ASYMPTOTIC CONES OF FINITELY GENERATED GROUPS

SIMON THOMAS AND BOBAN VELICKOVIC

Abstract. Answering a question of Gromov [7], we shall present an example of a finitely generated group Γ and two non-principal ultrafilters A, B such that the asymptotic cones Con_{A} Γ and Con_{B} Γ are not homeomorphic.

1. Introduction

Let Γ be a finitely generated group equipped with a fixed finite generating set and let d be the corresponding word metric. Consider the sequence of metric spaces X_{n} = (Γ, d_{n}) for n ≥ 1, where d_{n}(g, h) = d(g, h)/n. In [5], Gromov proved that if Γ has polynomial growth, then the sequence (X_{n} | n ≥ 1) of metric spaces converges in the pointed Gromov-Hausdorff topology to a complete geodesic space Con_{∞}(Γ), the asymptotic cone of Γ. (Recall that a geodesic space is a metric space (X, d) such that for all points x, y ∈ X, there exists an isometric mapping from the interval [0, d(x, y)] to a path in X joining x and y.) In [4], van den Dries and Wilkie generalised the construction of asymptotic cones to arbitrary finitely generated groups. However, their construction involved the choice of a nonprincipal ultrafilter A on the set ℕ^{+} of positive natural numbers, and it was not clear whether the resulting asymptotic cone Con_{A}(Γ) depended on the choice of the ultrafilter A. In this paper, answering a question of Gromov [7], we shall present an example of a finitely generated group Γ and two non-principal ultrafilters A, B such that the asymptotic cones Con_{A} Γ and Con_{B} Γ are not homeomorphic.

Our group Γ will be chosen from the class of small cancellation groups which was introduced by Bowditch in [1]. For each n ∈ ℕ^{+}, let w_{n}(a, b) be the word (a^{n}b^{n})^{7} in the two letters a and b; and for each I ⊆ ℕ^{+}, let Γ_{I} be the group with presentation

⟨a, b | (w_{n}(a, b))_{n∈I}⟩.

Then Γ_{I} satisfies the C′(1/6) cancellation property; ie. whenever u is a common initial subword of two distinct elements r_{1}, r_{2} of the set R_{I} of all cyclic conjugates

The research of the first author was partially supported by NSF Grants.

2 SIMON THOMAS AND BOBAN VELICKOVIC

of relators and their inverses, then 6 length(u) < min{length(r_{1}), length(r_{2})}. This implies that if w is a reduced word in the letters a, b which represents the identity element in Γ_{I}, then there exists an element r ∈ R_{I} which has a common subword of length more length(r)/2 with w. (For example, see Section V.4 [8].) Notice that for such an element r, we must have that length(r) < 2 length(w). So an easy induction on length(w) yields the following result.

Lemma 1.1. Suppose that w is a (not necessarily reduced) word in the letters a, b which represents the identity element in Γ_{I}. Then w lies in the normal closure of the set of relators {w_{ℓ}(a, b) | ℓ ∈ I, ℓ < length(w)/7}.

□

In this section, after giving the definition of an asymptotic cone, we shall prove that if I is any infinite subset of ℕ^{+} and A is any non-principal ultrafilter which contains I, then the asymptotic cone Con_{A}(Γ_{I}) is not simply connected. Then in Section 2, we shall prove that if I is a suitably chosen sparse subset of ℕ^{+}, then there exists a non-principal ultrafilter B such that the asymptotic cone Con_{B}(Γ_{I}) is an ℝ-tree. A geodesic space (X, d) is said to be an ℝ-tree iff any two points of X are the endpoints of a unique topological arc (ie. the image of an injective continuous function from a closed interval of ℝ into X.) In particular, ℝ-trees are simply connected. Hence the asymptotic cones Con_{A}(Γ_{I}) and Con_{B}(Γ_{I}) are not homeomorphic.

Definition 1.2. A non-principal ultrafilter on ℕ^{+} is a collection A of subsets of ℕ^{+} satisfying the following properties.

(i) If A, B ∈ A, then A ∩ B ∈ A. (ii) If A ∈ A and A ⊆ B ⊆ ℕ^{+}, then B ∈ A. (iii) For all A ⊆ ℕ^{+}, either A ∈ A or ℕ^{+} ∖ A ∈ A. (iv) If F is a finite subset of ℕ^{+}, then F ∉ A.

Equivalently, if ω : P(ℕ^{+}) → {0, 1} is the function such that ω(A) = 1 iff A ∈ A, then ω is a finitely additive probability measure on ℕ^{+} such that ω(F) = 0 for all finite subsets F of ℕ^{+}. It is easily checked that if (r_{n}) is a bounded sequence of real numbers, then there exists a unique real number ℓ such that

{n ∈ ℕ^{+} | |r_{n} − ℓ| < ε} ∈ A

ASYMPTOTIC CONES OF FINITELY GENERATED GROUPS 3

for all ε > 0. We write ℓ = lim_{A} r_{n}.

Definition 1.3. Let Γ be a finitely generated group equipped with a fixed finite generating set and let d be the corresponding word metric. For each n ≥ 1, let X_{n} = (Γ, d_{n}) be the metric space defined by d_{n}(g, h) = d(g, h)/n. Let e be the identity element of Γ. Then X_{∞} is the set of all sequences (g_{n}) of elements of Γ such that there exists a constant c with d_{n}(g_{n}, e) ≤ c for all n ∈ ℕ^{+}.

Let A be a non-principal ultrafilter on ℕ^{+}. Define an equivalence relation ∼ on X_{∞} by (g_{n}) ∼ (h_{n}) iff lim_{A} d_{n}(g_{n}, h_{n}) = 0; and for each (g_{n}) ∈ X_{∞}, let (g_{n})_{A} denote the corresponding equivalence class. Then the asymptotic cone is defined to be

Con_{A}(Γ) = {(g_{n})_{A} | (g_{n}) ∈ X_{∞}}

endowed with the metric d_{A}((g_{n})_{A}, (h_{n})_{A}) = lim_{A} d_{n}(g_{n}, h_{n}).

Proposition 1.4 (van den Dries and Wilkie [4]). If Γ is a finitely generated group equipped with a fixed finite generating set and A is a non-principal ultrafilter on ℕ^{+}, then Con_{A}(Γ) is a complete geodesic space.

□

The asymptotic cone Con_{A}(Γ) can be understood intuitively as follows. (Our account borrows heavily from those in [7] and [2].) Imagine an observer who moves away from the Cayley graph of Γ. Then X_{n} is what he sees if he pauses to observe the Cayley graph when the points of Γ that are actually at a distance n apart appear to be only at a distance 1 apart. As the observer continues to move away, any finite configuration will eventually become indistinguishable from a single point; but he may observe certain finite configurations which resemble earlier configurations. The asymptotic cone Con_{A}(Γ) is a space which encodes all of these recurring finite configurations. For example, let I be an infinite subset of ℕ^{+} and let A be a nonprincipal ultrafilter which contains I. Then for each n ∈ I, the relator w_{n}(a, b) = (a^{n}b^{n})^{7} can be regarded as a loop of length 14n in the Cayley graph of Γ_{I}, and hence as a loop of length 14 in X_{n}. The sequence of loops (w_{n}(a, b) | n ∈ I) gives rise to a closed path of length 14 in Con_{A}(Γ_{I}) as follows. For each n ∈ I, write w_{n}(a, b) = x_{1} ⋯ x_{14n}, where each x_{i} ∈ {a, b}; and define f_{n} : [0, 14] → Γ_{I} by f_{n}(t) = x_{1} ⋯ x_{⌈tn⌉}. If n ∉ I, let f_{n} : [0, 14] → Γ_{I} be the function such that

4 SIMON THOMAS AND BOBAN VELICKOVIC

f_{n}(t) = e for all t ∈ [0, 14]. Now define f : [0, 14] → Con_{A}(Γ_{I}) by f(t) = (f_{n}(t))_{A}. Then it is easily checked that f is a closed path of length 14 in Con_{A}(Γ_{I}). We shall shortly prove that f is not null-homotopic, and hence that Con_{A}(Γ_{I}) is not simply connected.

Next we shall address the question of where the ultrafilter A comes into the intuitive picture of Con_{A}(Γ). As the definition of the closed path f indicates, if C ∉ A, then the observations of the metric spaces {X_{n} | n ∈ C} can safely be ignored. So we can imagine that our observer walks backwards away from the Cayley graph of Γ with his eyes tightly shut, only opening them at rare intervals. It will then hardly be surprising if two observers, who open their eyes at very different times, should get very different impressions of the asymptotic geometry of the Cayley graph of Γ. For example, suppose that I = {i_{m} | m ∈ ℕ} is an extremely sparse subset of ℕ^{+} and that a second observer only opens his eyes at various stages {t_{m} | m ∈ ℕ} such that i_{m} ≪ t_{m} ≪ i_{m+1} for some m ∈ ℕ. Suppose that N is a large natural number and that our second observer examines the ball of radius N around e in X_{t_{m}}, which corresponds to the ball of radius Nt_{m} around e in the Cayley graph of Γ_{I}. By choosing m sufficiently large, we can suppose that Nt_{m} ≪ i_{m+1}. This implies that the ball of radius Nt_{m} around e in the Cayley graph of Γ_{I} will be the same as that in the Cayley graph of the group Γ_{m}, which has the presentation

⟨a, b | w_{i_{0}}(a, b), …, w_{i_{m}}(a, b)⟩.

Since Γ_{m} is a finitely presented group which satisfies the C′(1/6) cancellation property, it follows that Γ_{m} is a hyperbolic group. Hence by Gromov [6], every asymptotic cone of Γ_{m} is an ℝ-tree. Since i_{m} ≪ t_{m}, this implies that the ball of radius N around e in X_{t_{m}} will already look very much like a tree to our second observer. Since N could be chosen to be arbitrarily large, our second observer will believe that the asymptotic cone of Γ_{I} is an ℝ-tree.

Now that we have described the intuition which motivates our construction, we are ready to present the details.

Theorem 1.5. If I is any infinite subset of ℕ^{+} and A is any non-principal ultrafilter which contains I, then the asymptotic cone Con_{A}(Γ_{I}) is not simply connected.

ASYMPTOTIC CONES OF FINITELY GENERATED GROUPS 5

Proof. (This argument is essentially just a proper subset of subsections 2.3 and 2.4 of Bridson [2].) Let S = [0, 1] × [0, 1] be the unit square and let ∂S be the boundary of S. Let f : [0, 14] → Con_{A}(Γ_{I}) be the closed path which was defined earlier. Then it is easy to convert this path into a corresponding continuous map h : ∂S → Con_{A}(Γ_{I}) which parametrizes the path proportionately to the length of ∂S. Suppose that h is null-homotopic. Then h can be extended to a continuous map H : S → Con_{A}(Γ_{I}). Let M be an integer such that if we subdivide S into M^{2} equally small squares with vertices {p_{i,j} | 0 ≤ i, j ≤ M}, then d_{A}(H(p_{i,j}), H(p_{i′,j′})) < 1/2 for every pair of adjacent vertices p_{i,j} and p_{i′,j′}. For each 0 ≤ i, j ≤ M, let (x_{i,j}^{n} | n ∈ ℕ^{+}) be a sequence of elements of Γ_{I} such that H(p_{i,j}) = (x_{i,j}^{n})_{A}, chosen so that if p_{i,j} ∈ ∂S and h(p_{i,j}) = f(t), then (x_{i,j}^{n}) = (f_{n}(t)). Let n ∈ I be an integer such that whenever p_{i,j} and p_{i′,j′} are adjacent vertices, then d_{n}(x_{i,j}^{n}, x_{i′,j′}^{n}) < 1 and hence d(x_{i,j}^{n}, x_{i′,j′}^{n}) < n. Notice that if p_{i,j}, p_{i′,j′} ∈ ∂S are adjacent vertices, then the edge between p_{i,j} and p_{i′,j′} parametrizes an arc of length 14/4M of the closed path f : [0, 14] → Con_{A}(Γ_{I}). Consequently, if we choose M > 14 and n > 4, then the corresponding arc between x_{i,j}^{n}and x_{i′,j′}^{n} in the loop w_{n}(a, b) has length at most ⌈14n/4M⌉ + 1 < n. If the adjacent vertices p_{i,j} and p_{i′,j′} do not both lie in ∂S, then choose a geodesic segment between x_{i,j}^{n} and x_{i′,j′}^{n} in the Cayley graph of Γ_{I}. Thus to the boundary of each of the M^{2} small squares in S, we have now associated a loop of length less than 4n in the Cayley graph of Γ_{I}. For each such small square S′, the labels on the edges of the associated loop yield a word w′ of length less than 4n which represents the identity element in Γ_{I}. Applying Lemma 1.1, we see that each such word w′ lies in the normal closure of the set of relators {w_{ℓ}(a, b) | ℓ ∈ I, ℓ < n}. But this implies that w_{n}(a, b) also lies in the normal closure of the set {w_{ℓ}(a, b) | ℓ ∈ I, ℓ < n}, which is a contradiction.

2. The construction

If (X, d) is a metric space and w ∈ X is a base point, then we define the Gromov inner product on X by

(x.y)_{w} = \frac{1}{2}(d(x, w) + d(w, y) − d(x, y)).

6 SIMON THOMAS AND BOBAN VELICKOVIC

Let δ ≥ 0. Then the metric space (X, d) is said to be δ-hyperbolic if for every w, x, y, z ∈ X, we have that

(x.y)_{w} ≥ min{(x.z)_{w}, (z.y)_{w}} − δ.

If Γ is a finitely generated group equipped with a fixed finite generating set, then Γ is said to be hyperbolic iff its Cayley graph is δ-hyperbolic for some δ ≥ 0. (It is well-known that the hyperbolicity of Γ does not depend on the choice of the finite generating set.) We shall make use of the following characterization of δhyperbolicity, a proof of which can be found in Section 1.1 [3].

Lemma 2.1. The metric space (X, d) is δ-hyperbolic iff

d(x, y) + d(z, w) ≤ max{d(x, z) + d(y, w), d(x, w) + d(y, z)} + 2δ

for all points w, x, y, z ∈ X.

□

We shall also make use of the following well-known characterization of ℝ-trees, a proof of which can be found in Section 3.4 [3].

Theorem 2.2. Let (X, d) be a geodesic space. Then (X, d) is an ℝ-tree iff (X, d) is 0-hyperbolic.

□

We are now ready to begin our construction. We shall define inductively two strictly increasing sequences of natural numbers ⟨i_{m} | m ∈ ℕ⟩ and ⟨t_{m} | m ∈ ℕ⟩. First we set i_{0} = 1. Now suppose that m ≥ 0 and that we have already defined ⟨i_{n} | n ≤ m⟩ and ⟨t_{n} | n < m⟩. Let Γ_{m} be the group with the presentation

⟨a, b | w_{i_{0}}(a, b), …, w_{i_{m}}(a, b)⟩.

Since Γ_{m} is a finitely presented group which satisfies the C′(1/6) cancellation property, it follows that Γ_{m} is a hyperbolic group. So there exists a constant δ_{m} ≥ 0 such that the Cayley graph of Γ_{m} is δ_{m}-hyperbolic. Let t_{m} be any integer such that t_{m−1} < t_{m} and 2δ_{m}/t_{m} < 1/m; and let i_{m+1} be any integer such that i_{m} < i_{m+1} and i_{m+1} > 8mt_{m}. Finally we let I = {i_{m} | m ∈ ℕ} and let B be any non-principal ultrafilter which contains the set T = {t_{m} | m ∈ ℕ}.

Theorem 2.3. Con_{B}(Γ_{I}) is an ℝ-tree.

Proof. Applying Lemma 2.1 and Theorem 2.2, we see that it is enough to show that

d_{B}(P_{1}, P_{2}) + d_{B}(P_{3}, P_{4})
≤ max{d_{B}(P_{1}, P_{3}) + d_{B}(P_{2}, P_{4}), d_{B}(P_{1}, P_{4}) + d_{B}(P_{2}, P_{3})}

for all points P_{1}, …, P_{4} ∈ Con_{B}(Γ_{I}). So let P_{1}, …, P_{4} be an arbitrary 4-tuple of elements of Con_{B}(Γ_{I}); and for each 1 ≤ k ≤ 4, let (P_{k}^{n} | n ∈ ℕ^{+}) be a sequence of elements of Γ_{I} such that P_{k} = (P_{k}^{n})_{B}. Let N be a natural number such that d_{B}(P_{k}, (e)_{B}) < N for each 1 ≤ k ≤ 4. Now fix some ε > 0 and let n_{0} be an integer such that n_{0} > N and 5/n_{0} < ε. Then there exists a set A ∈ B such that if n ∈ A, then the following conditions are satisfied.

(a) n > t_{n_{0}}.

(b) If 1 ≤ k, ℓ ≤ 4, then |d_{n}(P_{k}^{n}, P_{ℓ}^{n}) − d_{B}(P_{k}, P_{ℓ})| < 1/n_{0}.

(c) If 1 ≤ k ≤ 4, then d_{n}(P_{k}^{n}, e) < 2N.

Let B = A∩T. Then B ∈ B and so there exists an integer t_{m} ∈ B. Let 1 ≤ k, ℓ ≤ 4. Then d_{t_{m}}(P_{k}^{t_{m}}, e), d_{t_{m}}(P_{ℓ}^{t_{m}}, e) < 2N and so d(P_{k}^{t_{m}}, e), d(P_{ℓ}^{t_{m}}, e) < 2Nt_{m}. Let p_{k}, p_{ℓ} be geodesic paths from e to P_{k}^{t_{m}}, P_{ℓ}^{t_{m}} respectively in the Cayley graph of Γ_{I} and let p_{k,ℓ} be a geodesic path from P_{k}^{t_{m}} to P_{ℓ}^{t_{m}}. Then the labels on the edges of the loop p_{k} ∗ p_{k,ℓ} ∗ p_{ℓ}^{−1} yield a word w of length less than 8Nt_{m} which represents the identity element in Γ_{I}. Since 8Nt_{m} < 8n_{0}t_{m} < 8mt_{m} < i_{m+1}, Lemma 1.1 implies that w lies in the normal closure of the set of relators {w_{i_{0}}(a, b), …, w_{i_{m}}(a, b)}. It follows that p_{k,ℓ} is also a geodesic path from P_{k}^{t_{m}} to P_{ℓ}^{t_{m}} in the Cayley graph of the hyperbolic group Γ_{m}. Thus each pair of points P_{k}^{t_{m}}, P_{ℓ}^{t_{m}} is the same distance apart in the Cayley graphs of Γ_{I} and Γ_{m}. Consequently we must have that

d(P_{1}^{t_{m}},P_{2}^{t_{m}}) + d(P_{3}^{t_{m}}, P_{4}^{t_{m}})

≤ max{d(P_{1}^{t_{m}}, P_{3}^{t_{m}}) + d(P_{2}^{t_{m}}, P_{4}^{t_{m}}), d(P_{1}^{t_{m}}, P_{4}^{t_{m}}) + d(P_{2}^{t_{m}}, P_{3}^{t_{m}})} + 2δ_{m}.

Using the fact that 2δ_{m}/t_{m} < 1/m < 1/n_{0}, we now obtain that

d_{B}(P_{1}, P_{2}) + d_{B}(P_{3}, P_{4})

< d_{t_{m}}(P_{1}^{t_{m}}, P_{2}^{t_{m}}) + d_{t_{m}}(P_{3}^{t_{m}}, P_{4}^{t_{m}}) + 2/n_{0}

< max{d_{t_{m}}(P_{1}^{t_{m}}, P_{3}^{t_{m}}) + d_{t_{m}}(P_{2}^{t_{m}}, P_{4}^{t_{m}}), d_{t_{m}}(P_{1}^{t_{m}}, P_{4}^{t_{m}}) + d_{t_{m}}(P_{2}^{t_{m}}, P_{3}^{t_{m}})} + 3/n_{0}

< max{d_{B}(P_{1}, P_{3}) + d_{B}(P_{2}, P_{4}), d_{B}(P_{1}, P_{4}) + d_{B}(P_{2}, P_{3})} + 2/n_{0} + 3/n_{0}

< max{d_{B}(P_{1}, P_{3}) + d_{B}(P_{2}, P_{4}), d_{B}(P_{1}, P_{4}) + d_{B}(P_{2}, P_{3})} + ε.

Since ε > 0 was arbitrary, the result follows.

References

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[3] M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes: Les groupes hyperboliques de Gromov, Springer Lecture Notes 1441, 1990.

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