Limit groups, viewed by a logician Notes by Zoé Chatzidakis

The material presented in these notes, is certainly well-known to all specialists with some logic culture. It presents a model-theoretical proof of why a finitely presented group has finitely many maximal limit quotients.

Let G be a finitely presented group, generated by ā = (a_{1}, …, a_{m}), and with relations w_{1}(ā), …, w_{r}(ā). We let F be the free group on k generators.

I want to show that G has only finitely many maximal limit quotients. The strategy of the proof is simple: first of all, show that limit quotients of G correspond exactly to m-tuples in a (fixed) non-principal ultrapower of F which satisfy certain equations. Then, general logic results give that G only has finitely many limit quotients. If one requires that these limit quotients come from converging sequences of homomorphisms which are distinct modulo composition by elements of Aut(F), one can do so, by imposing that the m-tuples b̄ realise a certain type over F, see at the end. I first need the lemma on descending chains of closed sets (well-known, I believe).

Lemma. Consider the free group F on k generators, let ā be a tuple of elements of F, and Σ(x̄, ā) a system of group equations. Then there are formulas φ(x̄, ā), φ_{1}(x̄, ā), …, φ_{s}(x̄, ā), which are conjuctions of equations in (x̄, ā), and with the equations appearing in φ coming from Σ(x̄, ā), such that:

(1) If a tuple b̄ of F satisfies φ(x̄, ā) then it satisfies all equations of Σ(x̄, ā), and it also satisfies some φ_{i}(x̄, ā). If a tuple b̄ satisfies some φ_{i}(x̄, ā) then it satisfies φ(x̄, ā).

(2) Let U be a non-principal ultrafilter on ℕ, and F^{∗} = F^{ℕ}/U. For each i, there is a tuple b̄ in F^{∗} such that the equations w(x̄, ā) = 1 satisfied by b̄ are exactly those such that F ⊧ ∀x̄(φ_{i}(x̄, ā) → w(x̄, ā) = 1).

Proof. We know that F embeds in SL_{2}(ℤ) and therefore in SL_{2}(ℂ). Hence, F^{∗} embeds in (SL_{2}(ℂ))^{ℕ}/U = SL_{2}(ℂ^{ℕ}/U). In SL_{2}(ℂ^{ℕ}/U)^{m}, the Zariski closed set defined by the equations of Σ(x̄, ā) is in fact defined by a finite number of these equations (because the Zariski topology is Noetherian), and this gives us the formula φ(x̄, ā). Consider now the Zariski closure in SL_{2}(ℂ^{ℕ}/U)^{m} of the set of tuples of F^{m} which satisfy φ(x̄, ā). This set has finitely many irreducible components, say Z_{1}, …, Z_{s}. For each of these components, we consider the set Σ_{i}(x̄, ā) of group equations satisfied by the elements of Z_{i}. By Noetherianity, there a finite conjunction of equations φ_{i}(x̄, ā) which implies all equations of Σ_{i}(x̄, ā). The first assertion is therefore clear. Fix i. By irreducibility of Z_{i}, the union of finitely many Zariski-closed proper subsets of Z_{i} is a proper subset of Z_{i}, and therefore its complement in Z_{i} contains a point of F^{m}. By ω_{1}-saturation of F^{∗}, there is some b̄ ∈ F^{∗}, b̄ a generic of Z_{i} over ℚ(ā). This gives us the second assertion. Remark. This does not show that every finitely generated subgroup of F^{ℕ}/U is finitely presented, only that, modulo the axioms of T (the set of universal sentences true in F), finitely many equations suffice. The theory T will indeed contain many sentences of the form ∀x̄(w(x̄) = 1 → w′(x̄) = 1)), so that one given equation could imply infinitely many.

Let h_{n}, n ∈ ℕ, be a convergent sequence of homomorphisms h_{n} : G → F. The homomorphism h^{∗} : G → F^{∗} = F^{ℕ}/U, defined by h^{∗}(a_{j}) = (h_{n}(a_{j}))_{U} for j = 1, …, m, has kernel exactly ⋃_{m∈ℕ} ⋂_{n≥m} Ker(h_{n}). Hence the subgroup of F^{∗} generated by b̄ = h^{∗}(ā) is isomorphic to the limit group associated to the sequence (h_{n}).

Conversely, let b̄ be an m-tuple of elements of F^{∗} satisfying the equations w_{1}(b̄) = ⋯ = w_{r}(b̄) = 1. If b̄_{i}, i ∈ ℕ, are m-tuples of F such that b̄ = (b̄_{i})_{U}, then the set J = {i ∈ ℕ | w_{1}(b̄_{i}) = … = w_{r}(b̄_{i}) = 1} belongs to U. [It is here that we use the finite presentation of G. This can be avoided by going to an uncountable index set I, taking an ultrafilter F on I which is closed under countable intersection, and then working in F^{I}/F instead of F^{ℕ}/U]. Without loss of generality we may assume that J = ℕ. For each i ∈ ℕ we then have a homomorphism h_{i} : G → F, defined by h_{i}(ā) = b̄_{i}. Moreover, we have a homomorphism h^{∗} : G → F^{∗}, defined by h^{∗}(ā) = b̄.

It remains to extract from the h_{i}’s a converging subsequence. We fix an enumeration {w_{n}(ā) | n ∈ ℕ} of the elements of G. We will construct by induction on n, an infinite descending chain S_{n}, n ∈ ℕ, of subsets of ℕ, having infinite intersection. To ensure it has infinite intersection, we construct at the same time an increasing sequence I_{n}, n ∈ ℕ, with each I_{n} a subset of S_{n} of size n. We start with S_{−1} = ℕ, I_{−1} = ∅. Our induction hypothesis is the following:

(a) S_{n} ∈ U, I_{n} ⊆ S_{n} has size n and contains I_{n−1}

(b) if h^{∗}(w_{n}(ā)) = 1, then the set {i ∈ S_{n} | w_{n}(b̄_{i}) ≠ 1} is finite, and if h^{∗}(w_{n}(ā)) ≠ 1, then the set {i ∈ S_{n} | w_{n}(b̄_{i}) = 1} is finite, Assume S_{n} is defined, and consider w_{n+1}(ā). As S_{n} ∈ U, S_{n} is infinite , and we let

I_{n+1} be obtained by adjoining an element of S_{n} \ I_{n} to I_{n}. Consider the set U = {i ∈ ℕ | w_{n+1}(b̄_{i}) = 1}. If U ∈ U, we let S_{n+1} = (U ∩ S_{n}) ∪ I_{n+1}. If U ∉ U, then V = (ℕ \ U) ∈ U, and we let S_{n+1} = (V ∩ S_{n}) ∪ I_{n+1}. By induction hypothesis, S_{n} ∈ U, and this implies that S_{n+1} ∈ U.

We let S = ⋂_{n} S_{n}. It then contains ⋃_{n} I_{n}, and is therefore infinite. From the construction, it is clear that the sequence of homomorphisms {h_{i} | i ∈ S} is convergent, and that the limit group associated to this sequence is isomorphic to the subgroup of F^{∗} generated by b̄. We apply the lemma (without parameters) to the system of equations Σ(x̄) : w_{1}(x̄) = ⋯ = w_{r}(x̄) = 1, and will use the notation of the lemma. We fix m-tuples b̄_{1}, …, b̄_{s} of F^{∗}, with b̄_{i} satisfying the conclusion of (3) for Z_{i}, and let L_{i} be the subgroup of F^{∗} generated by b̄_{i}, f_{i} : G → L_{i} the homomorphism defined by f_{i}(ā) = b̄_{i}.

If h : G → L is a limit quotient of G, then we may, by the first step, assume that L ⊂ F^{∗}. Then h(ā) will belong to some Z_{i}, and therefore will satisfy all the group equations satisfied by b̄_{i}. This means that there is f : L_{i} → L such that h = f_{i}f. Thus the maximal quotients of G are among the groups L_{1}, …, L_{s}.

Note that the first step of the proof shows that the finitely generated subgroups of F^{∗} are ω-residually free, and that conversely every finitely generated ω-residually free group embeds in F^{∗}. If we are only interested in the limit groups originating from convergent sequences where the elements are distinct modulo composition by an inner automorphism of F, the trick is as follows. Let (h_{n}), n ∈ ℕ, be such a sequence of homomorphisms, and let h^{∗} : G → F^{∗} be defined by h^{∗}(ā) = (h_{i}(ā))_{U}. Composing each h_{i} by an inner automorphism of F is equivalent to conjugating the element h^{∗}(ā) = b̄ by an element of F^{∗}. Hence we may assume that for each i, the tuple h_{i}(ā) is closest to 1 among its conjugates (i.e., the max of the values of the distances to 1 of the tuples of h_{i}(ā) is minimal). Then the same is true in the 2-sorted structure (F^{∗}, ℕ^{∗}, d^{∗}) (where ℕ^{∗} = ℕ^{ℕ}/U, d^{∗}((g_{i})_{U}, (g′_{i})_{U}) = (d(g_{i}, g′_{i}))_{U}). As the sequence h_{i} is infinite and consists of distinct elements, we have that necessarily d^{∗}(1, b̄) is greater than all elements of ℕ ⊂ ℕ^{∗}.

Having made these observations, it is immediate that b̄ realises the following type Γ(x̄):

– ∀y d^{∗}(1, y^{−1}x̄y) ≥ d^{∗}(1, x̄).

– d^{∗}(1, x̄) ≥ c for all c ∈ ℕ. Note that each element c of ℕ is definable (in ℕ or in ℕ^{∗}), as being the c-th successor of the smallest element of ℕ (or of ℕ^{∗}).

Conversely, given an m-tuple b̄ of F^{∗}, satisfying Γ(x̄), one then defines as above homomorphisms h_{i} : G → F. Note that no conjugate of b̄ is in the copy of F sitting inside F^{∗}, and that for every c ∈ ℕ, the set T_{c} = {i | d(1, h_{i}(ā)) ≥ c} is in U. Hence, when building the decreasing sequence S_{n}, one makes sure that the elements of S_{n+1} \ I_{n} are in T_{n+1}. Then, from ⋂_{n} S_{n} one can extract an infinite subset J such that if i < j, then d(1, h_{i}(ā)) < d(1, h_{j}(b̄)). The sequence (h_{i})_{i∈J} is our desired convergent sequence.