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MAGNUS EMBEDDING AND ALGORITHMIC PROPERTIES OF GROUPS F/N(d) 5 1 FUNDAGUL,MAHMOODSOHRABI,ANDALEXANDERUSHAKOV 0 2 Abstract. In this paper we further study properties of Magnus embedding, n give a precise reducibility diagram for Dehn problems in groups of the form a F/N(d), and provide a detailed answer to Problem 12.98 in Kourovka note- J book. Wealsoshowthatmostofthereductionsarepolynomialtimereductions 5 andcanbeusedinpracticalcomputation. Keywords. Magnus embedding, word problem, power problem, conjugacy ] problem,freesolvablegroups. R 2010MathematicsSubject Classification. 20F19,20F10,20F65,03D15. G . h t a 1. Introduction m LetF =F(X)bethefreegroupongeneratorsX,N anormalsubgroupofF,N′ [ the derived subgroup of N, and N(d) the dth derived subgroup of N. The Magnus 1 embeddingisthemaintooltostudygroupsoftheformF/N′. Itwasintroducedin v [7] by W. Magnus who showed that the elements of F/N′ can be encoded by 2×2 1 matrices: 0 g π 0 M(X;N)= g ∈F/N, π ∈F , 1 (cid:26)(cid:18) 0 1 (cid:19)(cid:12) Γ(cid:27) (cid:12) 0 where F is a free module overthe groupri(cid:12)ng ZF/N. In this paper we study algo- Γ (cid:12) . 1 rithmicpropertiesofMagnusembeddinganddecidabilityofthefollowingproblems 0 for groups of the form F/N(d). 5 1 WP(G), word problem in G. Given a word w in the generators of G decide if : w=1 in G, or not. v i CP(G), conjugacy probleminG. Givenwordsu,vinthegeneratorsofGdecide X if u∼v in G, or not. r a PP(G), power problem in G. Given words u,v in the generators of G decide if v =uk in G for some k ∈Z, or not. The Magnus embedding provedto be especially robust in the study of free solvable groups. Indeed, free solvable group naturally appear in the context because F/F(d) =F/(F(d−1))′ is the free solvable group of rank n and degree d. It immediately follows from the work of Magnus that decidability of the word problem in F/N implies decidability of the word problem in F/N′. The conjugacy problem in groups of the type F/N′ was first approachedby J. Matthews in [8] who proved that: Date:January7,2015. ThethirdauthorhasbeenpartiallysupportedbyNSAMathematicalSciencesProgramgrant numberH98230-14-1-0128. 1 2 F.GUL,M.SOHRABI,ANDA.USHAKOV (a) u,v ∈F/N′ areconjugate(forfreeabelianF/N)ifandonlyiftheirimages under Magnus embedding are conjugate in M(X;N); (b) conjugacy problem in M(X;N) is decidable if and only if conjugacy problem in F/N is decidable and power problem in F/N is decidable. These two facts imply that free metabelian groups have decidable conjugacy problem. Later Remeslennikov and Sokolov in [15] extended (a) to any torsion free group F/N, showed that the power problem is decidable in free solvable groups, and deduced that free solvable groups have decidable conjugacy problem. Finally, C. Gupta in [4] proved that (a) holds for groups with torsion as well and hence decidability of the conjugacy and power problems in F/N implies decidability of the conjugacy problem in F/N′. We use the following notation for reducibility of decision problems in the sequel: CP(F/N) ⇒CP(F/N′). (cid:26) PP(F/N) In the light of these results, V. Shpilrain raised the following questions in [9, Problem 12.98]. Is it correct that: (a) WP(F/N) is decidable if and only if WP(F/N′) is decidable. (b) CP(F/N) is decidable if and only if CP(F/N′) is decidable. (c) WP(F/N′) is decidable if and only if CP(F/N′) is decidable. It was shown by Anokhin in [1] that only 12.98(a) has an affirmative answer. He constructed a group F/N such that CP(F/N) is decidable and CP(F/N′) is undecidable. Such a groupis clearlya counterexampleto both 12.98(b)and12.98(c). We also would like to mention several results related to practical computations in free solvable groups in which the Magnus embedding plays a crucial role. S. Vassileva showed in [19] that the power problem in free solvable groups can be solved in O(rd(|u|+|v|)6) time and used that result to show that the Matthews- Remeslennikov-SokolovapproachcanbetransformedintoapolynomialtimeO(rd(|u|+ |v|)8) algorithm for the conjugacy problem. In [18] those complexity bounds were further improved and randomized algorithms were developed. Another generaliza- tion was done by Lysenok and Ushakov in [6]. It was shown that the Diophantine problem for spherical quadratic equations, i.e., equations of the form: z−1c z ...z−1c z =1, 1 1 1 k k k infree metabeliangroupsisdecidable. Recallthatforeveryn≥2theDiophantine problem in free metabelian groups is undecidable (see [16]). Recently Vassileva proved in [20] that the Magnus embedding is a quasi-isometry. 1.1. Our contribution. Here we shortly outline the main results of the paper (somewhat simplifying the statements). Let F be a free group of rank at least 2 and N a recursively enumerable normal subgroup of F. Theorem 6.10. WP(F/N)⇒PP(F/N′). Theorem 6.13. PP(F/N)⇒CP(F/N′). Theorem 6.15. PP(F/N)⇐CP(F/N′). Theorem 6.17. CP(F/N)6⇒CP(F/N′). MAGNUS EMBEDDING AND ALGORITHMIC PROPERTIES OF GROUPS F/N(d) 3 Our results give the following reducibility diagram for the decision problems in groups of the form F/N(d): WP(F/N) WP(F/N′) WP(F/N′′) ... PP(F/N) PP(F/N′) PP(F/N′′) ... CP(F/N) CP(F/N′) CP(F/N′′) ... In particular, the following theorem holds. Corollary 6.16. For every recursively enumerable N EF the following holds. (a) PP(F/N′)⇔PP(F/N′′). (b) CP(F/N′′)⇔CP(F/N′′′). (c) WP(F/N′′)⇔PP(F/N′′)⇔CP(F/N′′). (cid:3) Furthermore, most of the reductions are polynomial time computable. Denote by P the class of decision problems decidable in polynomial time. Theorem. Suppose that WP(F/N) ∈ P. Then the problems WP(F/N(d)), PP(F/N(d)), and CP(F/N(d)) are in P for every d ≥ 2. Moreover, each of those problems has a unique polynomial bound that does not depend on d. (cid:3) Finally, inSection6.5we considertwocombinatorialproblemsforgroupsF/N′: subset sum problem SSP(F/N′) and acyclic graph problem AGP(F/N′). The main results of that sections are: Theorem6.20. SSP(F/N′)∈PifandonlyifWP(F/N)∈PandeitherN ={1} or [F :N]<∞. (cid:3) Theorem 6.18. If WP(F/N)∈P, N 6={1}, and [F :N]=∞, then SSP(F/N′) and AGP(F/N′) are NP-complete. (cid:3) Corollary 6.21. AGP(F/N′) ∈ P if and only if WP(F/N) ∈ P and either N ={1} or [F :N]<∞. (cid:3) 2. Preliminaries: X-digraphs Let X be a set (called an alphabet) and F = F(X) the free group on X. By X− we denote the set of formal inverses of elements in X and put X± =X∪X−. An X-labeled directed graph Γ (or an X-digraph) is a pair of sets (V,E) where the set V is called the vertex set and the set E ⊆ V ×V ×X± is called the edge set. An element e=(v ,v ,x)∈E designates an edge with the origin v (also denoted 1 2 1 by α(e)), the terminus v (also denoted by ω(e)), labeled with x (also denoted by 2 x µ(e)). Weoftenusenotationv →v todenotetheedge(v ,v ,x). ApathinΓisa 1 2 1 2 sequenceofedgesp=e ,...,e satisfyingω(e )=α(e )foreveryi=1,...,k−1. 1 k i i+1 The origin α(p) ofp is the vertexα(e ), the terminus ω(p)is the vertexω(e ), and 1 k the label µ(p) of p is the word µ(e )...µ(e ). We say that an X-digraph Γ is: 1 k • rooted if it has a special vertex, called the root; • folded (or deterministic) if for every v ∈V and x∈X there exists at most one edge with the origin v labeled with x; 4 F.GUL,M.SOHRABI,ANDA.USHAKOV • X-complete (or simply complete) if for every v ∈ V and x ∈ X± there 1 x exists an edge v →v ; 1 2 x • inverseifwitheveryedgee=g →g thegraphΓalsocontainstheinverse 1 2 edge g x→−1 g , denoted by e−1. 2 1 AllX-digraphsinthispaperareconnected. Amorphism oftworootedX-digraphs is a graph morphism which maps the root to the root and preserves labels. For more information on X-digraphs we refer to [17, 5]. Example 2.1. LetF =F(X)andH ≤F. TheSchreier graph ofthesubgroupH, denoted by Sch(X;H), is an X-digraph (V,E), where V is the set of right cosets V ={Hg |g ∈F} and E ={Hg →x Hgx|g ∈F, x∈X±}. Bydefinition,Sch(X;H)isafoldedcompleteinverseX-digraph. Wealwaysassume thatH istherootofSch(X;H). AspecialcaseoftheSchreiergraphiswhenHEF, called a Cayley graph of the group F/H denoted by Cay(X;H). (cid:3) Let Γ=(V,E) be an inverse X-digraph. The set of edges E can be split into a disjoint union E =E+⊔E−, where E+ ={e∈E |µ(e)∈X} is called the set of positive edges, and E− ={e∈E |µ(e)∈X−}. is called the set of negative edges. Clearly, (E+)−1 =E− and (E−)−1 =E+. Therank r(Γ)ofaninverseX-digraphΓisdefinedas|E+|−|T|,whereT isany spanning subtree of Γ. The fundamental group π (Γ) is the group of labels of all 1 cycles at the root; it is naturally a subgroup of F(X) of the rank r(Γ), see [5]. 3. Preliminaries: Computational model and data representation All computations are assumed to be performed on a random access machine. We use base 2 positional number system in which presentations of integers are converted into integers via the rule: (a ...a a a a ) =a 2k−1+...+a 22+a 2+a , k−1 3 2 1 0 2 k−1 2 1 0 where we assume that a = 1. The number k is called the bit-length of the k−1 presentation. LetGbeagroupgeneratedbyafinitesetX ={x ,...,x }. Weformallyencode 1 n the word problem for G as a subset of {0,1}∗ as follows. We first encode elements of the set X± = {x±,...,x±} by unique bit-strings of length ⌈log n⌉+1. The 1 n 2 code for a word w=w(X±) is a concatenation of codes for letters and, formally: r WP(F/N)={code(w)|w∈N}. Thus, the bit-length of the representation for a word w ∈F is: |code(w)| =|w|(⌈log n⌉+1). 2 Weencodethepowerandconjugacyproblemsinasimilarfashion. Forbothofthese problemsinstancesarepairsofwordsandtheencodingcanbedonebyintroducing a new letter “,” into the alphabet X±. MAGNUS EMBEDDING AND ALGORITHMIC PROPERTIES OF GROUPS F/N(d) 5 3.1. Quasi-linear time complexity. An algorithm is said to run in quasi-linear timeifits time complexityfunctionisO(nlogkn)forsomeconstantk ∈N. We use notationO˜(n)todenotequasi-lineartimecomplexity. Quasi-lineartimealgorithms arealsoo(n1+ε)foreveryε>0,andthus runfasterthananypolynomialinnwith exponentstrictlygreaterthan1. See[13]formoreinformationonquasi-lineartime complexity theory. Similarly, one can define quasi-quadratic O˜(n2), quasi-cubic O˜(n3) time complexity as O(n2logkn), O(n3logkn), etc. 4. Flows on inverse X-digraphs Let Γ = (V,E) be an inverse X-digraph. We say that a function f : E → Z is balanced if: (F1) f(e)=−f(e−1) for any e∈E. Allfunctions inthis paperarebalanced. Afunctionf :E →Zdefinesthe function N :V →Z: f N (v)= f(e), f X α(e)=v calledthenet-flow functionoff. Wesaythatf isaflow ifitsatisfiestheconditions (F1), (F2), and (F3). (F2) f has a finite support supp(f)={e∈E |f(e)6=0}. (F3) Either N (v)=0 for every v ∈V in which case we say that f is a circula- f tion, or there exist s,t ∈V such that N (v)= 0 for all v ∈V \{s,t}, and f N (s) = 1 and N (t) = −1 and we say that f is a flow from the source s f f to the sink t. Define F to be the set of all balanced integralfunctions on E with finite support: Γ F ={f :E →Z}. Γ For f,g ∈F define f +g ∈F as follows: Γ Γ (f +g)(e)=f(e)+g(e). Clearly, (F ,+) is an abelian group. The function k·k:F →Z defined by: Γ Γ kπk= |π(e)| eX∈E+ iscalledanorm onF . ItiseasytoseethateveryX-digraphmorphismϕ:Γ→∆ Γ induces a homomorphism of abelian groups ρ :F →F defined as follows: ϕ Γ ∆ ρ (f)(e′)= f(e), ϕ ϕ(Xe)=e′ for f ∈F and e′ ∈E(∆). Clearly, kπk≥kρ (π)k for every π ∈F . Γ ϕ Γ 4.1. Flows defined by words. Let Γ = (V,E) be an rooted folded complete inverse X-digraph and w =xε1...xεk ∈F(X). The word w defines a unique path i1 ik p in Γ: w xε1 xε2 xε3 xεk v →i1 v →i2 v →i3 ... →ik v 0 1 2 k where v is the rootof Γ, and a function πΓ :E →Z which associatesto anedge e 0 w the number of times e is traversed minus the number of times e−1 is traversed by p . It is easy to check that πΓ is a flow in Γ. We call πΓ the flow of w in Γ. w w w 6 F.GUL,M.SOHRABI,ANDA.USHAKOV ◦ 1 0 1 ◦ 0 ◦ 1 −1 −1 0 −1 −1 • ◦ • ◦ ρ 0 −1 1 ◦ ρ 0 ◦ 0 1 Figure 1. The Cayley graph of S = ha,bi with |a| = 3 and 3 |b| = 2 and the Schreier graph of H = hbi ≤ S . The straight 3 edges correspond to b and dashed ones to a. The values of π for w w =[a2,b] are shown on the edges. Lemma 4.1 (See [11, Lemma 2.5]). For any flow π : E(Γ) → Z there exists w∈F(X) satisfying π =πΓ. (cid:3) w Ingeneral,ifΓisnotcomplete,thensomewordscannotbetracedinΓ. Suppose thata reducednontrivialwordw canbe tracedinΓ. The setof edges traversedby w in Γ forms a connected X-digraph called the support graph of w in Γ. Lemma 4.2. Let Γ be a rooted folded inverse X-digraph and m the length of a shortest cycle in Γ (not necessarily at the root). Suppose that a reduced nontrivial word w can be traced in Γ and πΓ =0. Then |w|≥3m. w Proof. It follows fromour assumption π =0 that the path p is a cycle in Γ. Let w w ∆ be the support graph of w in Γ. The rank of ∆ can not be 0 (w is not reduced in this case) and can not be 1 (either w is not reduced or π 6=0). Therefore, the w rankof∆isatleast2. Eachedgeof∆istraversedbyw atleasttwice. Hence,itis sufficient to prove that 2|E(∆)|≥3m. Let ∆′ be a minimal subgraphof ∆ ofrank exactly 2. There are exactly two distinct configurations possible for ∆′, shown in Figure 2. a b a c b c Figure 2. Two configurations for support graphs in Lemma 4.2. MAGNUS EMBEDDING AND ALGORITHMIC PROPERTIES OF GROUPS F/N(d) 7 Let a,b,c be the lengths of arcs as shown in the figure. Since, the length of a shortest cycle in Γ is m, we get the following bounds for our cases: a+b≥m, b≥m,  a+c≥m,  b+c≥m, (cid:26) c≥m. In both cases we have 2(a+b+c) ≥ 3m which proves that 2|E(∆)| ≥ 3m. Thus, |w|≥3m. (cid:3) 4.2. Flows on Schreier graphs. In this section we study properties of flows on Schreier graphs. The following lemma is the most important tool in the study of groupsofthe type F/N′ andis the foundationofthe Magnusembedding discussed in Section 5. It is a well-known result and can be proven using algebraic topol- ogy techniques. Here we provide a proof using elementary properties of Stallings’ graphs. Lemma 4.3. Let H ≤ F, ∆=Sch(X;H), and w ∈F. Then π∆ =0 if and only w if w ∈[H,H]. Proof. “⇐” If w ∈[H,H], then π =0. w “⇒” Assume that π∆ =0. Then w ∈H. Taking a spanning tree in Sch(X;H) w wecanchoosea good(perhaps infinite) setofgeneratorsY forH correspondingto the cycles defined by positive edges outside of the spanning tree. The word w can be (uniquely) expressedas a wordw =u(Y) in the generatorsY. Since π∆ =0 we w should have σ (u)=0 (algebraic sum of powers of y’s in the expression for u is 0) y for every y ∈Y. This means that u can be expressed as a product of commutators of elements from hYi=H. Hence w ∈[H,H]. (cid:3) Corollary 4.4. w=1 in F/N′ if and only if πΓ =0 where Γ=Cay(X;N). (cid:3) w Corollary 4.5. Let H ≤ F and m is the length of a shortest nonempty word in H. Then the length of a shortest nonempty word in [H,H] is at least 3m. Proof. By Lemma 4.3 if w ∈ [H,H]\{ε}, then π∆ = 0 in ∆ = Sch(X;H). By w Lemma 4.2, |w|≥3m. (cid:3) Lemma4.6. LetNEF,w ∈F,and∆=Sch(X;hN,wi). Ifπ∆ =0,thenw ∈N. w Proof. If π∆ = 0 then, by Lemma 4.3, w ∈ [hN,wi,hN,wi]. Hence, w can be w expressedasaproductofcommutatorsoverhN,wi. Thatexpressesw asaproduct of elements from N and w’s with the trivial algebraic sum of powers for w. That product belongs to N because N is normal in F. (cid:3) Corollary 4.7. Let N EF, w ∈F \N, and ∆=Sch(X;hN,wi). Then π∆ 6=0. w 4.3. Flows on Cayley graphs. Let X = {x ,...,x }, F = F(X), N EF, G = 1 n F/N, and Γ=Cay(X;N). The group G acts on its Cayley graph Γ by shifts: eg =g−1h→x g−1hx. for g ∈ G and e = h →x hx ∈ E. The following action of ZG on F turns Γ the later into a ZG-module. For c g + ... + c g ∈ ZG and f ∈ F define 1 1 k k Γ f′ =(c g +...+c g )f on e=g →x gx to be: 1 1 k k f′(e)=c f(eg1)+...+c f(egk). 1 k Denote π by π for i=1,...,n. The next lemma is straightforward. xi i 8 F.GUL,M.SOHRABI,ANDA.USHAKOV Lemma 4.8. F is a free ZG-module of rank n with a free basis {π ,...,π }. In Γ 1 n particular, every π ∈ F can be uniquely expressed as a ZG linear combination of Γ π ,...,π . (cid:3) 1 n The set of circulation C in F is closed under addition and the scalar ZG- Γ Γ multiplication and hence C is a ZG-submodule of F . Γ Γ Lemma 4.9. If G is finitely presented, then C is a finitely generated ZG-module. Γ Proof. If N =ncl (r ,...,r ), then C =hπ ,...,π i. (cid:3) F 1 k Γ r1 rk There exists an algebraic way to define the net-flow function of π ∈ F . Define Γ a map N :F →ZG by: Γ n N π =α π +...+α π 7→ α (1−x ). 1 1 n n i i Xi=1 If π = n a gπ , then the coefficient for g in N(π) is i=1 g∈G g,i i P P ag,1+...+ag,n−agγ(x1)−1,1−...−agγ(xn)−1,n which is exactly the value of the net flow at g defined by π. Hence, N(π) is a description of N as an element of ZG. The next propositions are obvious. π Proposition 4.10. N :F →ZG is a ZG-module homomorphism. (cid:3) Γ Proposition 4.11. For any π ∈F the following holds. Γ (a) π ∈C if and only if N(π)=0. Γ (b) π is a flow on Γ if and only if N(π)=1−g for some g ∈G. (cid:3) Consider the augmentation map ǫ:ZG→Z, defined by: ǫ α g 7→ α . g g gX∈G gX∈G Recall that ǫ is a ring homomorphism and therefore a homomorphism of ZG- modules once Z is assumed to be a ZG-module with trivial G action. Lemma 4.12. The sequence F →N ZG→ǫ Z is an exact sequence of ZG-modules. Γ Proof. The inclusion im(N)⊆ker(ǫ) can be easily shownby induction onkπk. To show the opposite inclusion pick an arbitrary α g ∈ker(ǫ). Then g P α g = α g− α = α (g−1). g g g g X X X X If G∋g =xε1...xεk, then g−1 can be written as follows: i1 ik (xε1...xεk −xεi...xεk−1)+(xεi...xεk−1 −xεi...xεk−2)+...+(xε1 −1) i1 ik i1 ik−1 i1 ik−1 i1 ik−2 i1 k−1 = xε1...xεj(xεj+1 −1). i1 ij ij+1 Xj=0 Therefore, − α g = α (1−g) is a linear ZG-combination of the elements of g g the form (1−Pxi) and hPence belongs to im(N). Thus, αgg ∈im(N). (cid:3) P MAGNUS EMBEDDING AND ALGORITHMIC PROPERTIES OF GROUPS F/N(d) 9 5. Magnus embedding Let X ={x ,...,x }, F =F(X), N EF, and Γ=Cay(X;N). In this section 1 n we study relations between groups G = F/N and F/N′. It is easy to check that the set of matrices: g π M(X;N)= g ∈G, π ∈F (cid:26)(cid:18) 0 1 (cid:19)(cid:12) Γ(cid:27) (cid:12) (cid:12) forms a groupwith respect to the matrix mul(cid:12)tiplication which can be easily recog- nized as the wreath product ZnwrG. Let : F → F/N be the canonical epimorphism. Define a homomorphism µ:F →M(X;N) by: (1) x 7→µ xi πi , x−1 7→µ x−i 1 −x−i 1πi . i (cid:18) 0 1 (cid:19) i (cid:18) 0 1 (cid:19) It is easy to check by induction on |w| that: w π (2) µ(w)= w . (cid:18) 0 1 (cid:19) Proposition 5.1. For any w∈F if πΓ =0 then w=1. w Proof. Assume that w 6= 1 in F/N. Tracing w in Γ we obtain a path p from w 1 to wN 6= 1. The path p is not a circuit and the corresponding flow is not a w circulation, i.e. N (1)= πΓ(e)=1. πw w X α(e)=1 Therefore, πΓ(e)6=0 for some edge e adjacent to 1. Thus, πΓ 6=0. (cid:3) w w Now note that for every w ∈F: w ∈ker(ϕ)⇔πΓ =0 and w=1 w ⇔πΓ =0 (by Proposition 5.1) w ⇔w ∈N′ (by Lemma 4.3), which proves the following theorem. Theorem 5.2 (See [7]). Let F =F(x ,...,x ), N EF, and :F →F/N be the 1 n canonical epimorphism. The homomorphism µ:F →M(X;N) defined by x7→µ x πi (cid:18) 0 1 (cid:19) satisfies ker(µ) = N′. Therefore, F/N′ ≃ µ(F) ≤ M(X;N). The induced embedding F/N′ →M(X;N) is called the Magnus embedding. (cid:3) 5.1. Properties of Magnus embedding. The following proposition was proved in [15] using Fox derivatives. Let g ∈G, π = n α π ∈F , and i=1 i i Γ g n Pα π A= i=1 i i . (cid:18) 0 P 1 (cid:19) Proposition 5.3 (See [15, Theorem 2]). The following holds. (a) A∈µ(F) if and only if n α (1−x )=1−g. i=1 i i (b) A∈µ(N) if and only ifPni=1αi(1−xi)=0. P 10 F.GUL,M.SOHRABI,ANDA.USHAKOV Proof. Follows from (2) and Proposition 4.11. (cid:3) For a nontrivial g ∈G define a map τ :F →F by: g Γ Γ π 7→τg (1−g)π. DenoteSch(X;hN,gi)by∆. Thenaturalprojectionϕ:Γ→∆inducesanabelian group homomorphism ρ : F → F . Properties of the functions τ and ρ are g Γ ∆ g g very important in the study of the conjugacy problem in F/N′. Lemma5.4. ThesequenceF →τg F →ρg F is an exactsequenceof abelian groups. Γ Γ ∆ Proof. The image of τ is a subgroup of F generated by the elements (1−g)hπ g Γ i for h∈G and i=1,...,n. Clearly, ρ ((1−g)hπ )=0. g i Conversely, assume that π′ = ρ (π) = 0. Let H = hN,gi. Consider any edge g e′ =Hh→xi Hhx in ∆. By definition of ρ we get i g 0=π′(e′)= π(e)= π(Ngjh→xi Ngjhx ), i ϕ(Xe)=e′ Xgj where gj ranges over all distinct powers of g. It is easy to see that such π is an integral linear combination of the elements (1−g)hπ and, hence, belongs to i im(τ ). (cid:3) g Lemma 5.5 (Cf. [4, Lemma 4]). ker(τ ) is not trivial if and only if |g|=k <∞, g in which case it is an abelian subgroup of F : Γ ker(τ )= (1+g+...+gk−1)hπ |h∈G and i=1,...,n . g i (cid:10) (cid:11) Proof. Pickanyπ ∈F suchthat(1−g)π =0. Thengjπ =πforeveryj ∈Z,which Γ can not happen if |g| = ∞ since π has finite support. Assume that |g| = k < ∞. It is straightforwardto check that (1+g+...+gk−1)hπ ∈ker(τ ). On the other i g hand if gjπ = π for every j ∈ Z, then the coefficients α are constant on right h,i hgi-cosets and hence are linear combinations of the generators. (cid:3) Lemma 5.6. Let π ∈ F and 1 6= g ∈ G. If (1−g)π ∈ C , then there exists Γ Γ π∗ ∈C satisfying (1−g)π =(1−g)π∗. Γ Proof. If (1−g)π ∈C , then Γ N((1−g)π)=N(π)−gN(π)=0. Therefore, N(π)=gjN(π) for every j ∈Z. Case-I. Assume that |g| = ∞. Since N(π) has finite support, it must be the case that N(π)=0. Thus, π ∈C . Γ Case-II. Assume that |g|=k <∞. Then: N(π)=gN(π)=...=gk−1N(π), i.e., N(π) is constant on right hgi-cosets. Hence, N(π)=(1+g+...+gk−1)N′, for some N′ ∈ZG. By Lemma 4.12, we have: 0=ǫ(N(π))=ǫ(1+g+...+gk−1)ǫ(N′)=kǫ(N′) in Z, which implies that ǫ(N′)=0. Hence N′ ∈ker(ǫ) and by Lemma 4.12: N′ = α (1−x ) for some α ∈ZG. i i i X