ON SMALL CANCELLATION K-GENERATED GROUPS WITH (K-1)-GENERATED SUBGROUPS ALL FREE

INNA BUMAGINA

Abstract. We give a set of effective sufficient conditions for a group G with small cancellation, generated by k elements, to have (k−1)−generated subgroups all free. We also prove that the group G has some more properties that in general do not hold in the whole class of the small cancellation groups.

1. Introduction

Let G be a k−generated group. We say that G has a (k − 1)− property if the (k − 1)−generated subgroups of G are all free. By a (k − 1)−generated subgroup of G we mean a subgroup generated by a set V of any k − 1 elements of G, not necessarily taken from the set of the generators of G.

1.1. The small cancellation hypothesis. Let F_{m} be the free group on a set X of generators where m = |X| is the cardinality of X. We will consider only finitely generated groups. A letter is an element of the set Y of generators and inverses of generators. A word w is a finite string of letters, w = y_{1} … y_{n}. We denote the identity of F by 1. Each element of F has a unique presentation as a reduced word w = y_{1} … y_{n} in which no two successive letters y_{j}y_{j+1} form an inverse pair x_{i}x_{i}^{−1} or x_{i}^{−1}x_{i} with x_{i} ∈ X. The integer n is the length of w, which we denote by |w|. A reduced word w is said to be cyclically reduced if y_{n} is not the inverse of y_{1}. We use “≡” to denote graphical identity of words.

A subset R̂ of F is said to be symmetrized if all the elements of R̂ are cyclically reduced and, for each R in R̂, all cyclically reduced conjugates of both R and R^{−1} also belong to R̂. The elements of R̂ are called relators.

2000 Mathematics Subject Classification. Primary: 20F05, 20F06; Secondary:

05C25.

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A subword b of a relator R ≡ u_{1}bu_{2} is said to be a piece if there is a relator S, with a factorization S ≡ bv, such that S^{−1}bu_{2}u_{1} is not freely equal to 1 or to a conjugate of a relator.

The small cancellation hypothesis says that the pieces are relatively small parts of the elements of R̂. One can formulate the small cancellation hypothesis in the metrical form C′(λ) where λ is a positive real number by the next way.

Condition C′(λ). If R ∈ R̂, R ≡ bc where b is a piece then |b| < λ|R|. We shall also use the non-metrical form C(p) where p is a natural

number.

Condition C(p). No element of R̂ consists of less than p pieces. Remark. If C′(λ) holds then C(p) for p < \frac{1}{λ} holds also. In a similar way, one can define notions of a piece and of small

cancellation in a free product of groups, which slightly differ from the corresponding notions defined above (see [14, Chapter V,§9] for more details); we will use these later on.

1.2. Known results. A well-known example of a group that has the (k − 1)−property is as follows. Example 1. G = ⟨a_{1}, …, a_{g}, b_{1}, …, b_{g} | [a_{1}, b_{1}] … [a_{g}, b_{g}] = 1⟩ where g > 1, and [a_{i}, b_{i}] = a_{i}b_{i}a_{i}^{−1}b_{i}^{−1} is the commutator of a_{i} and b_{i}. This is the fundamental group of the closed oriented manifold of genus g. In this group, each subgroup of an infinite index is free while the index of any (2g − 1)−generated subgroup is infinite.

Another example of a group with the (k − 1)−property can be obtained in a different way. Example 2. A presentation of the free group F_{k−1} that satisfies the small cancellation condition C′(\frac{1}{6}) :

G = ⟨x_{1}, …, x_{k} | x_{k} = x_{1} … x_{k−1} … x_{1}^{12} … x_{k−1}^{12}⟩ ≅ F_{k−1}

The (k−1)−property does not hold for all the groups with small cancellation; negative examples can easily be constructed. So the question is, what are sufficient conditions that ensure the (k − 1)−property?

One of the first results belongs to Guba [11] who proved that in a small cancellation group generated by 3 or more elements, with an additional condition on its relations, 2-generated subgroups are all free.

ON SMALL CANCELLATION K-GENERATED GROUPS 3

This result was generalized by Arzhantseva and Ol’shanski [1]. They gave a (non-effective) sufficient condition on relators of a k-generated group that ensures the (k − 1)-property for this group. Moreover, they proved that this condition is generic; as a result they obtained that a finitely presented group has the (k − 1)-property with probability close to 1, provided its relations are long enough. A generalization of the (k − 1)-property, and a sufficient condition that ensures the more general property are considered in [2]. The techniques we use in the present paper, are similar to the techniques used by Arzhantseva and Ol’shanski in [1]. We should emphasize that our research was carried independently. Moreover, on the contrary to their results, the conditions we obtain are effective.

1.3. The main results of the present paper. We give a set of effective sufficient conditions that ensure the (k −1)−property. In fact, we give a rule for writing down relations of a k−generated group G and prove that if we follow this rule then G will have the (k − 1)−property.

This rule is as follows. Let G = ⟨x_{1}, …, x_{k} | R_{i}, 1 ≤ i ≤ t⟩,(t ≥ 1) be a presentation of G where the alphabet X = {x_{1}, …, x_{k}} of G is given while the relations R_{i} (1 ≤ i ≤ t) are to be specified.

For each letter x_{i} of the alphabet we prepare a set S_{i}^{(1)} of Ak^{2} different exponents of this letter (A is a constant, A > 15):

S_{i}^{(1)} = {x_{i}^{α_{i,j}} | 1 ≤ j ≤ Ak^{2}}

where α_{i,l} ≠ α_{i,s} if l ≠ s. Now, a relation

R = x_{1}^{α_{1,1}}x_{2}^{α_{2,1}} … x_{k}^{α_{k,1}}x_{1}^{α_{1,2}}x_{2}^{α_{2,2}} … x_{k}^{α_{k,2}} … x_{1}^{α_{1,Ak^{2}}}x_{i}^{α_{2,Ak^{2}}} … x_{k}^{α_{k,Ak^{2}}}(1)

is a product of all these syllables taken in order from the sets S_{1}, S_{2}, …, S_{k}. So |R| = Ak^{3} where the length |w| of the word w means the number of syllables in it.

If we want to write down one more relation of G then we have to prepare new sets S_{i}^{(2)} of new exponents of each letter. In fact, some exponents appearing in the previous relation(s) can be repeated but it is necessary to keep small cancellation condition C′(λ) where λ < \frac{1}{10k^{2}} (1 − \frac{13}{A}). Also, relations longer than Ak^{3} can be considered; then one should fix λ < \frac{2}{15} \frac{(A−13)k}{|R_{max}|} where |R_{max}| is the length of the longest

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of the relations. Proceeding in this way we complete the set of the relations of G. Example 3. Let k = 3; we write two relations according to the above rule as follows.

R_{1} = x_{1}x_{2}^{3}x_{3}^{5}x_{1}^{7}x_{2}^{9}x_{3}^{11} … x_{1}^{801}x_{2}^{803}x_{3}^{805}
R_{2} = x_{1}^{2}x_{2}^{4}x_{3}^{6}x_{1}^{8}x_{2}^{10}x_{3}^{12} … x_{1}^{802}x_{2}^{804}x_{3}^{806}

Note that |R_{1}|, |R_{2}| > 400, and the length of a piece is equal 2 so that the small cancellation condition C′(λ) where λ < \frac{1}{200} is satisfied.

More precisely, this rule is formulated in the following Theorem. Theorem 1. Let G = ⟨x_{1}, …, x_{k} | R_{i} (1 ≤ i ≤ t)⟩ where k ≥ 3, t is any natural number and the following conditions hold:

(1) Each relation R_{i} is an ordered (in the sense of equation (1)) product of syllables of the type x_{j}^{α_{s}} such that all the k letters of alphabet are involved and each letter appears with Ak^{2} different exponents while all these exponents are unique; it follows that for all R_{i}, |R_{i}| = Ak^{3}, where A is an integer constant, A ≥ 25;

(2) R̂ = {R_{i}} (i = 1, …, t) is a small cancellation set that satisfies the condition C′(λ) where λ < \frac{1}{10k^{2}} (1 − \frac{13}{A});

Then (k−1)−generated subgroups of G are all free, and rank(G) = k. Remark. For k = 2, if R satisfies C′(\frac{1}{6}) and ∀i = 1, …, t, R_{i} ∈ R is

not a proper power in a free group then G is torsion free ([14]).

By rank(G) we mean the minimal number of the elements that can generate G. The problem of rank (namely, to prove that a k−generated group cannot be generated by a system of less than k elements) is unsolvable for small cancellation groups in general, as shown in [4]. In their example, a construction of Rips [15] is used. The groups that satisfy the conditions of this theorem have some additional nice properties that do not hold for small cancellation groups in general. Theorem 3 below asserts that in the group G, the (k − 1)−generated subgroups are quasiconvex (see Definition 14 of a quasiconvex subgroup in Section 4). Recall that there are small cancellation groups that have non-quasiconvex 2-generated subgroups ([15],[4],[18],[13]). As a subsequence of Theorem 3, we obtain the Howson property for (k − 1)−generated subgroups of G: an intersection of any two (k − 1)−generated subgroups of G is a finitely generated subgroup of G

ON SMALL CANCELLATION K-GENERATED GROUPS 5

(Corollary 3 Section 4). An example given in [18] shows that in general small cancellation groups do not have Howson property even for 2-generated subgroups. Again, in this example, the above mentioned construction of Rips [15] is used^{1}.

All our proofs are applicable to the subgroups of G which are generated by less than k − 1 elements. This means that all the above properties of (k − 1)−generated subgroups of G hold also for subgroups of G generated by less than k − 1 elements.

2. Graphs

We follow [19]. A graph Γ consists of two sets E and V (we will also write E(Γ) and V (Γ)), and two functions E → E and E → V : for each e ∈ E there is an element e^{−1} ∈ E, and an element o(e) ∈ V. The rules to be satisfied are just these: (e^{−1})^{−1} = e and e^{−1} ≠ e. An e ∈ E is a directed edge of Γ; e^{−1} is the reverse of e. The elements of V are called vertices of Γ; o(e) is the initial vertex of the edge e. We define the terminal vertex of e to be t(e) = o(e^{−1}). An orientation of Γ consists of a choice of exactly one edge in each pair {e, e^{−1}}. A map of graphs f : Γ → Δ consists of a pair of functions, edges to edges, vertices to vertices, preserving the structure. Given f, an orientation of the target Δ determines a unique orientation of Γ which is preserved by f, called the orientation of Γ induced by f. Thus, graphs and their maps form a category in which various categorical concepts may be discussed. For instance, pushout do not always exist. The necessary and sufficient condition for the pushout of a pair of maps of graphs, α_{1} : Γ → Δ_{1} and α_{2} : Γ → Δ_{2}, to exist is that there exist orientations of Γ, Δ_{1}, Δ_{2} which are preserved by α_{1} and α_{2}. There are two functors, “edges” and “vertices,” from graphs to the category of sets; if pushout exists it is preserved by these two functors.

2.1. Paths. Definition 1. A path p in Γ, of length n = |p|, with initial vertex u and terminal vertex v, is an n−tuple of edges of Γ, p = e_{1}e_{2} … e_{n}, such that

1When this paper had already been written, we were informed on the following result of Arzhantseva [3]: given a number L, one can define a class of small cancellation groups with all L-generated subgroups quasiconvex, which implies the Howson property for L-generated subgroups (L may be greater than the rank of the group).

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for i = 1, 2, …, n − 1, we have t(e_{i}) = o(e_{i+1}), and such that u = o(e_{1}) and v = t(e_{n}). For n = 0, given any vertex v, there is a unique path Λ_{v} of length 0 whose initial and terminal vertices coincide and are equal to v. A path p is called a cycle if its initial and terminal points coincide. We call the path p simple if o(e_{i}) ≠ o(e_{j}) for i ≠ j. If in a path p, o(e_{i}) = o(e_{j}) = v for some i ≠ j, we say that p crosses itself at the vertex v. Definition 2. A round-trip is a path of the form ee^{−1}. If a path p contains two adjacent edges forming a round-trip, then by deleting that round-trip we get a path p′ with the same initial and terminal vertices as p, and with |p′| = |p| − 2. We think of p′ as obtained from p by an elementary reduction. We define a reduced path to be a path containing no round-trips.

The equivalence relation on the set of all paths in Γ defined by elementary reduction is called homotopy and is denoted by ∼ . We denote by [p] the homotopy class of p. Let π(Γ) be a set of all homotopy classes of Γ. In the case of one-vertex graph Γ, the classical theory shows that each homotopy class of paths contains a unique reduced path. This is extended in [19] to general graphs. Definition 3. A graph is connected if any pair of vertices is joined by some path. A graph is forest if the only cycles have length 0. A tree is a connected forest.

In the present paper, we consider connected graphs only. If u and v are vertices in a tree T then there is a unique reduced path in T starting at u and ending at v. This path is denoted by [u, v]_{T} .

2.2. The fundamental group and basic cycles of a graph. Definition 4. Let μ_{1} = e_{1} … e_{n}, μ_{2} = f_{1} … f_{m} be two paths in Γ. We say that a sum of μ_{1} and μ_{2} is defined if t(μ_{1}) = o(μ_{2}). We denote this sum μ = μ_{1} ∘ μ_{2} and define it as follows: μ is a reduced path which is equivalent to μ_{1} followed by μ_{2} :

[μ] = [e_{1} … e_{n}f_{1} … f_{m}].

Sum of paths is compatible with homotopy. Each element of π(Γ) has an inverse: if Λ_{v} is the path of length 0 at v, let Λ_{v}^{−1} = Λ_{v}; if μ = λe where e is an edge, we define recursively μ^{−1} = e^{−1}λ^{−1}; then [μ]^{−1} = [μ^{−1}]. This justifies calling π(Γ) the fundamental groupoid of

ON SMALL CANCELLATION K-GENERATED GROUPS 7

Γ. The set of elements of π(Γ) starting and ending at a fixed vertex v forms a group, π_{1}(Γ, v), the fundamental group of Γ based at v.

Given a map f : Γ → Δ there is defined homomorphism, denoted by the same symbol, f : π_{1}(Γ, v) → π_{1}(Δ, f(v)).

Definition 5. Let Γ be a connected graph and let T be a maximal tree of Γ. Let H = {e_{1}, e_{2}, …, e_{n}} be a set of edges of Γ not contained in T. We fix a vertex v ∈ Γ and define basic cycles {c_{1}, c_{2}, …, c_{n}} of Γ as follows: c_{i} = [v, o(e_{i})]_{T} ∘ e_{i} ∘ [t(e_{i}), v]_{T} , i = 1, 2, …, n.

The cycles {c_{1}, c_{2}, …, c_{n}} are precisely Shreier’s free generators of the fundamental group π_{1}(Γ, v) of Γ [17]. So any cycle c ∈ π_{1}(Γ, v) at v in Γ is a sum of basic cycles or their inverses. The number of basic cycles in Γ does not depend on the choice of T. If there are n basic cycles in Γ then we will also say that Γ is a graph with n independent cycles and denote it by Γ_{n}. If we change the base point v for another vertex v_{0} then our basic cycles c_{i} will be changed for new basic cycles

ĉ_{i} = [v_{0}, v]_{T} ∘ c_{i} ∘ [v, v_{0}]_{T} .

Each basic cycle c_{i} can be considered a “conjugate” of some simple cycle λ_{i}. We call this simple cycle a basic loop. Basic loop λ_{i} necessarily contains e_{i}.

For instance, there are three distinct non-empty cycles in graph G starting at v_{1} (Fig. IC): μ_{1} = e_{1}e_{2}, μ_{2} = e_{2}^{−1}e_{3} and μ_{3} = e_{1}e_{3}. Any maximal tree consists of e_{4} and one of the edges e_{1}, e_{2}, e_{3} so there are two basic cycles. Let us choose maximal tree T = {e_{4}, e_{2}} and base point v_{1}. So our basic cycles are c_{1} = μ_{1} and c_{2} = μ_{2} while μ_{3} = c_{1} ∘ c_{2}.

Both basic cycles are simple so they coincide with the basic loops. If we change now the base point for v_{0} then new basic cycles will be as follows: ĉ_{1} = e_{4}e_{1}e_{2}e_{4}^{−1} and ĉ_{2} = e_{4}e_{2}^{−1}e_{3}e_{4}^{−1}. Both are not simple; the basic loops are λ_{1} = e_{1}e_{2} and λ_{2} = e_{2}^{−1}e_{3} and c_{1} = e_{4}λ_{1}e_{4}^{−1}, c_{2} = e_{4}λ_{2}e_{4}^{−1}.

The following Theorem 2 will be used later on. Before the theorem, we need one more definition. We follow [19]. Definition 6. A pair of edges (e_{1}, e_{2}) is said to be admissible if o(e_{1}) = o(e_{2}) and e_{1} ≠ e_{2}^{−1}.

In this case, we can identify t(e_{1}) to t(e_{2}), e_{1} to e_{2}, e_{1}^{−1} to e_{2}^{−1}, to obtain a graph denoted by Γ/[e_{1} = e_{2}], which we call the result of folding (e_{1}, e_{2}) in Γ. This is an instance of the pushout construction. Theorem 2 (Stallings). Suppose that

Γ ⟶^{α_{1}} Δ_{1}
α_{2}↓  ↓^{β_{1}}
Δ_{2} ⟶_{β_{2}} Θ

is a pushout diagram of graphs, and that Γ, Δ_{1}, Δ_{2} are connected. Let u be a vertex of Γ; call the images of u in Δ_{1}, Δ_{2}, Θ respectively, v_{1}, v_{2}, w. Then

π_{1}(Θ, w) = β_{1}π_{1}(Δ_{1}, v_{1}) ∨ β_{2}π_{1}(Δ_{2}, v_{2}).

Corollary 1. If (e_{1}, e_{2}) is an admissible pair of edges in a connected graph Γ̂, then the folding map Γ̂ → Γ = Γ̂/[e_{1} = e_{2}] is surjective on fundamental groups. Corollary 2. Let Γ̂ be a connected graph. If (e_{1}, e_{2}) is an admissible pair of edges in Γ̂ such that t(e_{1}) ≠ t(e_{2}) then the folding map Γ̂ → Γ = Γ̂/[e_{1} = e_{2}] is injective on fundamental groups.

ON SMALL CANCELLATION K-GENERATED GROUPS 9

2.3. Labeled graphs. Definition 7. Let G = ⟨X | R⟩ be a group with the set X of generators, and let Y = {x^{±1} | x ∈ X}. A labeled graph Γ is a graph Γ_{n} equipped with a function

lab : E(Γ_{n}) ⟶ Y

that satisfies the following conditions: ∀e ∈ Γ_{n}, lab(e) ≠ 1 and lab(e^{−1}) = lab(e)^{−1}. The element y_{i} = lab(e_{i}) ∈ Y is called the label of e_{i}. For a path μ = e_{1} … e_{n} in Γ_{n}, its label lab(μ) is defined as the product lab(μ) = lab(e_{1}) … lab(e_{n}) while for the path Λ_{v} of length 0 at a vertex v, lab(Λ_{v}) = 1. We identify the word lab(μ) with the corresponding element in G. Let O be a base point, and let μ_{1}, …, μ_{n} be the basic cycles of the underlying graph Γ_{n}; we call H = ⟨lab(μ_{1}), …, lab(μ_{n})⟩ the fundamental group of Γ, and denote it π_{1}(Γ, O).

Now, let G = ⟨x_{1}, …, x_{k}⟩ be the free group of rank k, and let V = {v_{1}, v_{2}, …, v_{n}} be a system of elements of G which we consider words in the alphabet Y. Denote H = ⟨v_{1}, …, v_{n}⟩ the subgroup of G generated by these elements. Construct a labeled graph Γ with the fundamental group π_{1}(Γ, O) = H by the following standard algorithm. Algorithm 1. With the notation above, assume that all v_{j} ∈ V (j = 1, …, k−1) are freely reduced. For each v_{j}, we form a cycle γ_{j} with lab(γ_{j}) = v_{j}. We form the graph Γ̂ by taking the cycles γ_{1}, …, γ_{k−1} and arranging these around a common base point O. Then π_{1}(Γ̂, O) = H. Now we obtain a new graph Γ identifying admissible edges whose labels are equal, so Γ will be a result of a series of finitely many foldings in Γ̂. Each folding corresponds to a free reduction in a product of elements of V so that H ≅ π_{1}(Γ, O).

2.4. Branch points and hanging vertices in a graph. Definition 8. If v is a vertex of the graph Γ, the star of v in Γ is the set of edges of Γ whose initial vertex is v : St(v, Γ) = {e ∈ E | v = o(e)}. The cardinality of St(v, Γ) is called the valence (or the degree) of v in Γ. We denote it by val(v, Γ). We call v a branch point if val(v, Γ) ≥ 3 and a hanging vertex if val(v, Γ) = 1. Definition 9. Let p be a path in Γ when its initial and terminal vertices are branch points or hanging vertices but for each inside vertex v of p, val(v) = 2. Then we call p a linear path (p is called a maximal arc,

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sometimes abbreviated to arc, in the terminology of [1]). For technical reasons, a simple cycle we also consider a linear path. Proposition 1. Let Γ_{n,s} be a graph with n independent cycles and s hanging vertices. Then Γ_{n,s} contains B_{n,s} = 2n − 2 + s branch points at most.

Proof. Without loss of generality, we can assume that every linear path in Γ_{n,s} has a single edge. The proof is going by induction. First, let n=1, s=0. Γ_{1,0} consists of a single simple cycle so it has no branch points. Also, by the formula, B_{1,0} = 0. Let us suppose that for Γ_{n,s} the present proposition holds. Let T_{n+1,s+1} be a maximal tree of Γ_{n+1,s+1}. There are n + 1 edges in Γ_{n+1,s+1} \ T_{n+1,s+1}. If we take off one of these, say, e then we will obtain Γ_{n,s+1} while the number of branch points will decrease by no more than 2 (when val(o(e), Γ_{n+1,s+1}) = val(t(e), Γ_{n+1,s+1}) = 3.) So B_{n+1,s+1} − B_{n,s+1} ≤ 2. If now we take off an edge f ∈ T_{n,s+1} with a hanging vertex as one of its endpoints then the number of branch points either will decrease by 1 or will be not changed at all. So B_{n,s+1} − B_{n,s} ≤ 1. Thus, B_{n+1,s+1} ≤ B_{n,s} + 2 + 1 ≤ 2n − 2 + s + 2 + 1 = 2(n + 1) − 2 + (s + 1) as claimed. □

3. Graphs with colored edges

In this section, we consider only those cycles which are not powers of other cycles. Definition 10. Let Σ_{m} = {σ_{1}, σ_{2}, …, σ_{m}} be a set of m distinct colors. A colored graph is a graph Γ equipped with a function

col : E(Γ) ⟶ Σ_{m}

where E(Γ) is a set of all the edges of Γ. We say that Γ is colored by m colors if the function col is onto. We say that an edge e_{i} has color σ_{j} if col(e_{i}) = σ_{j}. A path μ in Γ is one-colored if all its edges have the same color. Proposition 2. If there is in Γ a one-colored cycle then there is in Γ a simple one-colored cycle.

Proof. Let μ = f_{1}f_{2} … f_{l} be a one-colored cycle which is not simple. This means that μ crosses itself i.e., there exist in μ two edges, f_{i} and f_{j} with i ≠ j and o(f_{i}) = o(f_{j})(= t(f_{j−1})). Let i < j. We consider now an one-colored cycle μ_{1} = f_{i}f_{i+1} … f_{j−1}. If j = i + 1 then o(f_{i}) = t(f_{i})

ON SMALL CANCELLATION K-GENERATED GROUPS 11

so μ_{1} = f_{i} is the simple one-colored cycle. Otherwise, μ_{1} is shorter than μ; if μ_{1} is not a simple cycle then we find a one-colored cycle μ_{2} which is shorter than μ_{1}. Proceeding by this way we find the simple cycle. □

For instance, let μ = e_{1}e_{2}e_{3}e_{4} (fig.BC) be an one-colored cycle. μ is not simple because o(e_{2}) = o(e_{4}). Then by the above rule, we consider a shorter one-colored cycle μ_{1} = e_{2}e_{3} that turns out to be a simple cycle.

Let Γ_{n} denote a connected graph with n independent cycles.

Proposition 3. Let Γ_{n} be a graph colored by n + 1 or more different colors. Then in Γ_{n} there are one-colored cycles of n different colors at most.

Proof. The proof is going by induction. First, if n = 2 then in Γ_{2} with edges of three (or more) distinct colors there are one-colored cycles of no more than 2 different colors. Indeed, there are no more than 3 simple cycles in Γ_{2} : two basic loops λ_{1} and λ_{2} and their sum μ. The sum μ is a simple cycle only if λ_{1} and λ_{2} have an edge in common (like the graph on fig.IC). Because μ and λ_{1} as well as μ and λ_{2} have edges in common, one-colored simple cycles in Γ_{2} can be colored by no more than two different colors. In view of Proposition 2, the assertion holds for Γ_{2}.

Now, suppose that Proposition holds for Γ_{n} . We claim that in Γ_{n+1} with edges of n + 2 (or more) different colors, there are one-colored cycles of no more than n + 1 different colors.

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First, suppose there is a one-colored basic loop λ_{i} in Γ_{n+1}. With the notation of Definition 5, consider in Γ_{n+1} the subgraph Γ̃_{n} = Γ_{n+1} \ e_{i} obtained by removing the edge e_{i}. Hence c_{1}, … c_{i−1}, c_{i+1}, …, c_{n+1} are the basic cycles of Γ̃_{n} which justifies our notation. The edges of Γ̃_{n} are colored either by n+1 or more different colors. Hence by the induction hypothesis, in Γ̃_{n} there are one-colored cycles of l ≤ n different colors. Let Σ_{n} = {σ_{1}, …, σ_{l}} be the set of these colors. Suppose, in Γ_{n+1} there are one-colored cycles of m different colors when m > n. As for any one-colored cycle d ∈ Γ_{n+1} that has color col(d) ∉ Σ_{n} necessarily e_{i} ∈ d, the assertion follows.

Now, suppose there is no one-colored basic loop in Γ_{n+1}. If in Γ_{n+1} there is not any one-colored cycle then Proposition holds. Otherwise, by Proposition 2, there is in Γ_{n+1} a simple one-colored cycle c_{∗}, and c_{∗} is not a basic cycle. Hence, c_{∗} is a sum of two or more basic cycles or their inverses: c_{∗} = c_{i_{1}} ∘ c_{i_{2}} ∘ ⋯ ∘ c_{i_{m}}. As sum of cycles is a reduced path by Definition 4, we may assume that c_{i_{j+1}} ≠ c_{i_{j}}^{−1} for all j = 1 …, m − 1, so that with the notation of Definition 5, e_{i_{j}}^{ϵ} ∈ c_{∗} where ϵ = ±1. Furthermore, ∀j ≠ 1, c_{i_{j}} ≠ c_{i_{1}}^{ϵ}. For if a basic cycle c_{i_{1}}^{ϵ} is met in the sum twice then c_{∗} will be not a simple cycle (because it crosses itself at the vertex o(e_{i_{1}})). Then c_{i_{1}} = c_{∗}∘c_{i_{m}}^{−1}∘⋯∘c_{i_{2}}^{−1}. This let us consider a system of n+1 cycles in Γ_{n+1} as follows: C = {c_{1}, …, c_{i_{1}−1}, c_{∗}, c_{i_{1}+1}, …, c_{n+1}}. Each cycle in Γ_{n+1} is a sum of cycles of C or their inverses. The proof in this case is identical with the proof in case above while we regard c_{∗} as the one-colored basic loop and mean e_{i_{1}} by the edge e_{∗} which is to be removed. □

Let Γ_{k−1,1} be a connected graph with k − 1 independent cycles and a hanging vertex O. Assume that the edges of Γ_{k−1,1} are colored by k distinct colors, in particular blue color is among them. By Proposition 3 we can assume that in Γ_{k−1,1} there is no blue cycle. Let us also assume that in Γ_{k−1,1} each linear path is either one-colored or of the type “edges of color 1 - edges of color 2”. In what follows, by an FBSpath we mean a path of the form “ an edge of color σ_{f} - blue edges - an edge of color σ_{s}”, where the colors σ_{f}, σ_{s} are not necessarily different but both are different from blue color. Proposition 4. With the notation above, there are in Γ_{k−1,1} less than 13k^{2} various FBS-paths.

ON SMALL CANCELLATION K-GENERATED GROUPS 13

Proof. Consider a new graph Δ where the set V (Δ) of its vertices consists of the hanging vertex O, all branch points of Γ_{k−1,1} and vertices that represent two-colored linear paths in Γ_{k−1,1} :

V (Δ) = {O} ∪ {v ∈ V (Γ_{k−1,1}) | val(v, Γ_{k−1,1}) ≥ 3 or
val(v, Γ_{k−1,1}) = 2&(v = o(e_{i}) = t(e_{j})), col(e_{i}) ≠ col(e_{j})}.

Hence, |V (Δ)| ≤ 1+B_{k−1,1}+L_{k−1,1} where B_{k−1,1} denotes the number of the branch points, and L_{k−1,1} denotes the number of the linear paths in Γ_{k−1,1}. By Proposition 1, B_{k−1,1} ≤ 2k −3. Regard each linear path of Γ_{k−1,1} as an edge; easy induction shows that there are B_{k−1,1} edges in a maximal tree of Γ_{k−1,1}, and k − 1 more edges define the basic cycles of the graph. Whence L_{k−1,1} = B_{k−1,1} + k − 1 ≤ 3k − 4. It follows that |V (Δ)| ≤ 5k − 6.

Edges of Δ represent the one-colored linear paths of Γ_{k−1,1}, and inherit their colors. So Δ has no blue cycle just like Γ_{k−1,1}. Now, we remove from Δ all the edges that are not blue and obtain a new graph Δ̃. All edges of Δ̃ are blue so it is a forest. The number of FBS-paths in Γ_{k−1,1} is bounded by the number P of different paths in Δ̃, while

P ≤ \frac{|V (Δ)| × |V (Δ) − 1|}{2} ≤ \frac{(5k − 6)(5k − 7)}{2} < 13k^{2}

as claimed. □

4. Sufficient conditions that ensure the (k-1)-property

Let G = ⟨x_{1}, …, x_{k} | R_{i} (1 ≤ i ≤ t)⟩ be a group. We consider the free group of rank k: G̃ = ⟨x_{1}, x_{2}, …, x_{k}⟩ and its normal subgroup Ñ = ⟨R̂⟩G̃, generated by the set R̂ = {R_{i}} (i = 1, …, t). There is a homomorphism ϕ : G̃ ⟶ G onto G, with Ker(ϕ) = Ñ. Let W̃ be a finite system of elements of G̃, and let H̃ = ⟨W̃ ⟩ be the subgroup of G̃ generated by these elements. Let W = ϕ(W̃) ⊂ G be the image of W̃ under the natural homomorphism ϕ, and let H = ⟨W ⟩. Then H = ϕ(H̃). We consider the set Y = {x_{1}^{±1}, …, x_{k}^{±1}} the alphabet of both G̃ and G. With the notation above, let R̂ be a small cancellation set satisfying C′(λ)-condition with λ > 0.

We recall the definition of a word reduced by Dehn (see, for instance [9] for the relevant discussion).

Definition 11. An element g̃ ∈ G̃ is reduced by Dehn if there is no relator R_{i} ∈ R such that R_{i} = rq where r is a subword of g̃ = urv, and |r| > |q|.

In what follows, we use a somewhat weaker property. Definition 12. We call an element g̃ ∈ G̃ reduced by Rips if there is no relator R_{i} ∈ R such that R_{i} = rQ where r is a subword of g̃ = urv, and Q is either the empty word or a product of less than five pieces. We call W̃ a system reduced by Rips if each g̃ ∈ W̃ is reduced by Rips. We say that W is Rips-normalized if each h̃ ∈ H̃ is reduced by Rips.

Clearly, a word reduced by Dehn is necessarily reduced by Rips. It follows from the definition that a Rips-normalized system is reduced by Rips. The inverse implication is not necessarily true.

In this Section, we prove effective sufficient conditions that ensure that the group G has the (k-1)-property. We also show that these conditions ensure the rank property of G. The conditions are formulated in Theorem 1 (Section 1.3). One of the important ingredients of our proof is the following lemma (this version of the Greendlinger’s lemma follows from [14, Chapter V, Theorem 4.4], and also is implied by results obtained in [16]). Lemma 1 (Generalized Greendlinger’s lemma). Let us assume the following:

(1) F is the free group of rank k;

(2) R̂ is a symmetrized subset of the elements of F that satisfies

the small cancellation condition C′(λ) where λ ≤ \frac{1}{6}. Let N =< R̂ >^{F} be the normal subgroup of F generated by the elements of R̂, and let v ∈ F be a non-trivial element of F. If v ∈ N then v is not reduced by Rips.

As a corollary of Lemma 1 we have the following proposition. Proposition 5. With the notation above, let G̃ and R̂ satisfy the conditions of Lemma 1. Suppose, W̃ = {w̃_{1}, …, w̃_{l}} ⊂ G̃, where l ≤ k − 1, is a system reduced by Rips. If W̃ is Rips-normalized then H is a free subgroup of G. If H = G then W̃ is not Rips-normalized.

Proof. If W̃ is Rips-normalized then, by Lemma 1, the intersection H̃ ∩ Ñ is trivial, hence, H ≅ H̃/(H̃ ∩ Ñ) is free. Now, suppose that H = G. If G is not isomorphic to a free group then necessarily the intersection H̃ ∩ Ñ is not trivial, hence, by Lemma 1, W̃ is not Ripsnormalized. If G is isomorphic to a free group (cf. Example 2), then rank(G) < k. As the system W = {w_{1}, …, w_{l}} generates the whole group G, the elements x_{1}, …, x_{k} of G can be expressed as words in w_{1}, …, w_{l} as follows: x_{i} = f_{i}(w_{1}, …, w_{l}) (i = 1, …, k). Consider the elements x̃_{i} = f_{i}(w̃_{1}, …, w̃_{l}) of G̃. Obviously, ∀i, ϕ(x̃_{i}) = x_{i}. But the set {x̃_{1}, …, x̃_{k}} is different from the set X = {x_{1}, …, x_{k}} of the generators of G̃ because l < k so that X cannot be expressed via W̃ . This implies that for some i, the element f_{i}(w̃_{1}, …, w̃_{l}) is not reduced by Rips, hence W̃ is not Rips-normalized. □

The above proposition gives rise to the rule for writing down words R_{i} in the free group G̃ in order to ensure the (k − 1)−property for its factor group G. The relations R_{i} should be chosen in such a way that for any system W̃_{l} with cardinality l < k, the system W̃ is Ripsnormalized. The rules for writing relations down are formulated in Theorem 1 (Section 1.3).

Proof of Theorem 1. Proposition 5 implies that if any system W̃_{l} of l < k elements of G̃ is Rips-normalized then the conclusion of the present theorem holds. Let W̃ = {w̃_{1}, w̃_{2}, …, w̃_{k−1}} ⊂ G̃ be a freely reduced system that is reduced by Rips, and let H̃ = ⟨w̃_{1}, …, w̃_{k−1}⟩. We claim that W̃ is Rips-normalized. To prove this, we construct by Algorithm 1, a graph Γ_{k−1} with the fundamental group π_{1}(Γ_{k−1}, O) ≅ H̃. Suppose, W̃ is not Rips-normalized, then there exists a non-trivial element h ∈ H which is not reduced by Rips. h is a label of a nontrivial, reduced, closed path in Γ_{k−1}. According to Definition 12, h has a subword g such that R_{i} = gQ where R_{i} is a relator of G, and Q is either the empty word or a product of less than five pieces. Hence, there is a (not necessarily closed) reduced path P in Γ_{k−1} whose label g = lab(P) is a subword of R_{i} with

|g| ≥ (1 − 4λ)|R_{i}| = (1 − \frac{4}{10k^{2}} (1 − \frac{13}{A}))Ak^{3} > \frac{9}{10}Ak^{3}.

Now, assume that there is one-to-one correspondence between the set X = {x_{1}, …, x_{k}} of the generators of G and a set Σ_{k} = {σ_{1}, …, σ_{k}} of k colors, so that col(x_{i}) = σ_{i}. We use the expressions “e is labeled by x_{i}” and “e is colored by σ_{i}” as synonyms. Let μ = e_{1} … e_{m} be a subpath of a linear path in Γ_{k−1} such that col(e_{1}) = ⋯ = col(e_{m}) = σ_{i}

16 INNA BUMAGINA

for some i i.e., lab(e_{1}) = ⋯ = lab(e_{m}) = x_{i}; we regard μ as a single edge of color σ_{i}, labeled by x_{i}^{m}. We distinguish the following two cases:

(1) In the graph Γ_{k−1}, each linear path p is colored either by one or by two colors, and in the latter case the edges colored by different colors are separated (this means, there is precisely one pair (e_{1}, e_{2}) of edges in p such that t(e_{1}) = o(e_{2}), and col(e_{1}) ≠ col(e_{2})). In other words, this condition asserts that in Γ_{k−1}, the label of a linear path consists of two syllables at most (these labels are called short).

(2) There are in Γ_{k−1} linear paths that contain several pairs of adjacent edges of different colors. In other words, there is in Γ_{k−1} a linear path μ with the label lab(μ) = x_{i_{1}}^{m_{1}} … x_{i_{s}}^{m_{s}} when s ≥ 3 (these labels are called long). Note that syllables x_{i_{2}}^{m_{2}} … x_{i_{s−1}}^{m_{s−1}} in the middle of a long label lab(μ) do not change in any product of the elements of W̃ .

Consider the first case. By Proposition 3, there is an element x_{j} ∈ X such that in Γ_{k−1} there is no cycle with edges all labelled by x_{j}. Let M be the total number of different exponents of x_{j} that occur in the labels of paths in Γ_{k−1}. By Proposition 4, M ≤ 13k^{2}. In particular, if a subword u of a relation is the label of a path p in Γ_{k−1} : u = lab(p), then in u no more than 13k^{2} different exponents of x_{j} occur so that |u| ≤ 13k^{3}. Since for A > 15, \frac{9}{10}Ak^{3} > 13k^{3}, it follows that no path P as defined above, exists in Γ_{k−1}.

Now, consider the second case. Let B and E be number of branch points and linear paths in Γ_{k−1}, respectively. Notice that in Γ_{k−1}, there is one hanging vertex at most. By Proposition 1, B ≤ 2k − 3, hence

E = B + k − 1 ≤ 3k − 4 < 3k.  (2)

The first part of the proof shows that P must pass through linear paths with long labels. (Each linear path with a long label can be met only once in P because unique syllables of the relation in its label can never be repeated.) From the above calculations it follows that at least \frac{9}{10}Ak^{3} − 13k^{3} syllables of a relation R_{i} are contained in long labels. There are E paths with long labels, at most. Whence, there exists a linear path μ_{i} with the label u_{i} = lab(μ_{i}) which is a subword of R_{i}, and such that |u_{i}| ≥ \frac{|g|−13k^{3}}{E} . In view of (2), an easy calculation shows that for k ≥ 3, A ≥ 25 and λ < \frac{1}{10k^{2}} (1 − \frac{13}{A}),

|u_{i}| ≥ \frac{|g| − 13k^{3}}{E} > \frac{(\frac{9}{10}A − 13)k^{3}}{3k} > 5λ|R_{i}|.

We replace u_{i} by its supplement w_{i} in R_{i} everywhere it can be met as a subword of an element of W̃ . The new system W̃_{i} is R_{i}-reduced because u_{i} is a product of 5 pieces at least. Moreover, since in Γ_{k−1} there is only one linear path with the label u_{i} (namely, μ_{i}), the replacement in W̃ amounts to the following changes in Γ_{k−1} : delete this linear path μ_{i} and attach a new path ν_{i} labelled by w_{i}, instead. W̃_{i} is R_{i}−normalized because after these changes, no path in Γ_{k−1} has the five removed pieces in its label. Thus, in Γ_{k−1} there is no path with the label that is not reduced by Rips.

Now, assume that there exists a system W = {w_{1}, …, w_{k−1}} of k −1 elements that generates G. Let W̃ = {w̃_{1}, …, w̃_{k−1}} be a system of elements of G̃ such that for all i, ϕ(w̃_{i}) = w_{i}, hence, H = G. We can assume that W̃ is reduced by Rips, whence W̃ is Rips-normalized. Notice that G satisfies also the conditions of Lemma 1, so we get a contradiction with Proposition 5. Hence, the group G cannot be generated by less than k elements, and rank(G) = k.

5. Quasiconvexity and the Howson property

It turns out that the conditions of Theorem 1 ensure a few additional nice properties of G. To discuss these properties, we need some more definitions.

Let G be a group generated by a finite set X, and let F (X) denote a free group generated by X. Each reduced word w ∈ F (X) over the set X represents an element of G. We denote by l_{X}(w) the length of the word w in the meaning of the definition given in Section 1.1. The Cayley graph Γ(G, X) with respect to the generating set X is a geodesic metric space. Let w̄ be a geodesic of w in Γ(G, X). Definition 13. A word w over X is called (λ, ϵ) − quasigeodesic if for any subword v of w the inequality l_{X}(v) ≤ λl_{X}(v̄) + ϵ holds.

In general, a subset A of a geodesic metric space (X, d) is termed ϵ−quasiconvex in X if any geodesic [a_{1}, a_{2}] in X joining points a_{1}, a_{2} ∈ A is contained in an ϵ−neighborhood of A.

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Definition 14. If G is a finitely generated group and A is a subgroup of G, we say that A is quasiconvex in G if for some finite generating set X of G, A is a quasiconvex subset of the Cayley graph Γ(G, X).

The following Theorem 3 asserts that (k −1)-generated subgroups of the group G are quasiconvex. Before we state Theorem 3 let us recall that there are small cancellation groups that have non quasi-convex 2-generated subgroups ([13]). Theorem 3. Let a group G = ⟨x_{1}, …, x_{k} | R_{i} (1 ≤ i ≤ t)⟩ satisfy slightly strengthened conditions of Theorem 1:

(1) Each relation R_{i} is a product of syllables of the type x_{j}^{α_{i}} such that all the k letters of alphabet are involved and each letter appears with Ak^{2} different exponents while all these exponents are unique; it follows that for all R_{i}, |R_{i}| ≥ Ak^{3}, where A is a constant, A > 13;

(2) All the exponents in all the relations are unique. If H ⊂ G is generated by less than k elements then H is quasiconvex in G.

Proof. We are going to show that for any Rips-normalized system W_{k−1}, with |W_{k−1}| ≤ k − 1, all the Rips-reduced words may be reduced by Dehn in a finite number of steps. This provide the proof together with the following facts:

1) H ⊆ G is quasi-convex in G if and only if there are λ, ϵ such that all the X−geodesic words representing elements of H are (λ, ϵ)−quasi-geodesic.

2) If G = ⟨X | R⟩ is a small cancellation group then there exists a pair (λ, ϵ) such that all the Dehn-reduced words are (λ, ϵ)−quasigeodesic.

Let us be given a system W_{k−1} = {w_{1}, w_{2}, …, w_{k−1}} ⊂ G which is supposed to be normalized by Rips and Nielsen-reduced. Let an element w = (w_{1}, w_{2}, …, w_{k−1}) of G be not reduced by Dehn. We claim that Dehn reduction of w can be made in one step. This means, if for each subword u of w that makes up more than a half of a relation R_{i} = ut^{−1} (|u| > \frac{1}{2}|R_{i}|) we replace u by its complement t then the process will stop. In other words, having done all the possible replacements in w we obtain a Dehn-reduced word. These complements t cannot be

ON SMALL CANCELLATION K-GENERATED GROUPS 19

used for further Dehn-reductions. Indeed, W_{k−1} is a Rips-normalized system so every “short” sub-word t of a relation R_{i} contains 10 or more unique syllables of R_{i}. Assume, w = v_{1}uv_{2} where u is a maximal subword of R_{i} in the following meaning: for any non-empty sub-words s, r of w such that w = ṽ_{1}suv_{2} = v_{1}urṽ_{2}, neither su nor ur is not a sub-word of R_{i}. So v_{1}, v_{2} survive intact while Dehn reduction is made. After the Dehn reduction we obtain ŵ = v_{1}tv_{2}. Now we claim that any free as well as any Dehn reduction of ŵ (if either is possible) survive the sub-word t intact. Suppose a free reduction of ŵ that does not survive t can be made. Let

ŵ = v_{1}tv_{2} = (v̂_{1}α^{−1})(αt̂β)(β^{−1}v̂_{2}).

So before the Dehn reduction we had w = v̂_{1}α^{−1}uβ^{−1}v̂_{2}. But t = αt̂β so ut^{−1} = 1 means that u(αt̂β)^{−1} = 1 hence α^{−1}uβ^{−1}t̂^{−1} = 1. In other words, if either α or β is a non-empty word then α^{−1}uβ^{−1} is a subword of the relation R_{i} that contains u as a proper sub-word. This contradicts the maximality of u hence there is no free reduction as a result of the Dehn reduction. Now, let us suppose that some Dehn reduction does not survive t. Because t contains unique syllables of the relation R_{i} that never appear in other relations, this means that ŵ contains more than a half of R_{i} as a sub-word. Let

ŵ = v_{1}tv_{2} = (v̂_{1}γ)t(δv̂_{2})

where R_{i} = γtδy^{−1} = 1. Because by assumption R_{i} = ut^{−1} it follows that u = γ^{−1}yδ^{−1}. But the last equality means that before the Dehn reduction we had

w = v_{1}uv_{2} = (v̂_{1}γ)(γ^{−1}yδ^{−1})(δv̂_{2})

So if either γ or δ was non-empty then w would be not freely reduced and we would obtain a contradiction. So any Dehn reduction of ŵ has to survive t. Thus, having done all the possible Dehn reductions in w we obtain a word ŵ that is both freely and Dehn reduced.

Let now H ⊂ G be a subgroup generated by the elements of W_{k−1}. As we have shown, for every w ∈ H there is a Dehn-reduced word ŵ that represents w in G. By 2), ŵ is (λ, ϵ)− geodesic so by 1), H is quasi-convex. □

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Corollary 3. Let H ⊂ G and K ⊂ G be two subgroups generated by less than k elements each one. Then H ∩ K ⊂ G is finitely generated.

Proof. Since H and K are quasiconvex by Theorem 3, H ∩ K is quasiconvex by [18, Proposition 3]; whence by [18, Proposition 1] H ∩ K is finitely generated. □

There is a particular case when we can even determine the generators of H ∩ K. Theorem 4. Let G = ⟨x_{1}, x_{2}, …, x_{k} | R_{1}, R_{2}, …, R_{t}⟩ satisfy the conditions of Theorem 1. Let X = {x_{1}, x_{2}, …, x_{k}} be a generating system of G. Consider two subsets of X : V = {v_{1}, v_{2}, …, v_{n}} and W = {w_{1}, w_{2}, …, w_{m}}, and the subgroups H = gp(V) and K = gp(W) of G that are generated by these subsets. Then H ∩ K = gp(V ∩ W).

Proof. Clearly, gp(V ∩ W) ⊆ H ∩ K. Now, assume that g ∈ H ∩ K. We claim that g ∈ gp(V ∩ W). The element g can be written as a word in the elements of V as well as in the elements of W :

g = g_{1}(v_{1}, v_{2}, …, v_{n}) = g_{2}(w_{1}, w_{2}, …, w_{m}).

It follows that g_{1}g_{2}^{−1} = 1_{G}.

Consider the free group G̃ = ⟨x_{1}, x_{2}, …, x_{k} | −⟩ generated by the set X. Clearly, G = G̃/⟨R_{1}, R_{2}, …, R_{t}⟩G̃, and by Lemma 1, it follows that the word g̃ = g_{1}(v_{1}, v_{2}, …, v_{n})g_{2}(w_{1}, w_{2}, …, w_{m})^{−1} which represents the element g̃ ∈ G̃, is not reduced by Rips. Therefore, either g_{1} or g_{2} contains a subword r of a relation, so that |r| > \frac{1}{2}(Ak^{3} − 4). If both V and W are proper subsets of X then last inequality contradicts Proposition 4. Hence, one of the subsets, say V coincides with X : V = X, which implies V ∩ W = W. Moreover, in this case we have K ⊆ H = G, hence H ∩ K = K, so that our assumption g ∈ H ∩ K implies that g ∈ gp(W) = gp(V ∩ W) as claimed. The assertion of Theorem 4 follows. □

Note that by a theorem proved by D. Collins, in a finitely presented group G (with the notation of Theorem 4): H ∩ K = gp(V ∩ W) ∗ ⟨g⟩ for some (possibly, trivial) element g ∈ G. As the following example shows, a non-trivial free factor can appear even in the case when G is a one-relator hyperbolic group, and H and K are quasiconvex subgroups of G.

Example 4. Let A = ⟨v_{1}, v_{2}, v_{3} | −⟩ and B = ⟨w_{1}, w_{2}, w_{3} | −⟩ be two free groups of rank 3. Consider elements c_{1} = u(v_{1}, v_{2}) ∈ A and c_{2} = u(w_{1}, w_{2}) ∈ B, where the word u is chosen in such a way that the subgroups ⟨c_{1}⟩ and ⟨c_{2}⟩ are infinite cyclic and malnormal in A and in B, respectively: for instance, we set u(x, y) = xy^{25}xy^{26} … xy^{50} (a proof can be found in [5, Lemma 2]). We define the group

G = A ∗_{⟨c_{1}=c_{2}⟩} B = ⟨v_{1}, v_{2}, v_{3}, w_{1}, w_{2}, w_{3} | u(v_{1}, v_{2}) = u(w_{1}, w_{2})⟩

to be the amalgamated product of A and B over the infinite cyclic subgroup ⟨c⟩ (where c = c_{1} = c_{2} in G.) Then G satisfies the small cancellation condition C′(\frac{1}{8}), hence, is word hyperbolic. Let V = \{v_{1}, v_{2}\}, W = {w_{1}, w_{2}}, so that H = ⟨v_{1}, v_{2}⟩ and K = ⟨w_{1}, w_{2}⟩. It can be easily checked that H and K are quasiconvex in G. Now, V ∩ W = ∅ while H ∩ K = ⟨c⟩, which can be written as H ∩ K = gp(V ∩ W) ∗ ⟨c⟩.

Acknowledgements

I am deeply grateful to my thesis advisor Professor E. Rips for his valuable advice, help and encouragement. Theorem 3 and Corollary 3 are due to I. Kapovich, and Theorem 4 is due to A. Juhasz; I would like to thank them for these suggestions. Also, I wish to thank G. Arzhantseva for her valuable remarks on a preliminary version of this paper.

References

[1] G. N. Arzhantseva, A.Yu. Ol’shanskii, Generality of the class of groups in which subgroups with a lesser number of generators are free, (Russian) Mat. Zametki 59 (1996), no. 4, 489-496, 638; translation in Math. Notes 59 (1996), no. 3-4, 350-355. (Math Reviews 98k:20040)

[2] G. N. Arzhantseva, On the groups all of whose subgroups with fixed number of generators are free, Fundamental and Applied Mathematics, 3(3) (1997), 675-683 (in Russian).

[3] G. N. Arzhantseva, Generic properties of finitely presented groups and Howson’s theorem, Communications in Algebra, 26(11) (1998), 3783-3792.

[4] G. Baumslag, C.F. Miller and H. Short, Unsolvable problems about small cancellation and word hyperbolic groups, Bull London Math. Soc, 26 (1994), 97–101.

[5] I. Bumagin, Examples of amalgamated products, Israel J. Math., to appear.

[6] M. Coornaert, T. Delzant and A. Papadopoulos, Notes sur les groupes hyperboliques de Gromov, Lecture Notes in Mathematics, 1441, Springer-Verlag (1990).

[7] S.M. Gersten, Intersections of finitely generated subgroups of free groups and resolutions of graphs, Invent. Math., 71 (1983), 567–591.

[8] S.M. Gersten, H. Short, Rational subgroups of biautomatic groups, Annals of Math., (1992).

[9] E. Greendlinger, M. Greendlinger, On Dehn presentations and Dehn algorithms, Illinois J. Math., 30(1986), 360–363.

[10] V.S. Guba, A finite-generated simple group with free 2-generated subgroups (in Russian), Sib. Math. J., 27 (1986), 670–687.

[11] V.S. Guba, On conditions that insure 2-generated subgroups of small cancellation groups are free (in Russian), Izv. Vusov, Matematika, 7 (1986), 12–19.

[12] I. Kapovich, H. Short, Greenberg’s theorem for quasi-convex subgroups of word hyperbolic groups, Preprint, (1996).

[13] I. Kapovich, Amalgamated products and the Howson property, Canadian Math. Bull, 40(3), (1997), 330–340.

[14] R.C. Lyndon, P.E. Shupp, Combinatorial group theory, Springer-Verlag, Berlin, (1977).

[15] E. Rips, Subgroups of small cancellation groups, Bull London Math Soc, 14, (1982), 45–47.

[16] E. Rips, Generalized small cancellation theory and applications 1. The word problem. Israel Journal of Math, 41, (1982), 1–146.

[17] J.P. Serre, Trees, Springer-Verlag, Berlin and New York (1980).

[18] H. Short, Quasiconvexity and a Theorem of Howson, Group theory from a geometric viewpoint, Proc ICTP, (1991), Trieste, World Scientific, Singapore.

[19] J.R. Stallings, Topology of finite graphs, Invent. Math., 71 (1983), 551– 565.

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

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