A Rank Formula for Amalgamated Products with Finite Amalgam

A rank formula for amalgamated products with ﬁnite amalgam

Richard Weidmann

1 Introduction

Grushko’s theorem [4] says that the rank of a group is additive under free products, i.e. that rank A ∗ B = rank A + rank B. For amalgamated products no such formula exists unless we make strong assumptions on the amalgamation [11]; in fact for any n there exists an amalgamated product Gn = An ∗F2 Bn such that rank Gn = 2 and rank An, rank Bn ≥ n [10]. Of course F2, the free group of rank 2, is a very large group. Thus it makes sense to ask whether the situation is better if the amalgam is a small group. In this note we give a positive answer to this question in the case that the amalgam is finite. For a finite group C denote by l(C) the length of the longest strictly ascending chain of non-trivial subgroups. Thus l(C) = 1 iff C is a cyclic group of prime order. We show the following:

Theorem Suppose that G = A ∗C B with C ﬁnite. Then 1 rank G ≥ (rank A + rank B). 2l(C)+1

The theorem also holds for amalgamated products with inﬁnite amalgam provided that there is an upper bound for the length of any strictly ascending chain of subgroups of C. Such groups must be torsion groups; examples are the Tarski monsters exhibited by A. Yu Ol’shanskii [7]. The proof is related to the proof of Dunwoody’s version [3] of Linnell’s [6] accessibility result for ﬁnitely generated groups. However it is more involved as we not only keep track of the size of edge groups but also of the (relative) rank of vertex groups. A careful analysis of the proof suggests that a better bound might be possi- 1 1 ble namely that 2l(C)+1 can be replaced by 2l(C)+1 . The proof applies a folding sequence that terminates with the graph of groups corresponding to the amalgamated product. A similar formula for HNN-extensions can be found in the same way.

1 2 Approximating graphs of groups

In this section we briefly fix notations for graphs of groups and describe how a graph of groups can be approximated by a sequence of graphs of groups. What we describe follows immediately from the discussion of A-graphs in [5] which essentially is an alternative formulation of the combination of foldings and vertex morphisms discussed in [3] where M. Dunwoody builds on earlier work of J. Stallings [8], [9] and M. Bestvina and M. Feighn [1]. We will not describe the whole theory but only state some consequences. A trusting reader should find this sufficient to follow the arguments later on. A graph A consists of a set of vertices VA, a set of edges EA, an inversion −1 : EA → EA and maps α : EA → VA and ω : EA → VA such that α(e) = ω(e−1) for all e ∈ EA. We denote the Betti number of A by b(A). Recall that the Betti number of A is the number of edge pairs outside a maximal subtree of A.

A graph of groups A consists of a graph A, vertex groups Gv for every v ∈ VA, edge groups Ge for every edge e ∈ EA such that Ge = Ge−1 and boundary monomorphisms αe : Ge → Gα(e) and ωe : Ge → Gω(e) satisfying αe = ωe−1 ; thus only the maps αe need to be speciﬁed. An approximation of a graph of groups A is a pair (A¯, p¯) where A¯ is a graph of groups with underlying graph A¯ andp ¯ : A¯ → A is graph morphism such that ¯ ¯ Gv ⊂ Gp¯(v) for all v ∈ V A and Ge ⊂ Gp¯(e) for all e ∈ EA. Suppose now that A is a graph of groups with the following properties1: 1. A is a ﬁnite graph without loop edges, i.e. α(e) 6= ω(e) for all e ∈ EA.

2. n := rank π1(A) < ∞. 3. At least one vertex group is non-trivial and all edge groups are Noetherian.

It then follows immediately from the discussion in [5] that there exists a sequence of approximations

(A0, p0), (A1, p1),..., (Am, pm) of A (where Ai has underlying graph Ai) such that the following hold:

1. A0 is a subdivision of the wedge of n − 1 circles with base vertex v0. All

vertex and edge groups are trivial except Gv0 which is a cyclic group.

2. pm is a graph isomorphism, Gv = Gpm(v) for all v ∈ VAm and Ge = Gpm(e) for all e ∈ EAm. 1Note that we make the assumption that A has no loop edges as we would otherwise need to consider 6 moves instead of 3. The assumption that edge groups are Noetherian guarantees that the folding process can be done in such a way that all possible folds of type II are applied before any fold of type I or III. As a consequence folds of type I and III only identify edges with the same edge group. Finally the fact that at least one vertex group is non-trivial guarantees that any generating set of π1(A) is Nielsen equivalent to a another generating set containing an element from one of the vertex groups. This guarantees the existence of the ﬁrst approximation in the sequence of approximations.

2 3. The approximation (Ai, pi) is obtained from (Ai−1, pi−1) by on the three moves described below for 1 ≤ i ≤ m. Furthermore there exists a surjective graph morphism fi : Ai−1 → Ai such that pi−1 = pi ◦ fi.

In the following we will use the superscript i to indicate that the object in question belongs to the graph of groups Ai.

A move of type I: x, y, z ∈ VAi−1 are distinct vertices with pi−1(y) = pi−1(z) and e1, e2 ∈ EAi−1 are edges such that α(e1) = α(e2) = x, ω(e1) = y and ω(e ) = z. Furthermore αi−1(Gi−1) and αi−1(Gi−1) are conjugate in Gi−1. 2 e1 e1 e2 e2 x Ai is obtained from Ai−1 by identifying e1 with e2 and y with z and fi : Ai−1 → Ai is the quotient map. Put [y] = fi(y) = fi(z) and [e1] = fi(e1) = fi(e2). Thus we have VAi = (VAi−1 − {y, z}) ∪ {[y]} and EAi = (EAi−1 − ±1 ±1 ±1 {e1 , e2 }) ∪ {[e1] }. i i−1 i−1 −1 The vertex and edge groups of Ai are as follows: G[y] = hGy , hGz h i for some h ∈ G , Gi = Gi−1, Gi = Gi−1 for all v 6= y, z and Gi = Gi−1 pi−1(y) [e1] e1 v v f f for all f 6= e1, e2. i i−1 i i The boundary monomorphisms are as follows: α = α , α −1 = ω = [e1] e1 [e1] [e1] ωi−1. For all edges f with α(f) = z we deﬁne αi : Gi → Gi as αi (g) = e1 f f [y] f i−1 −1 i i−1 hαf (g)h . For all remaining edges e we put αe = αe . Furthermore U := ωi (Gi ) ⊂ hGi−1h−1 and U is in hGi−1h−1 conjugate to h(ωi−1(Gi−1))h−1. [e1] [e1] z z e2 e2

i−1 Gi−1 Gy e1 q Gi = Gi−1 Gi = hGi−1, hGi−1h−1i i−1 - x x [y] y z Gx P P i−1 i P G = Ge PP [e1] 1 q PP q q i−1 PP i−1 Ge2 Gz q

A move of type II: There exists an edge e ∈ Ai−1 with α(e) = x and ω(e) = y. We have Ai = Ai−1, fi is the identity and pi+1 = pi. i i−1 0 0 i i−1 We have Ge = hGe , h i for some element h ∈ Gpi−1(e) and Gf = Gf i i−1 for all other edges. Furthermore Gy = hGy , hi for some h ∈ Gpi−1(y) and i i−1 Gv = Gv for all other vertices. −1 i i−1 i i i For all f 6= e, e we have αf = αf and αe and ωe = αe−1 are such that i i−1 i i−1 i i−1 i−1 the restriction of αe to Ge ⊂ Ge is αe , the restriction of ωe to Ge is ωe i 0 and ωe maps h to h.

i i−1 0 i−1 Ge = hGe , h i i−1 Ge i−1 - Gx Gy Gi = Gi−1 Gi = hGi−1, hi q qx qx y q y

A move of type III: In this move there exist edges e1, e2 ∈ EAi−1 such that α(e1) = α(e2) = x and ω(e1) = ω(e2) = y. Ai is obtained from Ai by identifying e1 with e2 and fi is the quotient map. Denote fi(e1) = fi(e2) by [e1]. Thus we

3 ±1 ±1 ±1 have VAi = VAi−1 and EAi = (EAi−1 − {e1 , e2 }) ∪ {[e1] }. The map pi is simply the restriction of pi−1 to Ai. We have Gi = hGi−1, hi for some h ∈ G satisfying hωi−1(Gi−1)h−1 = y y pi(y) [e1] e1 ωi−1(Gi−1) and Gi = Gi−1 for all other vertices. Also Gi = Gi−1 and [e2] e2 v v [e1] e1 i i−1 Gf = Gf for all other edges. i i−1 i i i−1 i i−1 We further have α = α , ω = α −1 = ω and α = α for all [e1] e1 [e1] [e1] e1 f f −1 f ∈ EAi − {[e1], [e1] }.

i−1 Ge 1 i i−1 G[e ] = Ge1 i−1 i−1 - 1 Gx Gy i i−1 i i−1 Gx = Gx Gy = hGy , hi qi−1 q q q Ge2

3 Generation complexities

Let A be a graph of groups. Let P(EA) be the power set of EA. Suppose we have functions

r : VA → N0, p : VA → N0 and E : VA → P(EA) such that the following hold: 1. E(v) ⊂ EA is a set of cardinality p(v) with α(e) = v for all e ∈ E(v).

2. There exist elements g1, . . . , gr(v) ∈ Gv and elements he ∈ Gv for each e ∈ E(v) such that

−1 Gv = {g1, . . . , gr(v)} ∪ {heαe1 (Ge)he | e ∈ E(v)} .

2 Put w = (r, p): VA → N0 ×N0. We call the pair (w, E) a generation complexity and w¯(v) = r(v) + max(0, p(x) − 1) the weight of v.

It is clear that Gv = 1 for any vertex v with w(v) = (0, 0) and that any vertex v with w(v) = (0, 1) is inessential, i.e. there exists an edge e with α(e) = v such that αe is surjective. However the opposite is not necessarily true as we make no minimality assumption. In particular no essential vertex lies in the set

S = S(A, w) := {v ∈ VA | w(v) = (0, 0) or w(v) = (0, 1)}.

2 The function wG : VA → N0 × N0 deﬁned as wG(v) = (rank Gv, 0) clearly is a generation complexity (with E(v) = ∅ for all v ∈ VA). It is this function that generation complexities in general attempt to generalize by also taking into account elements that have been pushed through edge groups into a vertex group. Note that the generation complexity wG would be the right one to prove Grushko’s theorem.

4 4 The proof of the theorem

Suppose now that G = A ∗C B with finite C. Thus G = π1(A) where EA = −1 {e, e }, VA = {v, w} with α(e) = v and ω(e) = w, Gv = A, Gw = B and Ge = C. Clearly A satifies the conditions spelt out in Section 2. Thus we can find a sequence of approximations (A0, p0),..., (Am, pm) and maps f1, . . . , fm as described in Section 2. Put l := l(C).

Deﬁne w0 : A0 → N0 × N0 as w0(v0) = (1, 0) and w0(v) = (0, 0) for v 6= v0 and E0 : VA0 → P(EA0) as E0(v) = ∅ for all v ∈ VA0. Clearly the pair (w0,E0) is a generation complexity for A0. The proof of the theorem follows from the following:

Proposition Let 1 ≤ i ≤ m. Suppose that (wi−1,Ei−1) is a generation complexity for Ai−1. Then there exists a generation complexity (wi,Ei) for Ai such that c(Ai, (wi,Ei)) ≤ c(Ai−1, (wi−1,Ei−1))

where the complexity c(Aj, (wj,Ej)) of the pair (Aj, (wj,Ej)) is deﬁned as

X l−l(Gj )+1 X l−l(Gj )+1 X b(Aj) + (2 e − 1) − (2 v − 1) + w¯j(v).

e∈EAj v∈Sj v∈VAj with Sj = S(Aj, wj) and w¯j the weight function corresponding to wj. Before we give a proof of the Proposition we show that it implies the Theo- rem: Proof of the Theorem: It clearly follows from the proposition that there exists a generation complexity (wm,Em) for Am such that

c(Am, (wm,Em)) ≤ c(A0, (w0,E0)). (1) It turns out that the Theorem follows immediately from this inequality once we have computed c(A0, (w0,E0)) and c(Am, (wm,Em)). l+1 Claim 1: c(A0, (w0,E0)) = (n − 1)2 + 1. Proof of Claim 1: Clearly b(A0) = n − 1 and #EA0 − #S0 = n − 1. It follows l−l(G0)+1 l−l(G0 )+1 l+1 that P (2 e − 1) − P (2 v − 1) = (n − 1)(2 − 1) as all e∈EA0 v∈S0 occuring edge and vertex groups are trivial and l(1) = 0. The last summand yields 1 asw ¯0(v0) = 1 andw ¯0(v) = 0 for v 6= vo. Thus we have c(A0, (w0,E0)) = (n − 1) + (n − 1)(2l+1 − 1) + 1 = (n − 1)2l+1 + 1. 2

Claim 2: c(Am, (wm,Em)) ≥ rank A + rank B − 2 rank C + 1. Proof of Claim 2: We have b(Am) = 0. Clearly v, w ∈ VAm are essential; P l−l(Gm)+1 thus Sm = ∅. We have (2 e − 1) = 1 as the only edge group is e∈EAm P l−l(Gm) C itself. Furthermore 2 v = 0 as Sm = ∅. As v and w are vertices v∈Sm of valence 1 we have pm(v), pm(w) ≤ 1 and it is clear thatw ¯m(v) = rm(v) ≥ rank A − rank C as at least rank A − rank C elements of A needed to generate A together with a homomorphic image of C; the same argument shows that w¯(w) ≥ rank B − rank C. This implies that c(Am, (wm,Em)) ≥ 1 + (rank A − rank C) + (rank B − rank C) = rank A + rank B − 2 rank C + 1. 2

5 The claims together with (1) gives (n − 1)2l+1 + 1 ≥ rank A + rank B − 1 l+1 2 rank C + 1 which implies n ≥ 2l+1 (rank A + rank B − 2 rank C + 2 ). As 1 l+1 rank C ≤ l(C) this yields rank G = n ≥ 2l+1 (rank A + rank B − 2l + 2 ). The assertion of the theorem follows as −2l + 2l+1 ≥ 0 for l ≥ 0. Proof of the Proposition: We will distinguish a variety of cases. In each situation we will deﬁne the pair (wi,Ei) and show that it is a generating complexity. We distinguish the cases by the type of move that is applied to (Ai−1, pi−1) to obtain (Ai, pi). Recall that Sj := {v ∈ VAj | wj(v) = (0, 0) or wj(v) = (0, 1)} and that wj(v) = (rj(v), pj(v)).

Before we deal with the different cases we explain how to define the maps wi and Ei for a vertex that is not affected by the move, i.e. a vertex different from x, y and z if the move is of type I and different from x and y if the move is of type II or III. For such a vertex we put wi(v) := wi−1(v) and Ei(v) := Ei−1(v). It is obvious that this is permitted by the definition of the generation complexity as around such vertices the graph of groups is unchanged. It follows in particular that for such a vertex v ∈ Si iff v ∈ Si−1. For simplicity we will write

j j j cj = c(Aj, (wj,Ej)) = b(Aj) + Σ1 + Σ2 + Σ3

j j j j j P l−l(Ge)+1 P l−l(Gv )+1 where Σ1 = (2 − 1), Σ2 = − (2 − 1) and Σ3 = e∈EAj v∈Sj P w¯j(v). v∈VAj i i i Note that for moves of type I or II we need to show that Σ1 + Σ2 + Σ3 ≤ i−1 i−1 i−1 Σ1 + Σ2 + Σ3 as b(Ai) = b(Ai−1).

Moves of type I: We need to explain wi and Ei for the vertices x and [y].

We ﬁrst deal with the vertex x: If x ∈ Si−1, i.e. if wi−1(x) = (0, 0) or wi−1(x) = (0, 1) we can clearly put wi(x) = wi−1(x) and Ei(x) = fi(Ei−1(x)). If x∈ / Si−1 and {e1, e2} 6⊂ Ei−1(x) we can again put wi(x) = wi−1(x) and Ei(x) = fi(Ei−1(x)). If x∈ / Si−1 and {e1, e2} ⊂ Ei−1(x) we have to work a little harder. Again we put Ei(x) = fi(Ei−1(x)) but contrary to the above case we have that the cardinality of Ei(x) is by one smaller than the cardinality of Ei−1(x), thus we have one less edge group to generate the vertex group. As α (Gi−1) and e1 e1 α (G−1) where in G conjugate, it is possible to make up for this missing edge e2 e2 x group by adding one element. Thus we can put wi(x) = (ri−1(x)+1, pi−1(x)−1). Note that our discussion implies thatw ¯i(x) =w ¯i−1(x) and that x ∈ Si−1 i−1 i iﬀ x ∈ Si. As further Gx = Gx it follows that the contribution of x to the complexity is unchanged. We next deal with the vertex [y]. We distinguish three situations:

Case 1: Suppose that y ∈ Si−1 and z ∈ Si−1, i.e. that ri−1(y) = ri−1(z) = 0 and that Ei−1(y) and Ei−1(z) contain at most one edge. We distinguish three subcases: −1 −1 (a) Suppose that Ei−1(y) = {e1 } or Ei−1(y) = ∅ (the case Ei−1(z) = {e2 } or Ei−1(z) = ∅ is analogous). In this case we put wi([y]) = (0, 1) and Ei([y]) = i i−1 fi(Ei−1(z)). Note that this implies that [y] ∈ Si. We clearly have Σ1 = Σ1 − l−l(Gi−1)+1 i i−1 l−l(Gi−1)+1 i i−1 i−1 (2 e1 − 1), Σ = Σ + (2 y − 1) and Σ = Σ . As G ∼ 2 2 3 3 e1 =

6 i−1 i−1 i−1 l−l(Ge )+1 l−l(Gy )+1 Gy it follows that 2 1 − 1 = 2 − 1. This implies that i i i i−1 i−1 i−1 Σ1 + Σ2 + Σ3 = Σ1 + Σ2 + Σ3 . −1 −1 (b) Suppose now that e1 ∈/ {e3} = Ei−1(y), e2 ∈/ {e4} = Ei−1(z) and that ωi−1 is surjective (the case that ωi−1 is surjective is analogous). We put e2 e1 i i i i−1 wi([y]) = (0, 1) and Ei([y]) = {e3}. As in (a) we get Σ1 + Σ2 + Σ3 = Σ1 + i−1 i−1 Σ2 + Σ3 . −1 −1 (c) Suppose now that e1 ∈/ {e3} = Ei−1(y), e2 ∈/ {e4} = Ei−1(z) and that neither ωi−1 nor ωi−1 is surjective. Put w ([y]) = (0, 2) and E ([y]) = {e , e }. e1 e2 i i 3 4 i−1 i−1 i i−1 l−l(Ge +1) i i−1 l−l(Gy )+1 We have Σ1 = Σ1 − (2 1 − 1), Σ2 = Σ2 + (2 − 1) + i−1 l−l(Gz +1) i i−1 (2 − 1) and Σe = Σ3 + 1. The non-surjectivity of ωi−1 and ωi−1 imply that l(Gi−1) > l(Gi−1) and e1 e2 y e1 i−1 i−1 l−l(Gi−1+1) l−l(Gi−1)+1 l(G ) > l(G ) and therefore (2 e1 − 1) ≥ (2 y − 1) + z e1 i−1 l−l(Gz )+1 i i i i−1 i−1 i−1 (2 − 1) + 1 this implies that Σ1 + Σ2 + Σ3 ≤ Σ1 + Σ2 + Σ3 . Case 2: Suppose now that y∈ / Si−1 and that z ∈ Si−1; the opposite case is analogous. If wi−1(z) = (0, 0) we can argue as in case (a) above; thus we can assume that wi−1(z) = (0, 1). Choose e3 ∈ EAi−1 such that Ei−1(z) = {e3}. We distinguish three subcases: −1 (a) Suppose that e2 = e3. We put wi([y]) = wi−1(y) and Ei([y]) = fi(Ei−1(y)). i i i i−1 i−1 i−1 As in (a) of case 1 it follows that Σ1 + Σ2 + Σ3 = Σ1 + Σ2 + Σ3 . (b) Suppose that e−1 6= e and that ωi−1 is surjective. We put w ([y]) = w (y) 2 3 e2 i i−1 i i i i−1 i−1 i−1 and Ei([y]) = fi(Ei−1(y)). Again we get Σ1 + Σ2 + Σ3 = Σ1 + Σ2 + Σ3 . (c) Suppose that e−1 6= e and that ωi−1 is not surjective. We put w ([y]) = 2 3 e2 i i i−1 (ri−1(y), pi−1(y) + 1) and Ei([y]) = fi(Ei−1(y)) ∪ {e3}. We have Σ1 = Σ1 − l−l(Gi−1)+1 i−1 e i i−1 l−l(Gz )+1 i i−1 (2 2 − 1), Σ2 = Σ2 + (2 − 1) and Σ3 = Σ3 + 1. The non i−1 i−1 i−1 l−l(Gi−1)+1 surjectivity of ω implies that l(G ) < l(G ) and therefore 2 e2 − e2 e2 z i−1 l−l(Gz )+1 i i i i−1 i−1 i−1 1 > 2 − 1. It follows that Σ1 + Σ2 + Σ3 ≤ Σ1 + Σ2 + Σ3 . Case 3: Finally suppose that y, z∈ / Si−1. We put Ei([y]) = fi(Ei−1(y) ∪ Ei−1(z)) and wi([y]) = (ri−1(y) + ri−1(z), pi([y])) with pi([y]) = #Ei([y]). i i−1 Clearly pi([y]) ≤ pi−1(y) + pi−1(z) and therefore Σ3 ≤ Σ3 + 1. As further i i−1 i i−1 i i i i−1 i−1 i−1 Σ1 < Σ1 and Σ2 = Σ2 it follows that Σ1 + Σ2 + Σ3 ≤ Σ1 + Σ2 + Σ3 . Moves of type II: Again dealing with the vertex x is simple as we can simply put wi(x) = wi−1(x) and Ei(x) = Ei−1(x). The contribution of x to the complexity does not change.

Case 1: Suppose that y ∈ Si−1. −1 (a) Suppose that e ∈ Ei−1(y) or that Ei−1(y) = ∅. We can put wi(y) = (0, 1) i−1 i −1 i i−1 l−l(Ge )+1 l−l(Ge)+1 and Ei(y) = {e }. We have Σ1 = Σ1 − (2 − 1) + (2 − 1), i−1 i i i−1 l−l(Gy )+1 l−l(Gy )+1 i i−1 i−1 ∼ i−1 Σ2 = Σ2 +(2 −1)−(2 −1) and Σ3 = Σ3 . As Ge = Gy i ∼ i i i i i−1 i−1 i−1 and Ge = Gy this implies that Σ1 + Σ2 + Σ3 = Σ1 + Σ2 + Σ3 . −1 e (b) Suppose now that e ∈/ {e3} = Ei−1(y) and that ωi−1 is not surjective. We i−1 −1 i i−1 l−l(Ge )+1 put wi(y) = (0, 2) and Ei(y) = {e , e3}. We have Σ1 = Σ1 − (2 − i i−1 l−l(Ge)+1 i i−1 l−l(Gy )+1 i i−1 1) + (2 − 1), Σ2 = Σ2 + (2 − 1) and Σ3 = Σ3 + 1. As i i−1 i−1 i−1 i i i l(Ge) > l(Ge ) and l(Gy ) > l(Ge ) this implies that Σ1 + Σ2 + Σ3 ≤ i−1 i−1 i−1 Σ1 + Σ2 + Σ3 .

7 −1 e (c) Suppose now that e ∈/ {e3} = Ei−1(y) and that ωi−1 is surjective. We −1 put wi(y) = (0, 1) and Ei(y) = {e }. The same calculation as in (a) shows i i i i−1 i−1 i−1 that Σ1 + Σ2 + Σ3 = Σ1 + Σ2 + Σ3 . i i−1 i i−1 Case 2: Suppose that y∈ / Si−1. We clearly have Σ1 < Σ1 and Σ2 = Σ2 . −1 (a) Suppose that e ∈ Ei−1(y). We put wi(y) = wi−1(y) and Ei(y) = Ei−1(y). i i−1 i i i i−1 i−1 i−1 It follows that Σ3 = Σ3 and therfore Σ1 + Σ2 + Σ3 ≤ Σ1 + Σ2 + Σ3 . −1 (b) Suppose that e ∈/ Ei−1(y). We put wi(y) = (ri−1(y), pi−1(y) + 1) and −1 i i−1 i i Ei(y) = Ei−1(y) ∪ {e }. It follows that Σ3 ≤ Σ3 + 1 and therfore Σ1 + Σ2 + i i−1 i−1 i−1 Σ3 ≤ Σ1 + Σ2 + Σ3 . Moves of type III: We deal with x as we did in the case of a move of type i i i i−1 i−1 i−1 I. We only need to prove that Σ1 + Σ2 + Σ3 ≤ Σ1 + Σ2 + Σ3 + 1 as b(Ai) = b(Ai−1)−1. We put Ei(y) = fi(Ei−1(y)) and wi(y) = (fi−1(y)+1, pi(y)) with pi(y) = #Ei(y) ≤ #Ei−1(y) = pi−1(y). In particular we have y∈ / Si. i−1 i i−1 l−l(Ge )+1 i i−1 i We have Σ1 = Σ1 − (2 1 − 1), Σ2 = Σ2 if y∈ / Si−1,Σ2 = i−1 l−l(Gi−1)+1 i−1 Σ + (2 y − 1) if y ∈ S and Σi ≤ Σ + 1. As l(Gi−1) ≥ l(Gi−1) 2 i−1 3 3 y e1 i i i i−1 i−1 i−1 this implies that Σ1 + Σ2 + Σ3 ≤ Σ1 + Σ2 + Σ3 + 1.

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