ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS

LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

Abstract. Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that ℝ-rank(S) ≥ 2 and let Γ be a uniform lattice in G.

(a) If CH holds, then Γ has a unique asymptotic cone up to homeomorphism.

(b) If CH fails, then Γ has 2^{2^{ω}} asymptotic cones up to homeomorphism.

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 [15], 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. In [12], 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 D over the set ω of natural numbers, and it was initially not clear whether the resulting asymptotic cone Con_{D}(Γ) depended on the choice of the ultrafilter D. In [31], answering a question of Gromov [16], Thomas and Velickovic constructed an example of a finitely generated group Γ and two nonprincipal ultrafilters A, B such that the asymptotic cones Con_{A}(Γ) and Con_{B}(Γ) were not homeomorphic. But this still left open the interesting question of whether there exists a finitely presented group with more than one asymptotic cone up to homeomorphism. It seems almost certain that such a group exists; and, in fact, it seems natural to conjecture that there exists a finitely presented group with 2^{2^{ω}} asymptotic cones

The research of the first and third authors was supported by Heisenberg Fellowships of the DFG. The research of the second author was supported by the Israel Science Foundation. This paper is number 818 in the cumulative list of the second author’s publications. The research of the fourth author was partially supported by NSF Grants.

2 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

up to homeomorphism. (Recall that there are exactly 2^{2^{ω}} distinct nonprincipal ultrafilters over ω.) The main result of this paper provides a confirmation of this conjecture, under the assumption that CH fails.

Suppose that G is a connected semisimple Lie group and let Γ be a uniform lattice in G; i.e. a discrete subgroup such that G/Γ is compact. (For the existence of such a subgroup Γ, see Borel [2].) Then it is well-known that Γ is finitely presented. (For example, see [35, Chapter 3], [18, Chapter V].) Furthermore, Γ is quasi-isometric to G; and hence for each ultrafilter D, the asymptotic cones Con_{D}(Γ) and Con_{D}(G) are homeomorphic. In [16, Section 2.2.B_{1}], Gromov suggested that “ it seems that these groups G have pretty looking finite dimensional cones Con_{D}(G) which are (essentially) independent of the choice of the ultrafilter D.” It turns out that the situation is more interesting. Theorem 1.1. Suppose that G is a connected semisimple Lie group with at least one absolutely simple factor S such that ℝ-rank(S) ≥ 2 and let Γ be a uniform lattice in G.

(a) If CH holds, then Γ has a unique asymptotic cone up to homeomorphism.

(b) If CH fails, then Γ has 2^{2^{ω}} asymptotic cones up to homeomorphism.

Here CH denotes the Continuum Hypothesis; i.e. the statement that 2^{ω} = ω_{1}. Of course, it is well-known that CH can neither be proved nor disproved using the usual ZF C axioms of set theory. (For example, see Jech [19].)

In the remainder of this section, we shall sketch the main points of the proof for the special case when G = SL_{m}(ℝ) for some m ≥ 3. In particular, we shall explain the unexpected appearance of the Robinson field ^{ρ}ℝ_{D} as a topological invariant of the asymptotic cone Con_{D}(Γ). (As we shall explain below, the Robinson field ^{ρ}ℝ_{D} is a valuation field closely related to the corresponding field ^{∗}ℝ_{D} of nonstandard real numbers.) We shall begin by recalling the definition of an asymptotic cone of an arbitrary metric space. Definition 1.2. A nonprincipal ultrafilter over the set ω of natural numbers is a collection D of subsets of ω satisfying the following conditions:

(i) If A, B ∈ D, then A ∩ B ∈ D.

(ii) If A ∈ D and A ⊆ B ⊆ ω, then B ∈ D.

(iii) For all A ⊆ ω, either A ∈ D or ω ∖ A ∈ D.

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 3

(iv) If F is a finite subset of ω, then F ∉ D.

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

{n ∈ ω | |r_{n} − ℓ| < ε} ∈ D

for all ε > 0. We write ℓ = lim_{D} r_{n}. Definition 1.3. Suppose that D is a nonprincipal ultrafilter over ω. Let (X, d) be a metric space and for each n ≥ 1, let d_{n} be the rescaled metric defined by d_{n}(x, y) = d(x, y)/n. Let e ∈ X be a fixed base point. Then X_{∞} is the set of all sequences (x_{n}) of elements of X such that there exists a constant c with

d_{n}(x_{n}, e) ≤ c

for all n ≥ 1. Define an equivalence relation ∼ on X_{∞} by

(x_{n}) ∼ (y_{n}) if and only if lim_{D} d_{n}(x_{n}, y_{n}) = 0;

and for each (x_{n}) ∈ X_{∞}, let (x_{n})_{D} be the corresponding equivalence class. Then the asymptotic cone of X is

Con_{D}(X) = {(x_{n})_{D} | (x_{n}) ∈ X_{∞}}

endowed with the metric

d_{D}((x_{n})_{D}, (y_{n})_{D}) = lim_{D} d_{n}(x_{n}, y_{n}).

If (X, d) and (X′, d′) are metric spaces, then a map f : X → X′ is a quasiisometry iff there exist constants L ≥ 1 and C ≥ 0 such that for all x, y ∈ X

• \frac{1}{L}d(x, y) − C ≤ d′(f(x), f(y)) ≤ Ld(x, y) + C; and for all z ∈ X′

• d′(z, f[X]) ≤ C. The following result is well-known. (For example, see Proposition 2.4.6 of Kleiner-Leeb [20].)

4 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

Proposition 1.4. Let D be a nonprincipal ultrafilter over ω and let f : X → X′ be a quasi-isometry of metric spaces. Then f induces a bilipschitz homeomorphism φ : Con_{D}(X) → Con_{D}(X′) defined by φ((x_{n})_{D}) = (f(x_{n}))_{D}.

From now on, fix some m ≥ 3 and let Γ be a uniform lattice in SL_{m}(ℝ); i.e. Γ is a discrete subgroup of SL_{m}(ℝ) such that SL_{m}(ℝ)/Γ is compact. By Theorem IV.23 [18], Γ is quasi-isometric to SL_{m}(ℝ), viewed as a metric space with respect to some left-invariant Riemannian metric. Thus if D is a nonprincipal ultrafilter over ω, then Con_{D}(Γ) is bilipschitz homeomorphic to Con_{D}(SL_{m}(ℝ)). However, instead of working directly with the Riemannian manifold SL_{m}(ℝ), it turns out to be more convenient to work with the corresponding symmetric space, obtained by factoring out the maximal compact subgroup SO_{m}(ℝ). In more detail, recall that SL_{m}(ℝ) acts transitively as a group of isometries on the symmetric space P (m, ℝ) of positive-definite symmetric m × m matrices with determinant 1, via the action

g · A = gAg^{t}.

Clearly the stabiliser of the identity matrix I under this action is the subgroup SO_{m}(ℝ); and since SO_{m}(ℝ) is compact, it follows that Γ also acts cocompactly on P (m, ℝ). (For example, see [35, Section 2].) Hence, applying Theorem IV.23 [18] once again, it follows that Γ is also quasi-isometric to the symmetric space P (m, ℝ). The invariant Riemannian metric d on P (m, ℝ) can be described as follows. First recall that if A, B ∈ P (m, ℝ), then there exists g ∈ SL_{m}(ℝ) such that gAg^{t} and gBg^{t} are simultaneously diagonal. Hence it suffices to consider the case when

A = (a_{i,j}) and B = (b_{i,j})

are both diagonal matrices; in which case,

d(A, B) = \sqrt{∑(log a_{i,i} − log b_{i,i})^{2}}.

Fix some nonprincipal ultrafilter D over ω. Let X = P (m, ℝ) and let I ∈ X be the base point. We next consider the question of which sequences of diagonal matrices lie in X_{∞}. For each n ≥ 1, let

A_{n} = (a_{i,j}^{(n)}) ∈ X

be a diagonal matrix. Then

d_{n}(I, A_{n}) = \frac{\sqrt{∑(log a_{i,i}^{(n)})^{2}}}{n}

and so (A_{n}) ∈ X_{∞} if and only if there exists k ≥ 1 such that

(Exp)  e^{−kn} < a_{i,i}^{(n)} < e^{kn}

for each 1 ≤ i ≤ m and n ≥ 1.

Before we can describe the structure of the asymptotic cone Con_{D}(X), we first need to recall the definition of the corresponding field ^{∗}ℝ_{D} of nonstandard reals. Definition 1.5. Let ℝ^{ω} be the set of all sequences (x_{n}) of real numbers. Define an equivalence relation ≡ on ℝ^{ω} by

(x_{n}) ≡ (y_{n}) if and only if {n ∈ ω | x_{n} = y_{n}} ∈ D;

and for each (x_{n}) ∈ ℝ^{ω}, let [x_{n}]_{D} be the corresponding equivalence class. Then the field of nonstandard reals is defined to be

∗ℝ_{D} = {[x_{n}]_{D} | (x_{n}) ∈ ℝ^{ω}}

equipped with the operations

[x_{n}]_{D} + [y_{n}]_{D} = [x_{n} + y_{n}]_{D}
[x_{n}]_{D} · [y_{n}]_{D} = [x_{n} · y_{n}]_{D}

and the ordering

[x_{n}]_{D} < [y_{n}]_{D} if and only if {n ∈ ω | x_{n} < y_{n}} ∈ D.

In order to simplify notation, during the next few paragraphs, we shall write ^{∗}ℝ instead of the more precise ^{∗}ℝ_{D}. It is well-known that ^{∗}ℝ is a nonarchimedean real closed field and that ℝ embeds into ^{∗}ℝ via the map r ↦ [r]_{D}, where (r) denotes the sequence with constant value r. (For the basic properties of ^{∗}ℝ, see Lightstone-Robinson [22].)

Now suppose that (A_{n}) ∈ X_{∞} is a sequence of diagonal matrices, where each A_{n} = (a_{i,j}^{(n)}). Then we can define a corresponding diagonal matrix

(A_{n})_{D} = (α_{i,j}) ∈ P (m, ^{∗}ℝ)

6 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

by setting α_{i,j} = [a_{i,j}^{(n)}]_{D}. Using equation (Exp), we see that there exists k ≥ 1 such that each α_{i,i} satisfies the inequality

ρ^{k} < α_{i,i} < ρ^{−k},

where ρ ∈ ^{∗}ℝ is the positive infinitesimal defined by

ρ = (e^{−n})_{D}.

This suggests that Con_{D}(X) should “essentially” be P (m, K) for some field defined in terms of ^{∗}ℝ and ρ. Definition 1.6. Let M_{0} be the subring of ^{∗}ℝ defined by

M_{0} = {t ∈ ^{∗}ℝ | |t| < ρ^{−k} for some k ≥ 1}.

Then M_{0} has a unique maximal ideal

M_{1} = {t ∈ ^{∗}ℝ | |t| < ρ^{k} for all k ≥ 1}

and the Robinson field ^{ρ}ℝ is defined to be the residue field M_{0}/M_{1}. By [22], ^{ρ}ℝ is also a real closed field.

If (A_{n}) ∈ X_{∞} is a sequence of diagonal matrices, then

(A_{n})_{D} = (α_{i,j}) ∈ P (m, M_{0}),

and so we can define a corresponding matrix

\overline{(A_{n})}_{D} = (\overline{α}_{i,j}) ∈ P (m, ^{ρ}ℝ),

where each \overline{α}_{i,j} ∈ ^{ρ}ℝ is the element naturally associated with α_{i,j} ∈ ^{∗}ℝ. Unfortunately, it is not always the case that if (A_{n}), (A′_{n}) ∈ X_{∞} correspond to the same element of Con_{D}(X), then \overline{(A_{n})}_{D} = \overline{(A′_{n})}_{D}. Thus, in order to obtain Con_{D}(X), we must first factor P (m, ^{ρ}ℝ) by a suitable equivalence relation. Definition 1.7. If 0 ≠ α ∈ M_{0}, then log_{ρ} |α| is a finite possibly nonstandard real and hence is infinitesimally close to a unique standard real denoted by st(log_{ρ} |α|). By Lightstone-Robinson [22, Section 3.3], if t ∈ M_{0} ∖ M_{1} and i ∈ M_{1}, then

st(log_{ρ} |t|) = st(log_{ρ} |t + i|).

Hence we can define a valuation υ : ^{ρ}ℝ → ℝ ∪ {∞} by

υ(α) = st(log_{ρ} |α|).

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 7

Let

|α|_{υ} = e^{−υ(α)}

be the associated absolute value.

Following Leeb and Parreau [23], the asymptotic cone Con_{D}(X) can now be described as follows. For each A ∈ P (m, ^{ρ}ℝ), define the corresponding norm

φ_{A} : V (m, ^{ρ}ℝ) → ℝ

by

φ_{A}(\overline{x}) = | \sqrt{\overline{x}^{t}A\overline{x}} |_{υ}

Then

Con_{D}(X) ≅ {φ_{A} | A ∈ P (m, ^{ρ}ℝ)}.

This space of norms is an instance of a classical construction of Bruhat-Tits [8], [9] and has a rich geometric structure. (The various geometric notions discussed in the remainder of this paragraph will be defined and discussed in more detail in Section 2.) More precisely, Con_{D}(X) is an affine ℝ-building; and the set F of apartments of Con_{D}(X) is precisely the collection of subspaces of Con_{D}(X) which are isometric to ℝ^{m−1}. Furthermore, the elements of F correspond naturally to the unordered frames {^{ρ}ℝv_{1}, ⋯, ^{ρ}ℝv_{m}} of the m-dimensional vector space V (m, ^{ρ}ℝ) over ^{ρ}ℝ. Let ∂ Con_{D}(X) be the associated spherical building at infinity of Con_{D}(X). Then the apartments of ∂ Con_{D}(X) are the boundaries at infinity of the apartments of Con_{D}(X); and so the apartments of ∂ Con_{D}(X) also correspond naturally to the unordered frames {^{ρ}ℝv_{1}, ⋯, ^{ρ}ℝv_{m}} of V (m, ^{ρ}ℝ). As this observation suggests, ∂ Con_{D}(X) is the usual spherical building associated with the natural BN-pair of SL_{m}(^{ρ}ℝ); i.e. ∂ Con_{D}(X) is the flag space of the vector space V (m, ^{ρ}ℝ). In particular, since m ≥ 3, the isomorphism type of the Robinson field ^{ρ}ℝ is determined by the isomorphism type of the spherical building ∂ Con_{D}(X). By the Kleiner-Leeb topological rigidity theorem for affine ℝ-buildings [20], the isomorphism type of the spherical building ∂ Con_{D}(X) is a topological invariant of Con_{D}(X). Consequently, the isomorphism type of the Robinson field ^{ρ}ℝ is also a topological invariant of Con_{D}(X).

From now on, we shall write ^{ρ}ℝ_{D} to indicate the possible dependence on D of the Robinson field. Since ^{ρ}ℝ_{D} is a topological invariant of Con_{D}(X), the following

8 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

result implies that if CH fails, then X = P (m, ℝ) has 2^{2^{ω}} asymptotic cones up to homeomorphism. Consequently, as the uniform lattice Γ is quasi-isometric to P (m, ℝ), it follows that Γ also has 2^{2^{ω}} asymptotic cones up to homeomorphism. Theorem 1.8. If CH fails, then there exists a set {D_{α} | α < 2^{2^{ω}}} of nonprincipal ultrafilters over ω such that

ρℝ_{D_{α}} ≇ ^{ρ}ℝ_{D_{β}}

for all α < β < 2^{2^{ω}}.

Now suppose that CH holds. In this case, it is well-known that if A, B are nonprincipal ultrafilters, then the corresponding fields ^{∗}ℝ_{A}, ^{∗}ℝ_{B} of nonstandard reals are isomorphic. In [32], Thornton used the Diarra-Pestov [11], [24] representation of ^{∗}ℝ_{A} as a Hahn field to prove that if CH holds, then the Robinson fields ^{ρ}ℝ_{A} and ^{ρ}ℝ_{B} are isomorphic as valuation fields. It follows that X = P (m, ℝ) has a unique asymptotic cone up to homeomorphism. In fact, Thornton proved that the result holds for uniform lattices in arbitrary connected semisimple Lie groups. Theorem 1.9 (Thornton [32]). Assume CH. If A, B are nonprincipal ultrafilters, then

ρℝ_{A} ≅ ^{ρ}ℝ_{B}.

Furthermore, if G is a connected semisimple Lie group and Γ is a uniform lattice in G, then Γ has a unique asymptotic cone up to homeomorphism.

Finally we should stress that it remains an open problem whether it can be proved in ZF C that there exists a finitely presented group with more than one asymptotic cone up to homeomorphism. However, when it comes to the question of whether there exists a finitely presented group with 2^{2^{ω}} asymptotic cones up to homeomorphism, then the case is altered. Theorem 1.10. If CH holds, then every finitely generated group Γ has at most 2^{ω} asymptotic cones up to isometry. Corollary 1.11. The following statements are equivalent.

(a) CH fails.

(b) There exists a finitely presented group Γ which has 2^{2^{ω}} asymptotic cones up to homeomorphism.

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 9

(c) There exists a finitely generated group Γ which has 2^{2^{ω}} asymptotic cones up to homeomorphism.

The rest of this paper is organised as follows. In Section 2, we shall discuss the notions of an affine ℝ-building and its spherical building at infinity; and we shall explain why ^{ρ}ℝ_{D} is a topological invariant of Con_{D}(Γ), whenever Γ is a uniform lattice in a connected semisimple Lie group G with at least one absolutely simple factor S such that ℝ-rank(S) ≥ 2. Sections 3 and 4 will be devoted to the proof of Theorem 1.8. Finally we shall prove Theorem 1.10 in Section 5.

Throughout this paper, unless otherwise stated, the term “Lie group” will always mean a real Lie group.

2. Spherical and euclidean buildings

In this section, we shall discuss the notions of an affine ℝ-building and its spherical building at infinity; and we shall explain why ^{ρ}ℝ_{D} is a topological invariant of Con_{D}(Γ), whenever Γ is a uniform lattice in a connected semisimple Lie group G with at least one absolutely simple factor S such that ℝ-rank(S) ≥ 2.

Suppose that V ≅ ℝ^{n} is a finite-dimensional Euclidean vector space, with inner product ⟨−, −⟩ : V × V → ℝ. For a nonzero vector v ∈ V, let σ_{v} : V → V denote the Euclidean reflection u ⟼ u − 2\frac{⟨u,v⟩}{⟨v,v⟩}v. A finite spanning set Φ ⊆ V of (nonzero) vectors is called a root system if σ_{u}v ∈ Φ and 2\frac{⟨x,v⟩}{⟨v,v⟩} ∈ ℤ for all u, v ∈ Φ, see [5, Ch. VI]. The group W generated by the σ_{v} is called the Weyl group of the root system. Associated to such a finite reflection group W is a certain simplicial complex Σ(W), its Coxeter complex [5, Ch. IV]: roughly speaking, Σ(W) is a W - invariant triangulation of the unit sphere 𝕊^{n−1} ⊆ V. (The triangulation is obtained from the intersections of 𝕊^{n−1} with the reflection hyperplanes v^{⊥} for v ∈ Φ).

For example, the root system of type A_{n} and its Coxeter complex are defined as follows. Let {e_{1}, …, e_{n+1}} denote the standard orthonormal basis for ℝ^{n+1} and let V be the subspace of ℝ^{n+1} defined by

V = {x ∈ ℝ^{n+1} | x_{1} + ⋯ + x_{n+1} = 0}.

Then Φ = {e_{i} − e_{j} | i ≠ j} ⊆ V is a root system of type A_{n}. As an abstract group, the Weyl group W is the symmetric group on n + 1 letters, acting by coordinate permutations; and as a poset, Σ(W) is the set of all ⊆-ordered chains consisting of

10 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

nontrivial subsets of {1, …, n + 1} (i.e. Σ(W) is the first barycentric subdivision of the boundary of an n + 1-simplex).

A spherical building is an abstract simplicial complex (a poset) (Δ, ≤) with a distinguished collection of subcomplexes Σ, called apartments, satisfying the following axioms. (B_{1}) Each apartment Σ ⊆ Δ is isomorphic to a (fixed) Coxeter complex Σ(W). (B_{2}) Any two simplices in Δ are contained in some apartment. (B_{3}) Given two apartments Σ_{1}, Σ_{2} ⊆ Δ, there exists an isomorphism Σ_{1} → Σ_{2} fixing Σ_{1} ∩ Σ_{2} simplex-wise. For more details, we refer to [7], [27], [33]. The standard examples of buildings are obtained from algebraic groups as follows.

Let G be a reductive algebraic group defined over a field k [3] [4] [30] and let Δ denote the poset of all k-parabolic subgroups of G, ordered by reversed inclusion. Then Δ = Δ(G, k) is a spherical building, the canonical building associated to G over k, and the group G(k) of k-points of G acts strongly transitively on Δ(G, k), see Tits [33, Ch. 5]. If G is absolutely simple (i.e. if G is simple over the algebraic closure k̄) and adjoint, then the building uniquely determines the field of definition k.

Theorem 2.1. [33, 5.8] Let G and G′ be adjoint absolutely simple algebraic groups of rank at least 2 defined over the fields k and k′. If there is a building isomorphism Δ(G, k) ≅ Δ(G′, k′), then the fields k and k′ are isomorphic. □

We next introduce the notion of an affine ℝ-building (also called a Euclidean building). Below we shall give Tits’ definition [34], as corrected in [27, App. 3]. (A different set of axioms was proposed in Kleiner-Leeb [20], based on nonpositive curvature and geodesics; in [23], Parreau showed that these two approaches are equivalent). Let W be the Weyl group of a root system Φ ⊆ V, and let W_{aff} denote the semidirect product of W and the vector group (V, +). Then in V, we obtain the corresponding reflection hyperplanes (the fixed point sets of reflections), half-spaces (determined by reflection hyperplanes), and Weyl chambers (the fundamental domains for W), see [5].

a reflection hyperplane  a half-space  a Weyl chamber

Definition 2.2. Fix W and V as above. A pair (I, F) consisting of a nonempty set I and a family F of injections ϕ : V → I is called an affine ℝ-building if it has the following properties. (AℝB_{1}) If w ∈ W_{aff} and ϕ ∈ F, then ϕ ∘ w ∈ F. (AℝB_{2}) For ϕ, ψ ∈ F, the preimage X = ϕ^{−1}ψ(V) is closed and convex (possibly empty), and there exists w ∈ W_{aff} such that ϕ and ψ ∘ w agree on X. (AℝB_{3}) Given x, y ∈ I, there exists ϕ ∈ F with {x, y} ⊆ ϕ(V). Any ϕ-image of V is called an apartment; the ϕ-image of a reflection hyperplane, a half-space, and a Weyl chamber is called a wall, a half-apartment, and a sector, respectively. A wall is thick if it bounds three distinct half-apartments, and a point is thick if every wall passing through it is thick. (AℝB_{4}) Given two sectors S_{1}, S_{2} ⊆ I, there exist subsectors S′_{i} ⊆ S_{i} and an apartment F ⊆ I with S′_{1} ∪ S′_{2} ⊆ F. (AℝB_{5}) If F_{1}, F_{2}, F_{3} are apartments having pairwise a half-apartment in common, then F_{1} ∩ F_{2} ∩ F_{3} ≠ ∅. The dimension of an affine ℝ-building is the vector space dimension of V.

It follows that I admits a unique metric d which pulls back to the Euclidean metric on V for every ϕ : V → I, and that (I, d) is a CAT(0)-space. (See [6] for the notion of a CAT(0)-space.) We say that I is complete if (I, d) is complete as a metric space (every Cauchy sequence converges) and that F is complete if every injection ϕ : V → I which is compatible with the axioms (AℝB_{1})–(AℝB_{5}) is already in F. It can be shown that every F admits a unique completion F̂ such that (I, F̂) is an affine ℝ-building; the metric completion of I, however, is in general not an affine ℝ-building.

Proposition 2.3. Suppose that (I, F) and (I′, F′) are affine ℝ-buildings, and that f : I → I′ is a homeomorphism. Let F ⊆ I be an apartment. If I′ and F′ are complete, then f(F) is an apartment in I′.

12 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

Proof. This was first proved by Kleiner-Leeb [20, Prop. 6.4.1]. A sheaf-theoretic proof was given in Kramer-Tent [21], using the fact that apartments can be viewed as certain global sections in the orientation sheaf of the topological space I, which can be characterised topologically. From this, one deduces that both buildings have the same dimension, and that f(F) is locally isometric to Euclidean space. Since I′ is complete, f(F) is also complete and thus – being contractible – globally isometric to Euclidean space. Such a subspace is always an apartment in the completion of F′. □

Proposition 2.3 and Proposition 2.4 below are false if I′ is not assumed to be complete.

An affine ℝ-building (I, F) has a spherical building at infinity, denoted by ∂(I, F). The chambers of this building are equivalence classes of sectors, where two sectors are equivalent if the Hausdorff distance between them is finite. If a basepoint x_{0} ∈ I is fixed, then every chamber of ∂(I, F) has a unique x_{0}-based sector as its representative.

The draft D(I, F) of an affine ℝ-building (I, F) is the pair (I, A) consisting of all points and all apartments of (I, F). As a direct consequence of 2.3, we have the following result. Proposition 2.4. Let (I, F) and (I′, F′) be affine ℝ-buildings such that I, I′, F and F′ are complete. Then each homeomorphism f : I → I′ induces an isomorphism of drafts D(I, F) ^{≅} D(I′, F′). □

The draft disregards the metric structure and the Weyl group of the affine ℝbuilding. In general, nonisomorphic affine ℝ-buildings can have isomorphic drafts. However, we have the following result. Proposition 2.5. Suppose that I contains a thick point x. If

f : D(I, F) ^{≅} D(I′, F′)

is an isomorphism of drafts of affine ℝ-buildings, then f induces an isomorphism

∂f : ∂(I, F) → ∂(I′, F′)

between the respective spherical buildings at infinity.

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 13

Proof. Fix some thick point x of I. For each y ∈ I, let cvx{x, y} denote the intersection of all apartments containing x and y. This set should be pictured as a diamond-shaped set with x as one tip. Since x has thick walls, cvx{x, y} is always contained in one of the x-based sectors of I.

x  y

Let F ⊆ I be an apartment containing x. In the poset ({cvx{x, y} | y ∈ F } , ⊆), the unions over the maximal chains are precisely the x-based sectors in F. Thus any isomorphism of drafts preserves x-based sectors. The intersections of the xbased sectors in F naturally form a poset isomorphic to Σ(W); and the union of these posets, where F runs through all apartments containing x, is canonically isomorphic to the spherical building ∂(I, F). The result now follows from the fact that the underlying poset of a spherical building completely determines the building itself. □

Let G be a semisimple algebraic group defined over the real closure ℚ_{ℝ} of ℚ. Then the ℝ-rank of G is the maximal dimension of a ℚ_{ℝ}-split torus S of G. If R is any real closed field (such as R = ℝ or R = ^{ρ}ℝ), then ℚ_{ℝ} ⊆ R and S is also a maximal R-split torus over R. In fact, G has the same structure, rank, Tits diagram and building type over R as over its field of definition ℚ_{ℝ}, see [17, Sec. 4]. Note that the group G = G(ℝ) of ℝ-points of G, endowed with the Hausdorff topology, is a real Lie group; and its (Hausdorff) connected component G^{∘} is a semisimple Lie group G such that [G : G^{∘}] < ∞. Furthermore, if G is any connected semisimple Lie group, then there exists a semisimple algebraic group defined over ℚ_{ℝ} such that G/Z(G) and G(ℝ)^{∘} are isomorphic as Lie groups; for example, see [14, 1.14.6]. (In fact, G can even be taken to be defined over ℚ; but in our setting, it is more convenient to work with real closed fields.) We then define ℝ-rank(G) to be the ℝ-rank of the algebraic group G; and we define G to be absolutely simple if and only if G is absolutely simple. (Equivalently, G is absolutely simple if and only if the complexification 𝔤 ⊗_{ℝ} ℂ of the Lie algebra 𝔤 of G remains simple.) Since Z(G)

14 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

is finite, the Riemannian manifolds G and G/Z(G) are quasi-isometric. Hence, in the remainder of this section, we can restrict our attention to connected semisimple Lie groups of the form G(ℝ)^{∘}.

Fix a semisimple algebraic group G defined over ℚ_{ℝ} of ℝ-rank m ≥ 1. Let S ⊆ G be a maximal ℚ_{ℝ}-split torus and N = Nor_{G}(S) its normaliser. The quotient W = N/S is the relative Weyl group for G (once again, over any real closed field R). Let D be a non-principal ultrafilter, let ε = (1/n)_{D}, and let ρ = e^{ε} = (e^{−n})_{D}. The Robinson field ^{ρ}ℝ has a unique maximal o-convex subring ^{ρ}O ⊆ ^{ρ}ℝ, corresponding to the canonical o-valuation ν : ^{ρ}ℝ → ℝ∪{∞}, which was defined in Definition 1.7. (For general results about real closed fields and o-valuations, see [25].) Since ^{ρ}O is o-convex, we have that ℚ_{ℝ} ⊆ ^{ρ}O ⊆ ^{ρ}ℝ. Therefore the groups G(^{ρ}O) ⊆ G(^{ρ}ℝ) and S(^{ρ}O) ⊆ S(^{ρ}ℝ) are defined, as well as the coset spaces

I = G(^{ρ}ℝ)/G(^{ρ}O) and V = S(^{ρ}ℝ)/S(^{ρ}O).

Moreover, V can be regarded in a natural way as an m-dimensional real vector space, equipped with a natural action of W as a finite reflection group. For example, if G = SL_{m+1} and S is the group of diagonal matrices in SL_{m+1}, then N consists of permutation matrices acting by coordinate permutations; and the resulting root system is the one of type A_{m} described at the beginning of this section.

If we extend the W -action by the translations, then we obtain an affine Weyl group W_{aff}. For g ∈ G(^{ρ}ℝ) and v = nS(^{ρ}O) ∈ S(^{ρ}ℝ)/S(^{ρ}O) = V, let ϕ_{g}(v) = gnG(^{ρ}O) ∈ I and put

F = {ϕ_{g} | g ∈ G(^{ρ}ℝ_{D})}.

Theorem 2.6. The pair (I, F) = Δ_{aff}(G,^{ρ}ℝ,^{ρ}O) is an affine ℝ-building such that I and F are complete. The building at infinity is the spherical building ∂Δ_{aff}(G,^{ρ}ℝ,^{ρ}O) = Δ(G,^{ρ}ℝ).

Proof. This is a special case of a much more general result on the affine Λ-buildings associated to arbitrary real closed valued fields, which was proved in Kramer-Tent [21]. The fact that I and F are complete follows from the ω_{1}-saturatedness of countable ultrapowers. □

Now we shall explain how asymptotic cones fit into the picture. Let G be a semisimple Lie group of rank m ≥ 1, endowed with a left-invariant Riemannian

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 15

metric d. Let K ⩽ G be a maximal compact subgroup. The Riemannian symmetric space X = G/K carries a natural metric (unique up to homothety), and it is not difficult to show that the natural map G → G/K is a quasi-isometry. Thus X and G have bilipschitz homeomorphic asymptotic cones. As above, let G be an algebraic group over ℚ_{ℝ} with G(ℝ)^{∘} = G. Proposition 2.7. The asymptotic cone Con_{D}(X) is isometric to the point space I of the building Δ_{aff}(G,^{ρ}ℝ_{D},^{ρ}O_{D}).

Proof. This was proved in Kramer-Tent [21] and also independently in Thornton [32]. □

The fact that Con_{D}(X) is an affine ℝ-building was proved first by Kleiner-Leeb [20]. (A vague conjecture pointing in this direction was made by Gromov in [16, p. 54]). However, Kleiner and Leeb did not determine the building which one obtains. As we mentioned above, the fact that Con_{D}(X) can be identified as a metric space with the quotient G(^{ρ}ℝ)/G(^{ρ}O) was proved independently by Thornton [32], but he did not identify the affine building or the spherical building at infinity. This was done by Leeb and Parreau [23] for the special case of G = SL_{m+1}; and one should also mention Bennett’s general result [1] on the affine Λ-buildings related to SL_{m+1}(F) for arbitrary valued fields F. Theorem 2.8. Suppose that G is a connected absolutely simple Lie group such that ℝ-rank(G) ≥ 2 and let Γ be a uniform lattice in G. Let D and D′ be nonprincipal ultrafilters. If Con_{D}(Γ) and Con_{D′}(Γ) are homeomorphic, then the Robinson fields ρℝ_{D} and ^{ρ}ℝ_{D′} are isomorphic.

Proof. Let G be an absolutely simple algebraic group defined over ℚ_{ℝ} such that ℝ-rank(G) ≥ 2 and G = G(ℝ)^{∘}. Let K ⩽ G be a maximal compact subgroup. Since Γ is quasi-isometric to the symmetric space X = G/K, the asymptotic cones Con_{D}(Γ) and Con_{D′}(Γ) are bilipschitz homeomorphic to Con_{D}(X), Con_{D′}(X) respectively. Consequently, if Con_{D}(Γ) and Con_{D′}(Γ) are homeomorphic, then Con_{D}(X) and Con_{D′}(X) are also homeomorphic. Hence, by Propositions 2.7 and 2.4, the affine ℝ-buildings Δ_{aff}(G,^{ρ}ℝ_{D},^{ρ}O_{D}) and Δ_{aff}(G,^{ρ}ℝ_{D′},^{ρ}O_{D′}) have isomorphic drafts. Applying Proposition 2.5, it follows that the corresponding buildings at infinity Δ(G,^{ρ}ℝ_{D}) and Δ(G,^{ρ}ℝ_{D′}) are also isomorphic. Finally, by Theorem 2.1, the building Δ(G,^{ρ}ℝ_{D}) determines the field ^{ρ}ℝ_{D} up to isomorphism, and so ρℝ_{D} ≅ ^{ρ}ℝ_{D′} . □

We should make a few comments concerning the hypotheses on the Lie group G in the statement of Theorem 2.8. If G has ℝ-rank 1, then Con_{D} Γ is a homogeneous ℝ-tree with uncountable branching at every point. Furthermore, the isometry type of this ℝ-tree is independent of the choice of the ultrafilter D (and even of the Lie type of G.) For example, see Dyubina-Polterovich [13]. Thus the hypothesis on the ℝ-rank of G is certainly necessary. To see the necessity of the hypothesis that G is absolutely simple, consider the complex Lie group G = SL_{n}(ℂ) for some n ≥ 3. By embedding SL_{n}(ℂ) as an algebraic subgroup G(ℝ) of GL_{2n}(ℝ), we can regard G as a real simple Lie group of ℝ-rank n − 1. However, G is not absolutely simple, since G(ℂ) ≅ SL_{n}(ℂ) × SL_{n}(ℂ). (More generally, it turns out that a real simple Lie group G is absolutely simple if and only if G/Z(G) is not isomorphic to a complex Lie group.) When we consider the spherical building at infinity of the corresponding affine ℝ-building, then we are only able to recover the algebraic closure ^{ρ}ℂ_{D} = ^{ρ}ℝ_{D}(\sqrt{−1}) rather than the Robinson field ^{ρ}ℝ_{D} itself. Since ^{ρ}ℂ_{D} is an algebraically closed field of cardinality 2^{ω}, it follows that ^{ρ}ℂ_{D} ≅ ℂ for every nonprincipal ultrafilter D over ω. Thus the fields ^{ρ}ℂ_{D} cannot be used to distinguish between the (possibly different) asymptotic cones of SL_{n}(ℂ); and it remains an open question whether the asymptotic cones of complex simple Lie groups depend on the chosen ultrafilter. However, there is an obvious generalisation of Theorem 2.8 to semisimple Lie groups.

Corollary 2.9. Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that ℝ-rank(S) ≥ 2 and let Γ be a uniform lattice in G. Let D and D′ be nonprincipal ultrafilters. If Con_{D}(Γ) and Con_{D′}(Γ) are homeomorphic, then the corresponding Robinson fields ^{ρ}ℝ_{D} and ^{ρ}ℝ_{D′} are isomorphic.

This follows from Theorem 2.8, together with the fact that the buildings at infinity of Con_{D}(Γ) and Con_{D′}(Γ) decompose into products of the buildings corresponding to the simple factors of G.

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 17

3. Invariants of linear orders

The next two sections will be devoted to the proof of Theorem 1.8. We will begin by reducing Theorem 1.8 to an analogous statement concerning the linearly ordered sets ω^{ω}/D, where D is a nonprincipal ultrafilter over ω. Definition 3.1. A filter over ω is a collection D of subsets of ω satisfying the following conditions:

(i) ω ∈ D.

(ii) If A, B ∈ D, then A ∩ B ∈ D.

(iii) If A ∈ D and A ⊆ B ⊆ ω, then B ∈ D. The filter D is nontrivial if and only if

(iv) ∅ ∉ D.

Let D be a nontrivial filter over ω. Then ≡_{D} is the equivalence relation defined on ω^{ω} = {f | f : ω → ω} by

f ≡_{D} g if and only if {n ∈ ω | f(n) = g(n)} ∈ D.

For each f ∈ ω^{ω}, we denote the corresponding ≡_{D}-equivalence class by f/D; and we let

ω^{ω}/D = {f/D | f ∈ ω^{ω}}

equipped with the partial order defined by

f/D < g/D if and only if {n ∈ ω | f(n) < g(n)} ∈ D.

As usual, we identify each natural number ℓ ∈ ω with the corresponding element c_{ℓ}/D ∈ ω^{ω}, defined by c_{ℓ}(n) = ℓ for all n ∈ ω. If D is a nonprincipal ultrafilter over ω, then ω^{ω}/D is a linear order. In this case, we define

(ω^{ω}/D)^{∗} = {g/D ∈ ω^{ω}/D | ℓ < g/D for all ℓ ∈ ω}.

As we shall now explain, Theorem 1.8 is an easy consequence of Theorem 3.3, which we shall prove in Section 4. Definition 3.2. Let L_{1}, L_{2} be linear orders.

(a) L_{1} ≈_{f} L_{2} if and only if L_{1} and L_{2} have nonempty isomorphic final segments.

(b) L_{1} ≈_{i} L_{2} if and only if L_{1} and L_{2} have nonempty isomorphic initial segments.

Theorem 3.3. If CH fails, then there exists a set {D_{α} | α < 2^{2^{ω}}} of nonprincipal ultrafilters over ω such that

(ω^{ω}/D_{α})^{∗} ≉_{i} (ω^{ω}/D_{β})^{∗}

for all α < β < 2^{2^{ω}}.

Proof of Theorem 1.8. For each nonprincipal ultrafilter D over ω, let

ρℝ_{D}^{∞} = {a ∈ ^{ρ}ℝ_{D} | a > ℓ for every ℓ ∈ ω};

and let E_{D} be the convex equivalence relation defined on ^{ρ}ℝ_{D}^{∞} by

a E_{D} b if and only if |a − b| < ℓ for some ℓ ∈ ω.

Let L_{D} = ^{ρ}ℝ_{D}^{∞}/E_{D}, equipped with the quotient linear ordering; and regard L_{D} ×ℤ as a linear ordering with respect to the usual lexicographical ordering, defined by (a_{1}, z_{1}) < (a_{2}, z_{2}) if and only if either:

• a_{1} < a_{2}; or

• a_{1} = a_{2} and z_{1} < z_{2}.

Then it is easily checked that

L_{D} × ℤ ≅ {g/D ∈ (ω^{ω}/D)^{∗} | g/D < ρ^{−n} for some n ≥ 1}.

Now suppose that A, B are nonprincipal ultrafilters over ω and that f : ^{ρ}ℝ_{A} → ^{ρ}ℝ_{B} is a field isomorphism. Since ^{ρ}ℝ_{A}, ^{ρ}ℝ_{B} are real closed, it follows that f is also orderpreserving. It is also clear that f[^{ρ}ℝ_{A}^{∞}] = ^{ρ}ℝ_{B}^{∞} and that f maps the equivalence relation E_{A} to the equivalence relation E_{B}. It follows that the ordered sets L_{A} and L_{B} are isomorphic; and hence that

(ω^{ω}/A)^{∗} ≈_{i} (ω^{ω}/B)^{∗}.

Consequently, if {D_{α} | α < 2^{κ}} is the set of nonprincipal ultrafilters over ω given by Theorem 3.3, then

ρℝ_{D_{α}} ≇ ^{ρ}ℝ_{D_{β}}

for all α < β < 2^{κ}. □

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 19

A similar argument shows that the corresponding fields {^{∗}ℝ_{D_{α}} | α < 2^{κ}} of nonstandard reals are also pairwise nonisomorphic. This improves a result of Roitman [26], who proved that it is consistent that there exist 2^{ω} pairwise nonisomorphic fields of nonstandard reals, each of the form ^{∗}ℝ_{D} for some nonprincipal ultrafilter D over ω.

Most of the remainder of this section will be devoted to the construction of a collection of extremely nonisomorphic linear orders, which will later be used as suitable “invariants” in the proof of Theorem 3.3. This construction is a special case of the more general techniques which are developed in Chapter III of Shelah [29]. In order to make this paper relatively self-contained, we have provided proofs of the relevant results. We assume that the reader is familiar with the basic properties of regular cardinals, singular cardinals, and stationary subsets of regular cardinals. (For example, see Sections 6 and 7 of Jech [19].) Definition 3.4. Suppose that I is a linear order and that ∅ ≠ A ⊆ I.

(a) A subset B ⊆ A is said to be cofinal in A if for all a ∈ A, there exists an element b ∈ B such that a ≤ b. The cofinality of A is defined to be

cf(A) = min{|B| | B is a cofinal subset of A}.

(b) A subset B ⊆ A is said to be coinitial in A if for all a ∈ A, there exists an element b ∈ B such that b ≤ a. The coinitiality of A is defined to be

coi(A) = min{|B| | B is a coinitial subset of A}.

Definition 3.5. Suppose that I is a linear order and that λ, θ ≥ ω are regular cardinals. Then (I_{1}, I_{2}) is a (λ, θ)-cut of I if the following conditions hold:

(a) I = I_{1} ∪ I_{2} and s < t for all s ∈ I_{1}, t ∈ I_{2}.

(b) cf(I_{1}) = λ.
(c) coi(I_{2}) = θ.

Definition 3.6. Suppose that J, L are linear orders. Then the order-preserving map φ : J → L is an invariant embedding if whenever (J_{1}, J_{2}) is a (λ, θ)-cut of J for some λ, θ > ω, then there does not exist an element x ∈ L such that φ(s) < x < φ(t) for all s ∈ J_{1}, t ∈ J_{2}.

In this case, φ is said to be an invariant cofinal embedding if φ[J] is cofinal in L and φ is said to be an invariant coinitial embedding if φ[J] is coinitial in L

20 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

Lemma 3.7. If λ > ω_{1} is a regular cardinal, then there exists a set {I_{α} | α < 2^{λ}} of linear orders satisfying the following conditions:

(a) cf(I_{α}) = |I_{α}| = λ.

(b) If α ≠ β and φ_{α} : I_{α} → L, φ_{β} : I_{β} → L′ are invariant cofinal embeddings,

then L ≉_{f} L′.

Proof. Applying Solovay’s Theorem, let {S_{τ} | τ < λ} be a partition of the stationary set

S = {δ < λ | cf(δ) = ω_{1}}

into λ pairwise disjoint stationary subsets. (For example, see Jech [19, 7.6].) Fix some subset X ⊂ λ. Then for each α < λ, we define

λ_{α}^{X} = \begin{cases}ω_{2}, & if α ∈ ⋃_{τ∈X} S_{τ} \\ ω_{1}, & otherwise; \end{cases}

and we define the linear order

I_{X} = {(α, β) | α < λ and β < λ_{α}^{X}}

by setting (α_{1}, β_{1}) < (α_{2}, β_{2}) if and only if either:

• α_{1} < α_{2}, or

• α_{1} = α_{2} and β_{1} > β_{2}.

Suppose that X ≠ Y ⊆ λ. Let L, L′ be linear orders and let φ_{X} : I_{X} → L, φ_{Y} : I_{Y} → L′ be invariant cofinal embeddings. Suppose that L ≈_{f} L′ and let ψ : M → M′ be an isomorphism between the final segments M, M′ of L, L′ respectively. For each δ < λ, let

M_{δ} = {m ∈ M | m < φ_{X}(γ, 0) for some γ < δ}

and

M′_{δ} = {m′ ∈ M′ | m′ < φ_{Y} (γ, 0) for some γ < δ}.

Then there exists a club C ⊆ λ such that ψ[M_{δ}] = M′_{δ} for all δ ∈ C. Without loss of generality, we can suppose that there exists an ordinal τ ∈ X ∖ Y. Choose δ ∈ C ∩ S_{τ} such that M_{δ} ≠ ∅. Since φ_{X} is an invariant embedding, it follows that coi(M ∖ M_{δ}) = ω_{2}. Similarly, coi(M′ ∖ M′_{δ}) = ω_{1}. But this is impossible since ψ[M ∖ M_{δ}] = M′ ∖ M′_{δ}. □

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 21

Note that in the statement of Theorem 3.8, κ is not necessarily regular. In Section 4, we shall apply Theorem 3.8 in the case when κ = 2^{ω} > ω_{1}. Theorem 3.8. If κ > ω_{1}, then there exists a set {J_{α} | α < 2^{κ}} of linear orders satisfying the following conditions:

(a) |J_{α}| = κ.

(b) coi(J_{α}) = cf(κ) + ω_{2}.

(c) If α ≠ β and φ_{α} : J_{α} → L, φ_{β} : J_{β} → L′ are invariant coinitial embeddings,

then L ≉_{i} L′.

Proof. Let ⟨κ_{i} | i < cf(κ)⟩ be a sequence of regular cardinals with each κ_{i} > ω_{1} such that:

(a) if κ is singular, then κ = sup_{i<cf(κ)} κ_{i}; and

(b) if κ is regular, then κ_{i} = κ for all i < cf(κ) = κ. In either case, we have that ∏_{i<cf(κ)} 2^{κ_{i}} = 2^{κ}. (For the case when κ is singular, see Jech [19, 6.5].) Let θ = cf(κ) + ω_{2} and let {S_{τ} | τ < cf(κ)} be a partition of the stationary set

S = {δ < θ | cf(δ) = ω_{1}}

into cf(κ) pairwise disjoint stationary subsets. Let h : θ → cf(κ) be the function defined by:

• δ ∈ S_{h(δ)} for all δ ∈ S; and

• h(ξ) = 0 for all ξ ∈ θ ∖ S.

For each i < cf(κ), let {I_{i,α} | α < 2^{κ_{i}}} be the set of linear orders of cardinality κ_{i} given by Lemma 3.7. For any ν ∈ ∏_{i<cf(κ)} 2^{κ_{i}}, we define the linear order

J_{ν} = {(α, x) | α < θ and x ∈ I_{h(α),ν(h(α))}}

by setting (α_{1}, x_{1}) < (α_{2}, x_{2}) if and only if either:

• α_{1} > α_{2}, or

• α_{1} = α_{2} and x_{1} < x_{2}.

Arguing as in the proof of Lemma 3.7, we see that the set {J_{ν} | ν ∈ ∏_{i<cf(κ)} 2^{κ_{i}}} of linear orders satisfies our requirements. □

Now suppose that 2^{ω} = κ > ω_{1} and let {J_{α} | α < 2^{κ}} be the set of linear orders given by Theorem 3.8. Then clearly Theorem 3.3 would follow if we could construct

22 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

a set {D_{α} | α < 2^{κ}} of nonprincipal ultrafilters over ω such that for each α < 2^{κ}, there exists an invariant coinitial embedding

φ_{α} : J_{α} → (ω^{ω}/D_{α})^{∗}.

Unfortunately this direct approach leads to serious technical difficulties which we have not yet been able to overcome. In order to avoid these difficulties, in the next section, we shall instead construct a set {D_{α} | α < 2^{κ}} of nonprincipal ultrafilters over ω such that the following condition is satisfied:

• For each α < 2^{κ} and each initial segment L of (ω^{ω}/D_{α})^{∗}, there exists an

invariant embedding φ : ω_{1} + J_{α} → L.

(Here ω_{1} +J_{α} is the linear order consisting of a copy of the ordinal ω_{1} followed by a copy of J_{α}. In particular, (ω_{1}, J_{α}) is an (ω_{1}, cf(κ) + ω_{2})-cut of ω_{1} + J_{α}.) Of course, this is not enough to ensure that

(ω^{ω}/D_{α})^{∗} ≉_{i} (ω^{ω}/D_{β})^{∗}

for all β ≠ α, since the above condition does not rule out the possibility that there also exists an invariant embedding

ψ : ω_{1} + J_{β} → (ω^{ω}/D_{α})^{∗}.

Fix some α < 2^{κ} and let C_{α} be the set of β < 2^{κ} such that there exists an invariant embedding

ψ_{β} : ω_{1} + J_{β} → (ω^{ω}/D_{α})^{∗}.

For each β ∈ C_{α}, let (A_{β}, B_{β}) be the (ω_{1}, cf(κ) + ω_{2})-cut of (ω^{ω}/D_{α})^{∗} defined by

A_{β} = {g/D_{α} ∈ (ω^{ω}/D_{α})^{∗} | g/D_{α} < ψ_{β}(t) for some t ∈ ω_{1}}

and

B_{β} = {g/D_{α} ∈ (ω^{ω}/D_{α})^{∗} | g/D_{α} > ψ_{β}(t) for some t ∈ J_{β}}.

Then Theorem 3.8 implies that (A_{β}, B_{β}) ≠ (A_{γ}, B_{γ}) for all β ≠ γ ∈ C_{α}. Since the following result implies that the number of (ω_{1}, cf(κ) + ω_{2})-cuts of (ω^{ω}/D_{α})^{∗} is at most 2^{ω} = κ, it follows that |C_{α}| ≤ κ. This implies that there exists a subset W ⊆ 2^{κ} of cardinality 2^{κ} such that

(ω^{ω}/D_{α})^{∗} ≉_{i} (ω^{ω}/D_{β})^{∗}

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 23

for all α ≠ β ∈ W.

While the basic idea of Theorem 3.9 is implicitly contained in Section VIII.0 of Shelah [28] and Chapter III of Shelah [29], the result does not seem to have been explicitly stated anywhere in the literature. Theorem 3.9. Suppose that I is a linear order and that θ ≠ λ are regular cardinals. Then the number of (λ, θ)-cuts of I is at most |I|.

Proof. We shall just consider the case when λ < θ. Suppose that I is a counterexample of minimal cardinality and let {(A_{i}, B_{i}) | i < |I|^{+}} be a set of |I|^{+} distinct (λ, θ)-cuts of I. Let cf(|I|) = κ and express I = ⋃_{γ<κ} I_{γ} as a smooth strictly increasing union of substructures such that |I_{γ}| < |I| for all γ < κ.

First suppose that κ ≠ θ, λ. Then for each i < |I|^{+}, there exists an ordinal γ_{i} < κ such that A_{i} ∩ I_{γ_{i}} is cofinal in A_{i} and B_{i} ∩ I_{γ_{i}} is coinitial in B_{i}. It follows that there exists a subset X ⊆ |I|^{+} of cardinality |I|^{+} and a fixed ordinal γ < κ such that γ_{i} = γ for all i ∈ X. But this means that {(A_{i} ∩ I_{γ}, B_{i} ∩ I_{γ}) | i < X} is a set of |I|^{+} distinct (λ, θ)-cuts of I_{γ}, which contradicts the minimality of |I|.

Next suppose that κ = λ. Once again, for each i < |I|^{+}, there exists an ordinal γ_{i} < κ such that B_{i} ∩ I_{γ_{i}} is coinitial in B_{i}; and there exists a subset X ⊆ |I|^{+} of cardinality |I|^{+} and a fixed ordinal γ < κ such that γ_{i} = γ for all i ∈ X. Arguing as in the previous paragraph, we can suppose that for each i ∈ X, A_{i} ∩ I_{γ} is not cofinal in A_{i}. For each i ∈ X, choose an element a_{i} ∈ A_{i} ∖ I_{γ} such that s < a_{i} < t for all s ∈ A_{i} ∩ I_{γ} and t ∈ B_{i}. Suppose that i ≠ j ∈ X. Then we can suppose that B_{i} ⊊ B_{j}. Since B_{j} ∩ I_{γ} is coinitial in B_{j}, it follows that there exists an element

c ∈ (B_{j} ∖ B_{i}) ∩ I_{γ} ⊆ A_{i} ∩ I_{γ}.

But this means that a_{j} < c < a_{i} and so {a_{i} | i ∈ X} is a set of |I|^{+} distinct elements of I, which is a contradiction. A similar argument handles the case when κ = θ. This completes the proof of Theorem 3.9. □

4. Constructing ultrafilters

In this section, we shall prove Theorem 3.3. Our construction of the required set {D_{α} | α < 2^{2^{ω}}} of nonprincipal ultrafilters makes use of the techniques developed in Section VI.3 of Shelah [28].

24 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

Definition 4.1. Let D be a filter over ω and let

I_{D} = {X ⊆ ω | ω ∖ X ∈ D}

be the corresponding dual ideal. If A, B ⊆ ω, then we define

A ⊂ B mod D if and only if A ∖ B ∈ I_{D}

and

A = B mod D if and only if (A ∖ B) ∪ (B ∖ A) ∈ I_{D}.

Definition 4.2. Suppose that D is a filter over ω and that G ⊆ ω^{ω} is a family of surjective functions. Then G is independent mod D if for all distinct g_{1}, ⋯, g_{ℓ} ∈ G and all (not necessarily distinct) j_{1}, ⋯, j_{ℓ} ∈ ω,

{n ∈ ω | g_{k}(n) = j_{k} for all 1 ≤ k ≤ ℓ} ≠ ∅ mod D.

(Of course, this condition implies that D is a nontrivial filter.)

Suppose that G is independent mod D and that |G| = κ. Let I be any linear order of cardinality κ and suppose that G = {f_{t} | t ∈ I} is indexed by the elements of I. For each s < t ∈ I, let

B_{s,t} = {n ∈ ω | f_{s}(n) < f_{t}(n)}.

Then it is easily checked that D ∪ {B_{s,t} | s < t ∈ I} generates a nontrivial filter D^{+}. (For example, see the proof of Lemma 4.7.) It follows that if D ⊇ D^{+} is an ultrafilter, then we can define an order-preserving map φ : I → ω^{ω}/D by φ(t) = f_{t}/D. However, if we wish φ to be an invariant embedding, then we need to be able to control the behaviour of arbitary elements g/D ∈ ω^{ω}/D. The next few paragraphs will introduce the techniques which will enable us to accomplish this. Definition 4.3. Suppose that G ⊆ ω^{ω} is a family of surjective functions.

(a) F I(G) is the set of functions h satisfying the following conditions: (i) dom h is a finite subset of G; (ii) ran h ⊂ ω.

(b) For each h ∈ F I(G), let

A_{h} = {n ∈ ω | g(n) = h(g) for all g ∈ dom h}.

(c) F I_{s}(G) = {A_{h} | h ∈ F I(G)}.

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 25

Lemma 4.4. Suppose that D is a filter over ω and that G ⊆ ω^{ω} is a family of surjective functions.

(a) G is independent mod D if and only if A_{h} ≠ ∅ mod D for every h ∈ F I(G).

(b) If G is independent mod D, then there exists a maximal filter D^{∗} ⊇ D modulo which G is independent.

(c) If G is independent mod D and X ⊆ ω, then there exists a finite subset F ⊆ G such that G ∖ F is independent modulo either the filter generated by D ∪ {X} or the filter generated by D ∪ {ω ∖ X}.

Proof. These are the statements of Claims 3.15(4), 3.15(3) and 3.3 from Shelah [28, Chapter VI]. □

Now suppose that G ⊆ ω^{ω} is a family of surjective functions and that D is a maximal filter over ω modulo which G is independent. Then A is said to be a partition mod D if the following conditions are satisfied:

• A ≠ ∅ mod D for all A ∈ A;

• A ∩ A′ = ∅ mod D for all A ≠ A′ ∈ A;

• for all B ∈ P(ω) with B ≠ ∅ mod D, there exists A ∈ A such that

A ∩ B ≠ ∅ mod D. The subset B ⊆ ω is said to be based on A if for every A ∈ A, either A ⊆ B mod D or A ∩ B = ∅ mod D. By Claim 3.17(1) [28, Chapter VI], for every subset B ⊆ ω, there exists a partition A mod D such that

(i) B is based on A; and (ii) A ⊆ F I_{s}(G).

Furthermore, by Claim 3.17(5) [28, Chapter VI], A is necessarily countable and so there exists a countable subset G_{0} ⊆ G such that A ⊆ F I_{s}(G_{0}). In this case, we say that B is supported by F I_{s}(G_{0}) mod D.

The next lemma summarises the properties of supports that we shall require later in this section. Lemma 4.5. Suppose that G ⊆ ω^{ω} is a family of surjective functions and that D is a maximal filter over ω modulo which G is independent.

(a) F I_{s}(G) is dense mod D; i.e. for every B ⊆ ω with B ≠ ∅ mod D, there exists A_{h} ∈ F I_{s}(G) such that A_{h} ⊆ B mod D.

26 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

(b) For each B ⊆ ω, there exists a countable subset G_{0} ⊆ G such that B is supported by F I_{s}(G_{0}) mod D.

(c) Suppose that G = G_{1} ⊔ G_{2} and that A ⊆ ω is supported by F I_{s}(G_{1}) mod D. If h ∈ F I(G) and A_{h} ⊆ A mod D, then A_{h_{1}} ⊆ A mod D, where h_{1} = h ↾ G_{1}.

Proof. We have already discussed clause (b). Clauses (a) and (c) are the statements of Claims 3.17(1) and 3.17(4) from Shelah [28, Chapter VI]. □

The following lemma will ensure that our construction concentrates on the initial segments of (ω^{ω}/D)^{∗}. Lemma 4.6. Suppose that G ⊆ ω^{ω} is a family of surjective functions and that D is a maximal filter over ω modulo which G is independent. Suppose also that g ∈ ω^{ω} is a function such that ℓ < g/D for every ℓ ∈ ω. Then f/D < g/D for every f ∈ G.

Proof. This is Claim 3.19(1) from Shelah [28, Chapter VI]. □

Finally the next lemma is the key to our construction of the required set of ultrafilters. Lemma 4.7. Suppose that G ⊔ G^{∗} ⊆ ω^{ω} is a family of surjective functions and that D is a maximal filter over ω modulo which G ⊔ G^{∗} is independent. Suppose that I is a linear order and that G = {f_{t} | t ∈ I} is indexed by the elements of I. Then there exists a filter D^{+} ⊇ D over ω which satisfies the following conditions:

(a) If s ∈ I and ℓ ∈ ω, then ℓ < f_{s}/D^{+}.

(b) If s < t ∈ I, then f_{s}/D^{+} < f_{t}/D^{+}.

(c) Suppose that (I_{1}, I_{2}) is a (λ, θ)-cut of I such that λ, θ > ω. Then for every ultrafilter U ⊇ D^{+} over ω, there does not exist a function g ∈ ω^{ω} such that

f_{s}/U < g/U < f_{t}/U

for all s ∈ I_{1}, t ∈ I_{2}.

(d) D^{+} is a maximal filter over ω modulo which G^{∗} is independent.

Proof. For each t ∈ I and ℓ ∈ ω, let

A_{ℓ,t} = {n ∈ ω | ℓ < f_{t}(n)}.

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 27

For each pair s < t ∈ I, let

B_{s,t} = {n ∈ ω | f_{s}(n) < f_{t}(n)}.

Finally for each pair r < s ∈ I and each function g ∈ ω^{ω} such that g^{−1}(ℓ) is supported by F I_{s}(G^{∗} ⊔ {f_{t} | t ∈ I ∖ [r, s]}) mod D for all ℓ ∈ ω, let

C_{g,r,s} = {n ∈ ω | g(n) < f_{r}(n) or f_{s}(n) < g(n)}.

Let E be the filter on ω generated by D, together with all of the sets A_{ℓ,t}, B_{s,t}, C_{g,r,s} defined above. (Note that the next claim implies that E is a nontrivial filter.)

Claim 4.8. If h ∈ F I(G^{∗}), then A_{h} ≠ ∅ mod E.

Assuming Claim 4.8, we shall now complete the proof of Lemma 4.7. Applying Lemma 4.4(a), G^{∗} is independent mod E. Let D^{+} ⊇ E be a maximal such filter.

Since A_{ℓ,t} ∈ D^{+} for each t ∈ I and ℓ ∈ ω, it follows that ℓ < f_{s}/D^{+} and so clause 4.7(a) holds. Similarly, since B_{s,t} ∈ D^{+} for each s < t ∈ I, it follows that clause 4.7(b) holds. Finally suppose that (I_{1}, I_{2}) is a (λ, θ)-cut of I such that λ, θ > ω and that g ∈ ω^{ω}. By Lemma 4.5(b), for each ℓ ∈ ω, there exists a countable subset G_{ℓ} ⊆ G ⊔ G^{∗} such that g^{−1}(ℓ) is supported by F I_{s}(G_{ℓ}) mod D. Since λ, θ > ω, it follows that there exist r ∈ I_{1} and s ∈ I_{2} such that g^{−1}(ℓ) is supported by F I_{s}(G^{∗} ⊔ {f_{t} | t ∈ I ∖ [r, s]}) mod D for all ℓ ∈ ω and hence C_{g,r,s} ∈ D^{+}. This implies that if U ⊇ D^{+} is an ultrafilter over ω, then either g/U < f_{r}/U or f_{s}/U < g/U. Thus clause 4.7(c) also holds.

Thus it only remains to prove Claim 4.8. Suppose that h ∈ F I(G^{∗}). Then it is enough to prove that

A_{h} ∩ ⋂_{i≤a} A_{ℓ_{i},t_{i}} ∩ ⋂_{i<j≤a} B_{t_{i},t_{j}} ∩ ⋂_{k≤b} C_{g_{k},r_{k},s_{k}} ≠ ∅ mod D

in the case when the following conditions are satisfied:

• t_{0} < t_{1} < ⋯ < t_{a}.

• If k ≤ b, then r_{k}, s_{k} ∈ {t_{i} | i ≤ a}.

Let T = {f_{t_{i}} | i ≤ a}. We shall define a sequence of functions h_{m} ∈ F I(G ⊔ G^{∗}) inductively for m ∈ ω so that the following conditions are satisfied:

(1) h_{0} = h and h_{m} ⊆ h_{m+1}.

(2) dom h_{m} ∩ T = ∅.

28 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

(3) If h^{∗} ∈ F I(T ) and k ≤ b, then one of the following occurs for almost all m, (i) there exists ℓ ∈ ω such that A_{h_{m}∪h^{∗}} ⊆ g_{k}^{−1}(ℓ) mod D; or (ii) A_{h_{m}∪h^{∗}} ∩ g_{k}^{−1}(ℓ) = ∅ mod D for all ℓ ∈ ω.

(Clearly if (i) occurs, then there exists a fixed ℓ such that A_{h_{m}∪h^{∗}} ⊆ g_{k}^{−1}(ℓ) mod D for almost all m.) To see that the induction can be carried out, first fix an enumeration of the countably many pairs h^{∗}, k that must be dealt with. Now suppose that h_{m} has been defined and that we must next deal with the pair h^{∗}, k. There are two cases to consider. First suppose that there exists ℓ ∈ ω such that A_{h_{m}∪h^{∗}} ∩ g_{k}^{−1}(ℓ) ≠ ∅ mod D. By Lemma 4.5(a), there exists h̃ ∈ F I(G ⊔ G^{∗}) such that

Ah̃ ⊆ A_{h_{m}∪h^{∗}} ∩ g_{k}^{−1}(ℓ) mod D.

Clearly we must have that h_{m} ∪ h^{∗} ⊆ h̃; and in this case, we set

h_{m+1} = h̃ ↾ ((G ⊔ G^{∗}) ∖ T ).

Otherwise, we must have that A_{h_{m}∪h^{∗}} ∩ g_{k}^{−1}(ℓ) = ∅ mod D for all ℓ ∈ ω; and in this case, we set h_{m+1} = h_{m}.

Now fix some k ≤ b. Let r_{k} = t_{ι(k)} and s_{k} = t_{τ(k)}. Let

T_{k} = {f_{t_{i}} | i ∉ [ι(k), τ(k)]}.

Suppose that h^{∗} ∈ F I(T ). Then for all sufficiently large m, either:

(i) there exists ℓ ∈ ω such that A_{h_{m}∪h^{∗}} ⊆ g_{k}^{−1}(ℓ) mod D; or

(ii) A_{h_{m}∪h^{∗}} ∩ g_{k}^{−1}(ℓ) = ∅ mod D for all ℓ ∈ ω.

First suppose that (i) holds. Since g_{k}^{−1}(ℓ) is supported by

F I_{s}(G^{∗} ⊔ {f_{t} | t ∈ I ∖ [r_{k}, s_{k}]}),

Lemma 4.5(c) implies that:

(i)′ there exists ℓ ∈ ω such that A_{h_{m}∪(h^{∗}↾T_{k})} ⊆ g_{k}^{−1}(ℓ) mod D for almost all m. Similarly, if (ii) holds, then Lemma 4.5(c) implies that:

(ii)′ for almost all m, A_{h_{m}∪(h^{∗}↾T_{k})} ∩ g_{k}^{−1}(ℓ) = ∅ mod D for all ℓ ∈ ω. Let ψ_{k} : ω^{T_{k}} → ω ∪ {∞} be the function defined by

ψ_{k}(h^{∗} ↾ T_{k}) = \begin{cases}ℓ & if (i)′ holds; \\ ∞ & if (ii)′ holds. \end{cases}

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 29

For each infinite set W ⊆ ω, let W^{(T )} be the set of functions h^{∗} : T → W such that h^{∗}(f_{t_{i}}) < h^{∗}(f_{t_{j}}) for all i < j ≤ a; and for each k ≤ b, let φ_{k} : ω^{(T )} → 3 be the function defined by

φ_{k}(h^{∗}) = \begin{cases}0 & if ψ_{k}(h^{∗} ↾ T_{k}) < h^{∗}(f_{t_{ι(k)}}); \\ 1 & if ψ_{k}(h^{∗} ↾ T_{k}) > h^{∗}(f_{t_{τ(k)}}); \\ 2 & otherwise. \end{cases}

By Ramsey’s Theorem, there exists an infinite set W ⊆ ω such that φ_{k} ↾ W^{(T )} is a constant function for all k ≤ b.

Claim 4.9. For each k ≤ b, either φ_{k} ↾ W^{(T )} ≡ 0 or φ_{k} ↾ W^{(T )} ≡ 1.

Proof. Suppose that φ_{k} ↾ W^{(T )} ≡ 2, so that

h^{∗}(f_{t_{ι(k)}}) ≤ ψ_{k}(h^{∗} ↾ T_{k}) ≤ h^{∗}(f_{t_{τ(k)}})

for all h^{∗} ∈ W^{(T )}. Let |{j | ι(k) ≤ j ≤ τ(k)}| = p and let h′ : T_{k} → W be a strictly increasing function such that

|{w ∈ W | h′(f_{t_{ι(k)−1}}) < w < h′(f_{t_{τ(k)+1}})}| = 2p.

Then we can extend h′ to a function h^{∗} ∈ W^{(T )} such that either

ψ_{k}(h^{∗} ↾ T_{k}) = ψ_{k}(h′) < h^{∗}(f_{t_{ι(k)}})

or

ψ_{k}(h^{∗} ↾ T_{k}) = ψ_{k}(h′) > h^{∗}(f_{t_{τ(k)}}),

which is a contradiction. □

Choose an increasing sequence j_{0} < j_{1} < ⋯ < j_{a} of elements of W such that j_{i} > ℓ_{i} for each i ≤ a; and let h^{∗} ∈ F I(T ) be the function defined by h^{∗}(f_{t_{i}}) = j_{i} for each i ≤ a. To complete the proof of Claim 4.8, it is enough to show that for almost all m,

A_{h_{m}∪h^{∗}} ⊆ A_{h} ∩ ⋂_{i≤a} A_{ℓ_{i},t_{i}} ∩ ⋂_{i<j≤a} B_{t_{i},t_{j}} ∩ ⋂_{k≤b} C_{g_{k},r_{k},s_{k}} mod D.

It is clear that A_{h_{m}} ⊆ A_{h} for all m and that

A_{h^{∗}} ⊆ ⋂_{i≤a} A_{ℓ_{i},t_{i}} ∩ ⋂_{i<j≤a} B_{t_{i},t_{j}}.

30 LINUS KRAMER, SAHARON SHELAH, KATRIN TENT, AND SIMON THOMAS

Finally let k ≤ b. If φ_{k}(h^{∗}) = 0, then ψ_{k}(h^{∗} ↾ T_{k}) < h^{∗}(f_{t_{ι(k)}}) < ∞ and so for almost all m,

A_{h_{m}∪h^{∗}} ⊆ {n ∈ ω | g_{k}(n) = ψ_{k}(h^{∗} ↾ T_{k}) < h^{∗}(f_{t_{ι(k)}}) = f_{t_{ι(k)}}(n)} mod D.

Similarly, if φ_{k}(h^{∗}) = 1 and ψ_{k}(h^{∗} ↾ T_{k}) < ∞, then for almost all m,

A_{h_{m}∪h^{∗}} ⊆ {n ∈ ω | f_{t_{τ(k)}}(n) = h^{∗}(f_{t_{τ(k)}}) < ψ_{k}(h^{∗} ↾ T_{k}) = g_{k}(n)} mod D.

On the other hand, if φ_{k}(h^{∗}) = 1 and ψ_{k}(h^{∗} ↾ T_{k}) = ∞, then for almost all m,

A_{h_{m}∪h^{∗}} ∩ ⋃_{ℓ≤h^{∗}(f_{t_{τ(k)}} )} g_{k}^{−1}(ℓ) = ∅ mod D

and so

A_{h_{m}∪h^{∗}} ⊆ {n ∈ ω | f_{t_{τ(k)}}(n) = h^{∗}(f_{t_{τ(k)}}) < g_{k}(n)} mod D.

Hence, in every case, we have that for almost all m,

A_{h_{m}∪h^{∗}} ⊆ C_{g_{k},r_{k},s_{k}} mod D.

This completes the proof of Lemma 4.7. □

It is now straightforward to construct a set of ultrafilters satisfying the conclusion of Theorem 3.3.

Proof of Theorem 3.3. Suppose that 2^{ω} = κ > ω_{1}. Let {J_{α} | α < 2^{κ}} be the set of linear orders given by Theorem 3.8; and for each α < κ, let I_{α} = ω_{1} + J_{α}. Fix some α < κ. To simplify notation, let I_{α} = I. Then the corresponding ultrafilter D_{α} = D is constructed as follows.

Let F_{0} = {X ⊆ ω | |ω ∖X| < ω} be the Fréchet filter over ω. By Theorem 1.5(1) of Shelah [28, Appendix], there exists a family G ⊆ ω^{ω} of surjective functions of cardinality κ = 2^{ω} such that G is independent mod F_{0}. Let P(ω) = {X_{μ} | μ < κ} be an enumeration of the powerset of ω and “enumerate” G as {f_{ξ}^{μ} | μ, ξ < κ}. We shall define by induction on μ < κ

• a decreasing sequence of subsets G_{μ} ⊆ {f_{ξ}^{ν} | ξ < κ and μ ≤ ν < κ}; and • an increasing sequence of filters D_{μ} over ω

such that the following conditions are satisfied:

(a) G_{0} = G.

(b) |{f_{ξ}^{ν} | ξ < κ and μ ≤ ν < κ} ∖ G_{μ}| ≤ |μ| + ω.

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 31

(c) D_{μ} is a maximal filter modulo which G_{μ} is independent.

(d) Either X_{μ} ∈ D_{μ+1} or ω ∖ X_{μ} ∈ D_{μ+1}. When μ = 0, we let D_{0} ⊇ F_{0} be a maximal filter modulo which G_{0} = G is independent. If μ is a limit ordinal, then we define G_{μ} = ⋂_{ν<μ} G_{ν} and let D_{μ} ⊇ ⋃_{ν<μ} D_{ν} be a maximal filter modulo which G_{μ} is independent. Finally suppose that μ = ν + 1. By Lemma 4.4, there exists a finite subset F_{ν} ⊆ G_{ν} such that G_{ν} ∖ F_{ν} is independent modulo either the filter generated by D_{ν} ∪ {X_{ν}} or the filter generated by D_{ν} ∪ {ω ∖ X_{ν}}. Without loss of generality, suppose that G_{ν} ∖ F_{ν} is independent modulo the filter D′_{ν} generated by D_{ν} ∪ {X_{ν}}; and let E_{ν} ⊇ D′_{ν} be a maximal filter modulo which G_{ν} ∖ F_{ν} is independent. Let

G_{μ} = {f_{ξ}^{τ} ∈ G_{ν} ∖ F_{ν} | μ ≤ τ < κ}.

Note that

H_{ν} = {f_{ξ}^{τ} ∈ G_{ν} ∖ F_{ν} | τ = ν}

has cardinality κ. Hence we can re-index H_{ν} as H_{ν} = {f_{t}^{ν} | t ∈ I}. By Lemma 4.7, there exists a filter D_{μ} ⊇ E_{ν} which satisfies the following conditions:

(1) If s ∈ I and ℓ ∈ ω, then ℓ < f_{s}^{ν}/D_{μ}.

(2) If s < t ∈ I, then f_{s}^{ν}/D_{μ} < f_{t}^{ν}/D_{μ}.

(3) Suppose that (I_{1}, I_{2}) is a (λ, θ)-cut of I such that λ, θ > ω. Then for every ultrafilter U ⊇ D_{μ} over ω, there does not exist a function g ∈ ω^{ω} such that

f_{s}^{ν}/U < g/U < f_{t}^{ν}/U

for all s ∈ I_{1}, t ∈ I_{2}.

(4) D_{μ} is a maximal filter over ω modulo which G_{μ} is independent. Finally let D = ⋃_{μ<κ} D_{μ}. By clause (d), D is an ultrafilter.

Claim 4.10. If L is an initial segment of (ω^{ω}/D)^{∗}, then there exists μ < κ such that {f_{t}^{μ}/D | t ∈ I} ⊆ L.

Proof of Claim 4.10. Let g/D ∈ L. Then there exists μ < κ such that

A_{ℓ} = {n ∈ ω | ℓ < g(n)} ∈ D_{μ}

for all ℓ ∈ ω. Since D_{μ} is a maximal filter modulo which G_{μ} is independent, Lemma 4.6 implies that f/D_{μ} < g/D_{μ} for all f ∈ G_{μ}. Hence {f_{t}^{μ}/D | t ∈ I} ⊆ L. □

From now on, it is necessary to write D_{α}, I_{α}, etc.

Claim 4.11. Fix some α < 2^{κ}. Then the set

E_{α} = {β < 2^{κ} | (ω^{ω}/D_{α})^{∗} ≈_{i} (ω^{ω}/D_{β})^{∗}}

has cardinality at most κ.

Proof of Claim 4.11. Suppose that |E_{α}| ≥ κ^{+}. For each β ∈ E_{α}, let L_{β}, M_{β} be initial segments of (ω^{ω}/D_{β})^{∗}, (ω^{ω}/D_{α})^{∗} respectively such that there exists an isomorphism φ_{β} : L_{β} → M_{β}. By Claim 4.10, for each β ∈ E_{α}, there exists μ_{β} < κ such that

R_{β} = {f_{t}^{μ_{β}}/D_{β} | t ∈ I_{β}} ⊆ L_{β}.

Recall that I_{β} = ω_{1}+J_{β}. Let S_{β} = {f_{t}^{μ_{β}}/D_{β} | t ∈ ω_{1}} and T_{β} = {f_{t}^{μ_{β}}/D_{β} | t ∈ J_{β}}. Let θ = cf(κ) + ω_{2}. Then (φ_{β}[S_{β}], φ_{β}[T_{β}]) determines the (ω_{1}, θ)-cut (A_{β}, B_{β}) of (ω^{ω}/D_{α})^{∗} defined by

A_{β} = {g/D_{α} ∈ (ω^{ω}/D_{α})^{∗} | g/D_{α} < φ_{β}(s) for some s ∈ S_{β}}

and

B_{β} = {g/D_{α} ∈ (ω^{ω}/D_{α})^{∗} | g/D_{α} > φ_{β}(t) for some t ∈ T_{β}}.

By Theorem 3.9, there exist β ≠ γ ∈ E_{α} such that (A_{β}, B_{β}) = (A_{γ}, B_{γ}). But this is impossible, since we can define invariant coinitial embeddings of J_{β}, J_{γ} into B_{β} = B_{γ} by c ↦ φ_{β}(f_{c}^{μ_{β}}) and d ↦ φ_{γ}(f_{d}^{μ_{γ}}) respectively, which contradicts Theorem 3.8. □

Clearly Claim 4.11 implies that there exists a subset W ⊆ 2^{κ} of cardinality 2^{κ} such that

(ω^{ω}/D_{α})^{∗} ≉_{i} (ω^{ω}/D_{β})^{∗}

for all α ≠ β ∈ W. This completes the proof of Theorem 3.3. □

5. Asymptotic cones under CH

In this section, we shall prove Theorem 1.10. Let Γ be an infinite finitely generated group and let d be the word metric with respect to some finite generating

ASYMPTOTIC CONES OF FINITELY PRESENTED GROUPS 33

set. Then the main point is that each asymptotic cone Con_{D}(Γ) can be uniformly constructed from an associated ultraproduct

∏ M_{n}/D,

where each M_{n} is a suitable countable structure for a fixed countable first-order language L. If CH holds, then ∏ M_{n}/D is a saturated structure of cardinality ω_{1} and hence is determined up to isomorphism by its complete first-order theory T_{D}. Consequently, if CH holds, then since there are at most 2^{ω} possibilities for T_{D}, there are also at most 2^{ω} possibilities for ∏ M_{n}/D and hence also for Con_{D}(Γ). Definition 5.1. Let L be the first-order language consisting of the following symbols:

(a) the binary relation symbol R_{q} for each 0 < q ∈ ℚ; and

(b) the constant symbol e. Definition 5.2. For each n ≥ 1, M_{n} is the L-structure with universe Γ such that:

(a) M_{n} ⊨ R_{q}(x, y) if and only if d(x, y) ≤ qn; and

(b) e is the identity element of Γ. Definition 5.3. For each nonprincipal ultrafilter D over ω, let T_{D} be the complete first-order theory of ∏ M_{n}/D. Theorem 5.4. If D, D′ are nonprincipal ultrafilters over ω, then the following are equivalent.

(a) ∏ M_{n}/D ≅ ∏ M_{n}/D′.

(b) Con_{D}(Γ) is isometric to Con_{D′}(Γ).

Proof. We shall begin by describing how the asymptotic cone Con_{D}(Γ) can be uniformly constructed from the ultraproduct

∏ M_{n}/D = ⟨X; R_{q}, e⟩.

First define M_{D}^{0} to be the set of those x ∈ X such that

∏ M_{n}/D ⊨ R_{q}(x, e)

for some q > 0. Next define an equivalence relation ≈ on M_{D}^{0} by

x ≈ y if and only if ∏ M_{n}/D ⊨ R_{q}(x, y) for all q > 0.

For each x ∈ M_{D}^{0}, let ⟨x⟩ denote the corresponding ≈-class and let

C_{D} = {⟨x⟩ | x ∈ M_{D}^{0}}.

Then we can define a metric d_{D} on C_{D} by

d_{D}(⟨x⟩, ⟨y⟩) = inf{q | ∏ M_{n}/D ⊨ R_{q}(x, y)}.

It is easily checked that ⟨C_{D}, d_{D}⟩ is isometric to the asymptotic cone Con_{D}(Γ). Consequently, if ∏ M_{n}/D ≅ ∏ M_{n}/D′, then Con_{D}(Γ) is isometric to Con_{D′}(Γ).

It is also easily checked that ∏ M_{n}/D consists of 2^{ω} disjoint isomorphic copies of M_{D}^{0} and that each ≈-class has cardinality 2^{ω}. It follows that any isometry between Con_{D}(Γ) and Con_{D′}(Γ) can be lifted to a corresponding isomorphism between ∏ M_{n}/D and ∏ M_{n}/D′. □

Corollary 5.5. Assume CH. If D, D′ are nonprincipal ultrafilters over ω, then the following are equivalent.

(a) T_{D} = T_{D′}.

(b) Con_{D}(Γ) is isometric to Con_{D′}(Γ).

Proof. As we mentioned earlier, if CH holds, then ∏ M_{n}/D is a saturated structure of cardinality ω_{1} and hence is determined up to isomorphism by its complete firstorder theory T_{D}. (For example, see Chang-Keisler [10].) □

This completes the proof of Theorem 1.10.

References

[1] C. Bennett, Affine Λ-buildings. I., Proc. London Math. Soc. 68 (1994), 541–576.

[2] A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111–122.

[3] A. Borel, Linear algebraic groups, 2nd ed. Springer-Verlag, New York, 1991.

[4] A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150. Compléments à l’article: Groupes réductifs. Inst. Hautes Études Sci. Publ. Math. 41 (1972), 253–276.

[5] N. Bourbaki, Groupes et algèbres de Lie, ch. IV–VI, Paris, Hermann 1968.

[6] M. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Springer-Verlag, Berlin, 1999.

[7] K. Brown, Buildings, Springer-Verlag, New York (1989).

[8] F. Bruhat and J. Tits, Groupes réductifs dur un corp local, I. Données radicielles valuées, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–252.

[9] F. Bruhat and J. Tits, Schémas en groupes et immeubles des groupes classiques sur un corp local, Bull. Soc. Math. Fr. 112 (1984), 259–301.

[10] C. C. Chang and H. J. Keisler, Model Theory, North-Holland, Amsterdam, 1973.

[11] B. Diarra, Ultraproduits ultramétrique de corps valués, Ann. Sci. Univ. Clermont-Ferrand II Math. 22 (1984), 1–37.

[12] L. van den Dries and A. J. Wilkie, On Gromov’s theorem concerning groups of polynomial growth and elementary logic, J. Algebra 89 (1984), 349–374.

[13] A. Dyubina and I. Polterovich, Explicit constructions of universal ℝ-trees and asymptotic geometry of hyperbolic spaces, Bull. London Math. Soc. 33 (2001), 727–734.

[14] P. Eberlein, Geometry of nonpositively curved manifolds, Univ. Chicago Press, Chicago, 1996.

[15] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. IHES 53 (1981), 53–73.

[16] M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, vol. 2 (G. Niblo and M. Roller, eds.), LMS Lecture Notes Series 182, Cambridge University Press, 1993.

[17] F. Gosshans, Semi-simple algebraic groups defined over a real closed field, Amer. J. Math. 94 (1972), 473–485.

[18] P. de la Harpe, Topic in Geometric Group Theory, University of Chicago Press, Chicago and London, 2000.

[19] T. Jech, Set Theory, Academic Press, New York, 1978.

[20] B. Kleiner and B. Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 115–197.

[21] L. Kramer and K. Tent, Ultrapowers of Lie groups and symmetric spaces, preprint, Würzburg (2003).

[22] A. Lightstone and A. Robinson, Nonarchimedian fields and asymptotic expansions, North-Holland, New York, 1975.

[23] A. Parreau, Dégénérescences de sous-groupes discrets de groupes de Lie semisimples et actions de groupes sur les immeubles affines, Thèse, Université de Paris-Sud XI Orsay, 2000.

[24] V. Pestov, On a valuation field invented by A. Robinson and certain structures connected with it, Israel J. Math. 74 (1991), 65–79.

[25] S. Prieß-Crampe, Angeordnete Strukturen: Gruppen, Körper, projektive Ebenen, Springer-Verlag, Berlin, 1983.

[26] J. Roitman, Nonisomorphic hyper-real fields from nonisomorphic ultrapowers, Math. Z. 181 (1982), 93–96.

[27] M. Ronan, Lectures on buildings, Academic Press, Inc., Boston, MA, 1989.

[28] S. Shelah, Classification Theory and the Number of Non-Isomorphic Models, 2nd. Edition, North-Holland, Amsterdam, 1990.

[29] S. Shelah, Nonstructure Theory, in preparation.

[30] T.A. Springer, Linear algebraic groups, 2nd. ed. Birkhäuser Boston, Inc., Boston, MA (1998).

[31] S. Thomas and B.Velickovic, Asymptotic cones of finitely generated groups, Bull. London Math. Soc 32 (2000), 203–208.

[32] B. Thornton, Asymptotic cones of symmetric spaces, Ph.D. Thesis, University of Utah, 2002.

[33] J. Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Math. 386, Springer, Berlin, 1974.

[34] J. Tits, Immeubles de type affine, in: Buildings and the geometry of diagrams (Como, 1984), Lecture Notes in Math. 1181, Springer, Berlin, 1986, pp. 159–190.

[35] E. B. Vinberg, V. V. Gorbatsevich and O. V. Shvartsman, Discrete Subgroups of Lie Groups, in: Lie Groups and Lie Algebras II (A. L. Onishchik and E. B. Vinberg, eds.), Encyclopaedia of Mathematical Sciences 21, Springer, Berlin, 2000, pp. 1–123.

Mathematisches Institut, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany

   E-mail address: kramer@mathematik.uni-wuerzburg.de

Mathematics Department, The Hebrew University, Jerusalem, Israel

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

Mathematisches Institut, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany

   E-mail address: tent@mathematik.uni-wuerzburg.de

Mathematics Department, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA

   E-mail address: sthomas@math.rutgers.edu