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A Rank Formula for Amalgamated Products with Finite Amalgam

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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 amalga- mated 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 as- cending 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 amal- gamated 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 surjec- tive 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] }.
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