EXAMPLES OF AMALGAMATED PRODUCTS

INNA BUMAGINA

Abstract. The rank of a group G is the minimal number of elements that generate G. For any natural number n we construct two groups, G_{1} of rank r(G_{1}) = n and G_{2} of rank r(G_{2}) = 2n such that their amalgamated product over an infinite cyclic subgroup, malnormal in both factors, is generated by 2n = r(G_{1}) + r(G_{2}) − n elements. We also consider an example of amalgamated product of n factors: G = ∗_{i=1}^{n}_{A}G_{i} such that r(G) = n + 1, and r(A) ≥ 1. This example realizes the lower bound given in [W1] (see Theorem 2 in the present paper).

1. Introduction

By the celebrated Grushko’s theorem, the rank of the free product U_{1} ∗U_{2} of two arbitrary groups U_{1} and U_{2} is r(U_{1} ∗U_{2}) = r(U_{1})+r(U_{2}). However, no non-trivial analog of this theorem holds for amalgamated products. (See [L-S, I.11] and [Ro]). Kaufmann and Zieschang give in [K-Z] an example of an amalgamated product U = U_{1} ∗_{C} U_{2} such that r(U) = r(U_{1}) = r(U_{2}) = n, and the amalgamated subgroup C is cyclic of order two. They also prove the following.

Theorem 1 (R. Kaufmann, H. Zieschang). For every function d: ℕ ⟶ ℕ there exist groups U, V, W such that r(U ∗_{W} V) ≤ r(U) + r(V) − d(r(W)).

Weidmann [W2] shows that for any integer n, there are groups A, B such that r(A) ≥ n, r(B) ≥ n, and r(A ∗_{C} B) = r(C) = 2. [W2] and [K-Z] provide examples where the amalgamated subgroup is not malnormal. However, if we consider amalgamated products along a malnormal subgroup, the situation will change.

Definition 1. Let H be a subgroup of a group G. H is malnormal in G if for any g ∈ G \ H, we have gHg^{−1} ∩ H = {1}. We follow [W1] and

1991 Mathematics Subject Classification. Primary: 20F05, 20F06; Secondary:

05C25.

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say that H ⊂ G is premalnormal in G if there exists a proper subgroup U ⊂ G such that H ⊆ U and for all g ∈ G \ U the following condition holds: gHg^{−1} ∩ U = {1}. Premalnormality is a weaker condition than malnormality: if H is malnormal in G then putting U = H we see that H is premalnormal also.

The rank of amalgamated products over (pre)malnormal cyclics is of special interest in knot theory because of the connection between the tunnel number t(K) of a knot K and the rank of its fundamental group. In general, there is no uniform bound on the degeneration of tunnel numbers under connected sums, because using Morimoto’s example [M], Kobayashi [Ko] shows that for any positive integer n there exist knots K_{1} and K_{2} such that the tunnel number t(K_{1}#K_{2}) of their connected sum satisfies t(K_{1}#K_{2}) < t(K_{1}) + t(K_{2}) − n. However, some lower bounds are known. Scharlemann and Schultens [S-S] prove that the tunnel number of the sum of n knots is at least n. Weidmann [W1] proves the following group theoretical analog of their result:

Theorem 2 (R. Weidmann). Let G = ∗_{i=1}^{n}_{A}G_{i} and A ≠ {1}. If A is premalnormal in G_{i} for 1 ≤ i ≤ n then r(G) ≥ n + 1.

This theorem implies the result of Scharlemann and Schultens; on the other hand, it follows that the inequality r(G) ≥ n + 1 holds if A is malnormal also (which was proved in [K-S] for the case n = 2). It is not difficult to see that in case A is cyclic, the estimate of Theorem 2 is precise: just take G_{i} = ⟨a, g_{i}⟩ to be a copy of the free group of rank 2 for 1 ≤ i ≤ n, and A = ⟨a⟩. However, in the statement of Theorem 2, there is no restriction on the rank of the amalgamated subgroup A, while the above example (where r(A) = 1) cannot be extended to the general case. Nevertheless as our Example 2 shows, the equality in the estimate of Theorem 2 can be attained in case r(A) > 1 as well.

Another lower bound for r(G) is given by the following lemma.

Lemma 1. Let G = ∗_{i=1}^{n}_{A}G_{i} be an amalgamated product of n groups, A ≠ {1}, and suppose that J ⊂ {1, …, n} is a non-empty set of indices. Let H = ∗_{i∈J} _{A}G_{i} . If for all i ∉ J there is a map φ_{i}: G_{i} ⟶ A such that φ_{i}|_{A} = id_{A} then r(G) ≥ r(H).

Proof. We define a homomorphism φ : G ⟶ H in the following way:

φ|_{G_{i}} = \{ \begin{matrix}id_{G_{i}} & , i ∈ J \\ φ_{i} & , i ∉ J \end{matrix}

Clearly, φ is onto. Thus r(G) ≥ r(H).

It is well-known that every knot group is mapped onto the meridional subgroup by the abelianization, so Lemma 1 holds if we consider connected sum of n knots. In this case we get, in particular

r(G) ≥ max_{1≤i≤n} r(G_{i}).

Example 1 shows that equality can hold in the inequality of Lemma 1.

2. Examples of groups

Lemma 2. 1. Let a group S be defined as follows:

S = ⟨g, h, z | z = [g, h]⟩.

Then the subgroup Z = ⟨z⟩ ⊂ S is infinite cyclic and malnormal in S. Moreover, there exists a word w(x, y) such that in the group

R = ⟨z, b | b = w(z, bzb^{−1})⟩,

the subgroup Z = ⟨z⟩ ⊂ R is infinite cyclic and malnormal also.

2. Let W = ⟨c, d⟩ ≅ F_{2} be the free group of rank 2, and let a_{i} = w_{i}(c, d), i = 1, 2, …, k be elements of W. Then the words w_{i} can be chosen in such a way that A = ⟨a_{1}, …, a_{k}⟩ is a free subgroup of rank k which is malnormal in W.

Proof. 1. By [Gr, page 136], a maximal cyclic subgroup Z of a torsionfree word hyperbolic group G is malnormal. The statement for the free group S ≅ F_{2} follows immediately because in a free group, a commutator has no root. For the group R, the proof is constructive. We choose a word w so that R satisfies the small cancellation condition C′(\frac{1}{6}) (whence G is word hyperbolic), and so that there is a retract of R onto Z = ⟨z⟩ (which implies Z is a maximal cyclic subgroup of the torsion free group R). (See [Gr] and [L-S, Chapter V] for the definitions of a word hyperbolic group and of a group with small cancellation, respectively.) For instance, we set w(x, y) = xy^{−1}x^{2}yx^{3}y^{2}xy^{2}x^{2}y^{5}x^{6}yxy^{3}. Then any piece p(x, y) is either of the following words (or their subwords): xyx, xy^{2}x, x^{2}yx, x^{2}y^{2}, xy^{3}, x^{3}y, yxy^{2}, y^{2}x^{2}, yx^{3}, and it is

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not difficult to see that R satisfies condition C′(\frac{1}{7}), and hence C′(\frac{1}{6})). Indeed, regard w(x, y) and pieces p(x, y) as functions of x and y. Substitute (as suggested by the statement):

x = z, y = bzb^{−1}.  (1)

Then the relation r of R is as follows:

r = b^{−1}w(z, bzb^{−1}) =
b^{−1}zbz^{−1}b^{−1}z^{2}bzb^{−1}z^{3}bz^{2}b^{−1}zbz^{2}b^{−1}z^{2}bz^{5}b^{−1}z^{6}bzb^{−1}zbz^{3}b^{−1},

so that |r| = 46. It can be checked that if p is a piece relatively to {r} then |p| ≤ 6. Hence \frac{|p|}{|r|} ≤ \frac{6}{46} < \frac{1}{7} so that the given presentation of the group R satisfies C′(\frac{1}{7}) as claimed. The homomorphism φ: R ⟶ Z such that φ(z) = z and φ(b) = z^{29} is the retract.

2. Put a_{i} = w_{i}(c, d) = cd^{r_{i}}cd^{r_{i}+1} … cd^{s_{i}}, r_{i} = 25i, s_{i} = r_{i} + 24. Because [r_{i}, s_{i}] ∩ [r_{j}, s_{j}] = ∅ for i < j, a piece that can be cancelled in a product of a_{i}^{ϵ} and a_{j}^{−ϵ} (i ≠ j, ϵ = ±1) is either cd^{r_{i}} or cd^{s_{i}}. Hence {a_{1}, …, a_{k}} is a basis of a free subgroup of rank k.

Let X_{A} = {a_{1}^{±1}, …, a_{k}^{±1}} be an alphabet of A. We say that a word f = x_{1} … x_{l} in the elements of X_{A} is cyclically reduced in A if f is reduced as a word in the alphabet X_{W} = {c^{±1}, d^{±1}} of W and x_{1} ≠ x_{l}^{−1}. Notice that the word

a_{1}a_{2}^{−1} = cd^{r_{1}}cd^{r_{1}+1} … cd^{s_{1}−s_{2}}c^{−1} … d^{−(r_{2}+1)}c^{−1}d^{−r_{2}}c^{−1}

is cyclically reduced in A but is not cyclically reduced in the usual meaning.

Now, let 1 ≠ u, v ∈ A, and suppose that there exists g ∈ W \ A such that g^{−1}ug = v. We claim that necessarily g ∈ A which means malnormality of A. Let v = x_{1} … x_{n}, and let u = y_{1} … y_{m} when x_{1}, …, x_{n}, y_{1}, …, y_{m} ∈ X_{A}. First, suppose that v is cyclically reduced in A. If u is not cyclically reduced in A, we write u = s(u)ũs(u)^{−1} where s(u) ∈ A, and ũ (clearly, ũ ∈ A) is cyclically reduced in A. Now, gug^{−1} = g̃ũg̃^{−1} where g̃ = gs(u) so that g ∈ A if and only if g̃ ∈ A. Then g̃ũg̃^{−1}(= v) is a cyclic permutation of ũ. The exponents of d in the words w_{i} for all i are never repeated, and in any freely reduced product z of the elements of X_{A} the syllables d^{r_{i}+1}, …, d^{s_{i}−1} survive (as well as the very first and the very last exponents of d in z). Thus these syllables define both the factors and their order in z. The claim

EXAMPLES OF AMALGAMATED PRODUCTS 5

follows. If v = s(v)ṽs(v)^{−1} (where 1 ≠ s(v) ∈ A whence ṽ ∈ A, and ṽ is cyclically reduced in A) is not cyclically reduced in A then we consider the equality ṽ = ĝũĝ^{−1} where ĝ = s(v)^{−1}g̃ (g̃ is as above) so that ĝ ∈ A if and only if g ∈ A. This completes the proof of the malnormality of A.

2.1. Degeneration of the rank. For any n ≥ 3 we construct two groups, G_{1} of rank r(G_{1}) = n and G_{2} of rank r(G_{2}) = 2n such that their amalgamated product G = G_{1} ∗_{Z} G_{2} (where Z = ⟨z⟩ ≅ ℤ is malnormal in both factors G_{1} and G_{2}) is generated by r(G_{1}) + r(G_{2}) − n = 2n = r(G_{2}) elements.

Example 1. Let

G_{1} = ⟨z, b_{i}zb_{i}^{−1} | b_{i} = w(z, b_{i}zb_{i}^{−1}), 1 ≤ i ≤ n − 1⟩ = ∗_{i=1}^{n−1}_{Z}R_{i}
G_{2} = ⟨z, g_{1}, h_{1}, …, g_{n}, h_{n} | z = [g_{i}, h_{i}] for all i⟩ = ∗_{i=1}^{n}_{Z}S_{i},

where n ≥ 3, and R_{i} and S_{i} are copies of groups R and S respectively (R and S as in Lemma 2). Let G = G_{1} ∗_{Z} G_{2}, then

X = {g_{n}, h_{n}, b_{1}g_{1}b_{1}^{−1}, b_{1}h_{1}b_{1}^{−1}, …, b_{n−1}g_{n−1}b_{n−1}^{−1}, b_{n−1}h_{n−1}b_{n−1}^{−1}}

is a minimal system of generators of G.

Claim 1. With the notation of Example 1, r(G_{1}) = n, r(G_{2}) = 2n, Z is infinite cyclic and malnormal in G_{1} and G_{2}, and r(G) = 2n = #X.

Proof. The subgroup Z is infinite cyclic and malnormal in R and S, whence Z has the same properties in G_{1} and G_{2}. The rank property of G_{1} follows from Theorem 2. The group G_{2} is generated by the 2n elements g_{1}, h_{1}, …, g_{n}, h_{n}, and cannot be generated by a smaller number of elements, as the abelianization shows. All the generators of G_{1} and G_{2} (in the given presentations) can be expressed as words in the elements of X. Indeed, using the relations of G_{2} we write z = [g_{n}, h_{n}] and b_{i}zb_{i}^{−1} = [b_{i}g_{i}b_{i}^{−1}, b_{i}h_{i}b_{i}^{−1}] for all i = 1, …, n − 1. Using the relations of G_{1}, we express b_{1}, …, b_{n−1} in terms of z, b_{i}zb_{i}^{−1}. Now all the generators of G_{2} can be obtained from b_{i}g_{i}b_{i}^{−1}, b_{i}h_{i}b_{i}^{−1} ∈ X. Thus we have shown that r(G) ≤ #X = 2n. As 2n = r(G_{2}) = max{r(G_{1}), r(G_{2})} and the abelianization of G_{1} is a retract of G_{1} onto Z, in view of Lemma 1 we get r(G) ≥ 2n, whence r(G) = 2n(= r(G_{1}) + r(G_{2}) − n).

2.2. The lower bound by R. Weidmann can be attained in general. Our next example shows that equality in the statement of Theorem 2 can hold when r(A) > 1 also. This example is very much in the spirit of examples constructed by Richard Weidmann for different purposes (see, for instance [W3, Section 3]).

Example 2. We set G_{1} = W (W as in Lemma 2), and for all j = 2, …, n, G_{j} = ⟨a_{1}, …, a_{k}, g_{j}⟩ ≅ F_{k+1}, so that A = ⟨a_{1}, …, a_{k}⟩ ≅ F_{k} is malnormal in G_{i} for all i = 1, …, n. Then G is generated by c, d, g_{2}, …, g_{n}, and r(G) = n + 1.

Acknowledgments

I am grateful to my Ph.D. thesis advisor Professor Ilya Rips who helped me to realize the idea of Example 1. I also wish to thank the referee whose helpful suggestions enabled me to simplify the examples considerably and improve the exposition of the paper.

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Institute of mathematics, the Hebrew University, Jerusalem 91904, Israel

E-mail address: binna@math.huji.ac.il