Section 2.1) there exists 1 ≤ j ≤ k − (cid:2) h − j +1 c ( − j h j +1 , c ( − j +1 (cid:3) = 1 . Now Lemma 2.1 implies that h − j +1 ch j +1 ∈ C := C G s ( c ), s = 1 ,
2. Hence, due to Proposition2.2, h j +1 ∈ C . If k = 1, this and (3.1) imply that ˜ ϕ ( h ) = h h = 1, h ∈ C , and so h ∈ C as well. In particular, h = h ∗ h ∈ C ≤ H ∗ C H . Since ˜ ϕ | C is identity, h = 1 in thiscase.If k >
1, then h j c ( − j h j +1 c ( − j +1 h j +2 = h j h j +1 h j +2 ∈ G s , s = 3 + ( − j . Therefore h = ˜ h ∗ · · · ∗ ˜ h k − , where ˜ h i = h i if i = j, and ˜ h j := h j ∗ h j +1 ∗ h j +2 . UBGROUPS OF THE GROUP OF FORMAL POWER SERIES 7
Here ˜ h i − ∈ H , ˜ g i ∈ H , 1 ≤ i ≤ k − k − k = 1 we obtain that h = 1.This proves that ˜ ϕ is a monomorphism and, hence, it is an isomorphism (as ˜ ϕ is anepimorphism by definition). (cid:3) Thus we have proved that e H is a subgroup of G [[ r ]] isomorphic to H ∗ C H for C = { } .Now suppose that C = { } . Let us take c := exp( se + s e ) ∈ G [[ r ]] \ G σ ( C ) [[ r ]] and set¯ H := c − t H c t . Then similarly to Lemma 3.2 we get the following.
Lemma 3.4. ¯ H ∩ H = { } .Proof. If there exists some nontrivial u ∈ ¯ H ∩ H , then u = c − t vc t for some v ∈ H . Asin the proof of Lemma 3.2 this implies u = v ∈ H ∩ H . If H ∩ H = { } , then weobtain a contradiction. For otherwise, as in the proof above the separation condition andProposition 2.2 imply that u = c α for some nonzero α ∈ C . Hence, log( u ) = αse + αs e .Since the coefficients of the series expansion of log u belong to σ ( C ), the latter yields αs, αs ∈ σ ( C ); hence s = αs αs ∈ σ ( C ). This contradicts the algebraic independence of s over σ ( C ) and completes the proof of the lemma. (cid:3) Let e H ≤ G [[ r ]] be the subgroup generated by H and ¯ H . Consider the surjectivehomomorphism ϕ : H ∗ H → e H such that ϕ ( h ) = gH , h ∈ H , and ϕ ( h ) = c − t h c t , h ∈ H . Lemma 3.5. ϕ is an isomorphism.Proof. Let h ∈ Ker( ϕ ). Then h = h ∗ · · · ∗ h k for some h i − ∈ H , h i ∈ H , 1 ≤ i ≤ k (here ∗ stands for the product on H ∗ H ). Thus we have ϕ ( h ) = h c − t h c t · · · h k − c − t h k c t = 1 . Since t is algebraically independent over σ ( C ), arguing as in the proof of Lemma 3.3 weobtain that there exists 1 ≤ j ≤ k − (cid:2) h − j +1 c ( − j h j +1 , c ( − j +1 (cid:3) = 1 . Now Lemma 2.1 implies that h − j +1 ch j +1 ∈ C G [[ r ]] ( c ). Hence, due to Proposition 2.2, h j +1 ∈ C G [[ r ]] ( c ), i.e., h j +1 = c α for some α ∈ C . Then arguing as in the proof of Lemma 3.4 weobtain that α = 0. Hence, h j +1 = 1 and so h = ˜ h ∗ · · · ∗ ˜ h k − , where ˜ h i = h i if i = j and˜ h j := h j ∗ h j +1 ∗ h j +2 . Here ˜ h i − ∈ H , ˜ h i ∈ H , 1 ≤ i ≤ k − k − h = 1.This completes the proof of the lemma. (cid:3) Thus we have proved that in this case e H is a subgroup of G [[ r ]] isomorphic to H ∗ H .Finally, in both cases groups e H are BP by Theorem 4 and Corollary 6 of [KMS] whoseconditions are satisfied due to [KMS, Prop. 5] and our Proposition 2.2. UBGROUPS OF THE GROUP OF FORMAL POWER SERIES 8
This completes the proof of part (a) of the theorem.(b) Let G be a BP -subgroup of G [[ r ]] and C = C G ( u ) for a nontrivial u ∈ G . As in theproof of (a) we assume that G ≤ G F [[ r ]], where F is a proper subfield of C and s ∈ C \ F isalgebraically independent over F . Consider a subgroup e G ≤ G [[ r ]] generated by G and u s . Lemma 3.6. e G is isomorphic to the group G t := h G, t | [ c, t ] = 1 , c ∈ C G ( u ) i .Proof. Consider the epimorphism ϕ : G ∗ Z → e G such that ϕ ( g ) = g , g ∈ G , and ϕ ( n ) = u ns , n ∈ Z . Since [ ϕ (1) , c ] = 1, c ∈ C G ( u ), ϕ descends to an epimorphism ˜ ϕ : G t → e G . Let usshow that ˜ ϕ is a monomorphism. This will complete the proof of the lemma.Let g ∈ Ker( ˜ ϕ ). Then g = g ∗ t α ∗ · · · ∗ g k ∗ t α k , where g i ∈ G , α i ∈ Z , 1 ≤ i ≤ k (here ∗ is the product on G t ). Thus we have(3.2) ˜ ϕ ( g ) = g u α s · · · g k u α k s = 1 . If k = 1, then we obtain that g = u − α s . Since s is algebraically independent over F andthe coefficients of the series expansion of g belong to F , this implies that α = 0, hence, g = 1 and g = g ∗ t α = 1.For otherwise, by the same reason (3.2) implies that g u α n · · · g k u α k n = 1 , n ∈ Z . The expressions on the right belong to the BP -group G , hence, due to the separationcondition (see Section 2.1) there exists 1 ≤ i < k such that[ g − i +1 u α i g i +1 , u α i +1 ] = 1 . If both α i , α i +1 = 0, then arguing as in the proof of part (a) we obtain that g i +1 ∈ C G ( u ).This reduces the length of the word representing g from k to k −
1. The same is true if α i = α i +1 = 0 and i + 1 < k . Finally, if α k = 0, then the separation condition provides asimilar commutativity relation with a new i < k − g as well. Applying this reduction procedure k − k = 1, we get that g = 1, i.e. ˜ ϕ is an injection. (cid:3) To complete the proof of part (b) note that G t is a BP -group due to [KMS, Thm. 4]. (cid:3) Proof of Theorem 1.2.
Proof.
Let S ⊂ C be the transcendence basis of C over Q . We write S = S ⊔ S ⊔ S ⊔ S ,where all S i are of the cardinality of the continuum. Then a bijection S → S extends toan embedding σ : C ֒ → C such that S \ S is the transcendence basis of C over σ ( C ). Theisomorphism C ∼ = σ ( C ) induces an isomorphism G [[ r ]] ∼ = G σ ( C ) [[ r ]]. Thus without loss ofgenerality we may assume that G ≤ G σ ( C ) [[ r ]].Further, since the ordinal δ is of cardinality ≤ c , there exist injections τ i : δ → S i , 1 ≤ i ≤ λ ≤ δ , G λ is a BP -group and there is a monomor-phism ϕ λ : G λ → G F λ [[ r ]], where F λ ⊂ C is the minimal subfield containing σ ( C ) and all τ i ( γ ), γ ≤ λ , i = 1 , ,
3, such that ϕ λ | G α = ϕ α for all α < λ . UBGROUPS OF THE GROUP OF FORMAL POWER SERIES 9
For λ = 0 the result holds trivially with ϕ = id. Assuming that the result holds for allordinals < λ let us prove it for λ .First, assume that λ is a limit ordinal. By the definition, G λ := [ α<λ G α . Since all G α , α < λ , are BP -groups by the induction hypothesis, their union G λ is a BP -group as well.Now, we set ϕ λ ( g ) := ϕ α ( g ) , g ∈ G α , α < λ. Then due to the induction hypothesis, ϕ λ is a well-defined monomorphism of G λ to G [[ r ]].Moreover, the coefficients of the series expansions of elements of ϕ λ ( G λ ) belong to ∪ α<δ F α .Clearly, the latter is a subfield of F λ which proves the required statement in this case.Next, assume that λ is a successor ordinal, i.e., λ = α + 1 for an ordinal α < λ . Weapply Theorem 3.1 as follows.If G α +1 = G α ∗ C α F α , where F α is a nontrivial subgroup of G α , and either C α = { } or C α = C G α ( u ) = C F α ( u ) for some nontrivial u ∈ F α , then we choose in Theorem 3.1(a) H = H = ϕ α ( G α ), H = ϕ α ( F α ) and s = τ ( α + 1) ∈ S , t = τ ( α + 1) ∈ S . Thenthe proof of the theorem implies that G α +1 is embeddable to G [[ r ]] and the correspondingmonomorphism of Lemma 3.3 ˜ ϕ denoted in our case by ϕ α +1 extends ϕ α and is such thatthe coefficients of series expansions of elements of ϕ α ( G α +1 ) belong to the minimal subfieldof C containing F α and τ ( α + 1), τ ( α + 1) which is clearly a subfield of F α +1 .If G α +1 is an extension of a centralizer of G α , then we set in the proof of Theorem3.1(b), G = ϕ α ( G α ) and s = τ ( α + 1). Due to the theorem, G α +1 is embeddable to G C [[ r ]]and the corresponding monomorphism of Lemma 3.6 ˜ ϕ denoted now by ϕ α +1 extends ϕ α and is such that the coefficients of series expansions of elements of ϕ α ( G α +1 ) belong to theminimal subfield of C containing F α and τ ( α + 1) which is a subfield of F α +1 . Moreover,in both cases G α +1 is a BP -group. This completes the proof of the inductive step and,hence, of Theorem 1.2. (cid:3) Proof of Theorem 1.5
Repeating word-for-word the proof of Proposition 2.2 one obtains that any subgroupof the group G R [[ r ]] is CSA and, moreover, maximal abelian subgroups of a nontrivial H ≤ G R [[ r ]] have the form C H ( u ) = C G R [[ r ]] ( u ) ∩ H = h exp( λ log( u )) : λ ∈ R i ∩ H fornontrivial u ∈ H . One uses this to prove the following version of Theorem 3.1.Let F ⊂ R be a subfield such that the transcendence degree of R over F is at least two. Theorem 4.1. (a)
Let H and H be nontrivial subgroups of a BP -group H ⊂ G F [[ r ]] .Then the group H ∗ C H , where either C = { } or H ∩ H = { } and there is a nontrivial u ∈ H ∩ H such that C = C H ( u ) = C H ( u ) , is a BP -group embeddable to G R [[ r ]] . (b) An extension of a centralizer of a BP -subgroup of G F [[ r ]] is a BP -group embeddableto G R [[ r ]] . UBGROUPS OF THE GROUP OF FORMAL POWER SERIES 10
Proof.
Suppose S = S ⊔ S c ⊂ R is the transcendence basis of R over Q , where S isthe transcendence basis of F over Q . By the definition of F there exist some s, t ∈ S c algebraically independent over F . Starting with these elements we repeat literally theproof of Theorem 3.1 replacing σ ( C ) by F , C by R and G [[ r ]] by G R [[ r ]] to get the requiredstatement. (cid:3) Proof of Theorem 1.5.
Since the cardinality of G is less than c , the field F ⊂ R generatedby coefficients of series expansions of elements from G has the cardinality less than c aswell. Suppose S = S ⊔ S c ⊂ R is the transcendence basis of R over Q such that S is thetranscendence basis of F over Q . Since S is of the cardinality of the continuum, S c := S \ S is of the cardinality of the continuum as well. Hence, we can write S c = S ⊔ S ⊔ S ⊂ R where all S i , 1 ≤ i ≤
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