Compactifications of a Representation Variety

Benjamin M. S. Martin

February 17, 2000

Abstract Let F be a finitely generated group and let G be a linear algebraic group over an algebraically closed field k. Let R(F, G) be the variety of representations of F in G. Given a finite set of generators Δ for F, we define a compactification R_{Δ}(F, \overline{G}) of R(F, G). The compactification is highly dependent on the choice of generators. If F′ is another finitely generated group with a finite set of generators Δ′ and ϕ: F′ → F is a homomorphism, then there is an induced morphism of varieties ϕ^{#}: R(F, G) → R(F′, G). We prove that if ϕ(Δ′ ∪ {1}) ⊂ Δ ∪ {1}, then ϕ^{#} extends to a morphism from R_{Δ}(F, \overline{G}) to R_{Δ′}(F′, \overline{G}). We study the morphisms arising in this way from a group extension 1 → N → F → Q → 1.

1 Introduction

Let F be a finitely generated group and let G be a linear algebraic group. The set of representations (i.e. abstract group homomorphisms) from F to G is an affine variety called the representation variety R(F, G). If G is reductive then G acts on R(F, G) by conjugation, and the quotient variety C(F, G), called the character variety, is also affine. These varieties have been the subject of much study (see [8], [2], [6], for example). In this paper we describe a natural way to compactify representation varieties.

A homomorphism ϕ: F′ → F of finitely generated groups gives rise to a morphism ϕ^{#}: R(F, G) → R(F′, G), defined by ϕ^{#}(ρ) = ρ ∘ ϕ. If G is reductive then we also have a morphism \overline{ϕ^{#}} from C(F, G) to C(F′, G) determined by ϕ. The correspondences F ↦ R(F, G), ϕ ↦ ϕ^{#} and F ↦ C(F, G), ϕ ↦ \overline{ϕ^{#}} are contravariant functors. Maps of the form ϕ^{#} arise in many settings. In [9], the author showed that the real polarization map of Weitsman [13] may be interpreted using this formalism. Slodowy [12], §II.4 studied the fibres of i^{#}, where i is the inclusion of a Sylow p-subgroup of a finite group Γ into Γ.

We define a compactification of an affine variety V to be a projective variety W that contains V as an open subset. (Usually one requires also that V should be dense in W, but it will be convenient for us to adopt this more general definition. If V is dense then we shall call W a strict compactification.) If W_{1}, W_{2} are compactifications of V then we define a morphism (isomorphism) of compactifications to be a morphism (isomorphism) of varieties ψ: W_{1} → W_{2} that restricts to the identity on V. When W_{1}, W_{2} are strict compactifications, there can exist at most one morphism ψ: W_{1} → W_{2} (since any two must agree on the dense open subset V). If V carries some extra structure, such as the action of some group, then W can often be chosen to be compatible with this structure.

2000 Mathematics Subject Classification: 20C99

Compactifications of representation varieties and character varieties have been constructed before in a slightly different setting. Instead of R(F, G) and C(F, G), one may define real analytic varieties R(F, H) and C(F, H) for H a real reductive Lie group (see [2]). Let Π^{g} be the fundamental group of a compact orientable surface of genus g ≥ 2. Teichmüller space T_{g} may be identified with a subspace of C(Π^{g}, PSL_{2}(ℝ)), and several compactifications of T_{g} are known, including the Thurston compactification [5].

Our aim is to construct compactifications of R(F, G) and C(F, G) for arbitrary F that are compatible with the maps ϕ^{#} and \overline{ϕ^{#}}. One motivation for this is as follows. Casson’s invariant for an integral homology 3-sphere M is defined to be the topological intersection number of a certain pair of subvarieties V_{1} and V_{2} of C(Π^{g}, SU(2)) (see [1] for details). Each V_{i} is the image of \overline{p_{i}^{#}} for some epimorphism p_{i} from Π^{g} to a free group of rank g. A natural variation on this idea would be to consider the algebraic intersection number of the analogous pair of subvarieties in a compactification of C(Π^{g}, G), where G is a complex reductive algebraic group.

As a first step, in this paper we construct compactifications of the representation variety R(F, G). Unfortunately, the requirement that ϕ^{#} extend to a morphism of compactified representation varieties is too strong (see Theorem 5.7); to fulfill it, we are forced to consider not just a single compactification of R(F, G), but one for each finite set Δ = {γ_{1}, …, γ_{r}} of generators for F. First we choose a suitable compactification \overline{G} of G. The map ρ ↦ (ρ(γ_{1}), …, ρ(γ_{r})) gives an embedding of R(F, G) as a closed subset of G^{r}, and it is well known that the structure of affine variety that R(F, G) inherits is independent of the choice of generating set. Given a generating set Δ for F as above, we define R_{Δ}(F, \overline{G}) to be roughly the set of functions of the form ρ: U → \overline{G}, for some U with Δ ⊂ U ⊂ F, such that ρ is a homomorphism wherever this makes sense. Then R_{Δ}(F, \overline{G}) embeds as a closed subvariety of the projective variety \overline{G}^{r} and R(F, G) is open in R_{Δ}(F, \overline{G}). Moreover, given groups F, F′ with finite generating sets Δ, Δ′ respectively and a homomorphism ϕ: F′ → F such that ϕ(Δ′ ∪ {1}) ⊂ Δ ∪ {1}, the morphism ϕ^{#} extends to a morphism Φ^{#} from R_{Δ}(F, \overline{G}) to R_{Δ′}(F′, \overline{G}).

Let

1 → N → F →^{p} Q → 1

be an extension of groups, where F is finitely generated. Then Q is finitely generated and p^{#}: R(Q, G) → R(F, G) is a closed embedding with image consisting of the set of representations ρ such that ρ|_{N} is trivial. We may replace N by a suitable finitely generated subgroup K of N; for by Lemma 6.1, there exists K such that

p^{#}(R(Q, G)) = (j^{#})^{−1}(ρ_{0}^{K}),

where j is the inclusion of K in F and ρ_{0}^{K} denotes the trivial representation of K. We call such a subgroup K an approximate G-kernel of p.

We show how to carry this idea over to the case of compactified representation varieties. We define the notion of an approximate \overline{G}-kernel (Definition 6.4), and we prove that approximate \overline{G}-kernels always exist. They are highly dependent on the choice of generating sets for the finitely generated groups concerned.

The author is grateful for the support of the New Zealand Foundation for Science, Research and Technology (Postdoctoral Fellowship ANU601) and the Australian Research Council (Large Grant A69802270, Chief Investigator G. I. Lehrer).

2 Compactifying G

We begin with some notation. The symbol F denotes a finitely generated group. The identity of a group is denoted by 1, or by I if the group is a matrix group. We write F_{s} for the free group on s generators. The subgroup generated by a subset S of a group is denoted by ⟨S⟩. All varieties (including algebraic groups) are defined over a fixed algebraically closed field k, and we allow varieties to be reducible. We set k^{∗} = k\{0}. If V is a vector space over k with dim V < ∞ then ℙ(V) denotes the corresponding projective space. We denote the space of n × n matrices with entries from k by M_{n}(k).

We fix a linear algebraic group G. For material on representation varieties, see the article [8] of Lubotzky and Magid. Usually in the theory of representation varieties it is assumed that G is connected and reductive, but the basic results and definitions carry over to arbitrary G with simple modifications. However, the quotient variety C(F, G) is only defined for reductive G.

We choose a strict compactification \overline{G} of G as follows. Pick an embedding α of G as a closed subgroup of PGL_{n}(k). Such an embedding always exists, for we can find an embedding G → GL_{m}(k) for some m, and then follow this with the mapping GL_{m}(k) →^{ξ_{m}} GL_{m+1}(k) → PGL_{m+1}(k), where

ξ_{m}(A) = ( \begin{matrix}A & 0 \\ 0 & 1 \end{matrix} )

and the second arrow is the canonical projection. Regard PGL_{n}(k) as an open subvariety of ℙ(M_{n}(k)), and define \overline{G}(α) to be the closure of α(G) in ℙ(M_{n}(k)). Since α will be fixed, we shall suppress it and write \overline{G} instead of \overline{G}(α), and we shall identify G with the corresponding open subset of \overline{G}.

Notation 2.1 We shall call the identity embedding of PGL_{n}(k) into PGL_{n}(k) the standard embedding of PGL_{n}(k), and the embedding GL_{n}(k) →^{ξ_{n}} GL_{n+1}(k) → PGL_{n+1}(k), the standard embedding of GL_{n}(k).

Given 0 ≠ A ∈ M_{n}(k), let us denote its image in ℙ(M_{n}(k)) by [A]. Group multiplication on G×G extends to a morphism from an open subset of \overline{G}×\overline{G} into \overline{G}: we define [A][B] = [AB] on the set of pairs ([A], [B]) such that AB ≠ 0, and we call this morphism multiplication. The element [I] is a two-sided identity. The product [A][B] may be defined even when A, B are not invertible matrices. However [A] ∈ \overline{G} has an inverse in \overline{G} if and only if A is invertible.

Lemma 2.2 The set O of invertible elements of \overline{G} coincides with G.

Proof It is clear that O is an algebraic group, with open dense subgroup G. Lemma 7.4 of [4] implies that G = O. □

Multiplication on \overline{G} is associative wherever this makes sense, so we define multiplication on any finite product of copies of \overline{G} in the obvious way. We say that [A_{1}] ⋯ [A_{m}] is defined if ([A_{1}], …, [A_{m}]) is in the domain of the multiplication morphism, that is, if A_{1} ⋯ A_{m} ≠ 0.

More generally, let w be a word in letters γ_{1}, …, γ_{r}. We write 1 for the empty word. Let Σ = ([A_{1}], …, [A_{r}]) be an r-tuple of elements of \overline{G}, and let w be a reduced word in γ_{1}, …, γ_{r}. We say that w([A_{1}], …, [A_{r}]) or w(Σ) is defined if:

(i) whenever the letter γ_{i}^{−1} appears in w, [A_{i}] is invertible; and (ii) the element [B] obtained by substituting [A_{i}]^{±1} for γ_{i}^{±1} in w and then multiplying is defined. In this case we write w(Σ) = w([A_{1}], …, [A_{r}]) = [B]. For an arbitrary word w, we say that w([A_{1}], …, [A_{r}]) or w(Σ) is defined and equal to [B] if R(w)(Σ) is defined and equal to [B], where R(w) is the reduced word obtained from w. (We set 1(Σ) = [I].)

Example 2.3 (a) For any i, (γ_{i}γ_{i}^{−1})(Σ) = 1(Σ) = [I] is defined.

(b) For any i, γ_{i}(Σ) = [A_{i}] is defined.

(c) For any i, γ_{i}^{−1}(Σ) is defined if and only if [A_{i}]^{−1} is defined if and only if [A_{i}] ∈ G.

(d) If [A_{i}]^{−1} is not defined then (γ_{i}γ_{j}γ_{i}^{−1})(Σ) is not defined for j ≠ i, not even if [A_{j}] = [I].

Observation 2.4 If w is a word then the set

\{([A_{1}], …, [A_{r}]) ∈ \overline{G}^{r}| w([A_{1}], …, [A_{r}]) is defined\}

is open in \overline{G}^{r}. Moreover, the function Σ ↦ w(Σ) is a morphism from this set to \overline{G}.

Lemma 2.5 Let Σ = ([A_{1}], …, [A_{r}]) ∈ \overline{G}^{r} and let w_{1}, w_{2} be words such that w_{1}(Σ), w_{2}(Σ) and w_{1}(Σ)w_{2}(Σ) are defined. Then (w_{1}w_{2})(Σ) is defined and (w_{1}w_{2})(Σ) = w_{1}(Σ)w_{2}(Σ).

Proof Without loss of generality, assume that w_{1} and w_{2} are reduced. If w_{1}w_{2} is reduced then clearly we are done. Otherwise we can write w_{1} = w̃_{1}γ_{i}^{±1} and w_{2} = γ_{i}^{∓1}w̃_{2} for some i. Then [A_{i}] is invertible, w̃_{1}(Σ), w̃_{2}(Σ) are defined and we have w_{1}(Σ) = w̃_{1}(Σ)[A_{i}]^{±1}, w_{2}(Σ) = [A_{i}]^{∓1}w̃_{2}(Σ), so w_{1}(Σ)w_{2}(Σ) = w̃_{1}(Σ)w̃_{2}(Σ) and R(w_{1}w_{2}) = R(w̃_{1}w̃_{2}). The result follows by induction on the lengths of w_{1} and w_{2}. □

As a special case, suppose that w = γ_{m_{1}}^{η_{1}} ⋯ γ_{m_{s}}^{η_{s}} is a word (with each η_{i} ∈ {±1}), not necessarily reduced, such that [A_{m_{i}}] is invertible whenever η_{i} = −1 and such that [A_{m_{1}}]^{η_{1}} ⋯ [A_{m_{s}}]^{η_{s}} is defined. Then w([A_{1}], …, [A_{r}]) is defined and w([A_{1}], …, [A_{r}]) = [A_{m_{1}}]^{η_{1}} ⋯ [A_{m_{s}}]^{η_{s}}.

Example 2.6 Let S = {γ_{m_{1}}, …, γ_{m_{s}}} ⊂ {γ_{1}, …, γ_{r}}. Given a word w in γ_{1}, …, γ_{r}, let w_{S} denote the word in γ_{m_{1}}, …, γ_{m_{s}} obtained from w by deleting all letters γ_{i}^{±1} for all γ_{i} ∉ S. Suppose that [A_{i}] = [I] for all i such that γ_{i} ∉ S. It follows from the special case above that if w([A_{1}], …, [A_{r}]) is defined, then w_{S}([A_{1}], …, [A_{r}]) is also defined and w([A_{1}], …, [A_{r}]) = w_{S}([A_{1}], …, [A_{r}]).

3 Marked Groups

As noted in §1, the correspondence F ↦ R(F, G) is a contravariant functor from the category 𝔉𝔊 of finitely generated groups to the category of affine varieties. We need to keep track of sets of generators as well, so we work with a slightly different category.

Definition 3.1 A marked group F is a pair (F, Δ), where F is a finitely generated group and Δ is a finite set of generators for F. We call Δ a marking for F. If (F, Δ) and (F′, Δ′) are marked groups then a marked homomorphism Φ from (F′, Δ′) to (F, Δ) is a homomorphism ϕ: F′ → F such that ϕ(Δ′∪{1}) ⊂ Δ∪ {1}. Define the category 𝔐𝔊 of marked groups to be the category whose objects are the marked groups and whose morphisms are the marked homomorphisms.

Example 3.2 Let F = F_{s} for some s and let γ_{1}, …, γ_{s} be free generators for F_{s}. We call Δ = {γ_{1}, …, γ_{s}} the standard marking for F_{s}.

There is an obvious forgetful functor from 𝔐𝔊 to 𝔉𝔊.

Notation 3.3 If Φ is a marked homomorphism between marked groups then we denote the underlying homomorphism of finitely generated groups by |Φ|. Although Φ is completely determined by |Φ|, nevertheless the distinction between the two is important — see §5 below.

Remark 3.4 Given a pair of marked groups, there are only finitely many marked homomorphisms between them.

If F′ ⊂ F, Δ′ ⊂ Δ and |Φ| is inclusion, then we shall refer to Φ as inclusion and we shall call (F′, Δ′) a marked subgroup of (F, Δ).

Let Φ: (F′, Δ′) → (F, Δ) be a marked homomorphism. We say that Φ is a marked monomorphism if the homomorphism |Φ| is injective. We say that Φ is a marked epimorphism if |Φ|(Δ′ ∪ {1}) = Δ ∪ {1} (this condition of course implies that |Φ| is an epimorphism). A marked homomorphism is an isomorphism in the category 𝔐𝔊 if and only if it is both a marked monomorphism and a marked epimorphism.

Let (F, Δ) be a marked group and let N ⊴ F. Let Q = F/N and let p: F → Q be the canonical projection. Then p(Δ) is a finite generating set for Q and there is a marked epimorphism P: (F, Δ) → (Q, p(Δ)) induced by p. However, the inclusion of N in F cannot in general be made into a homomorphism of marked groups, even if N is finitely generated. We could have Δ ∩ N = ∅, for instance, when the only marked subgroup (K, Θ) of (F, Δ) such that K ⊂ N would be the trivial marked group (1, {1}).

4 Admissible Representations

Definition 4.1 Let F = (F, Δ) be a marked group, with Δ = {γ_{1}, …, γ_{r}}. An admissible representation of (F, Δ) into \overline{G} is a pair (U, ρ), where Δ ⊂ U ⊂ F and ρ: U → \overline{G} is a function with the following properties:

(a) for any γ ∈ F, γ ∈ U if and only if there is a word w in γ_{1}, …, γ_{r} representing γ such that w(ρ(γ_{1}), …, ρ(γ_{r})) is defined;

(b) for any γ ∈ U, for any word w in γ_{1}, …, γ_{r} representing γ such that w(ρ(γ_{1}), …, ρ(γ_{r})) is defined, we have ρ(γ) = w(ρ(γ_{1}), …, ρ(γ_{r})). We say that ρ(γ) is defined if and only if γ ∈ U. We denote the set of admissible representations by R_{Δ}(F, \overline{G}).

Remark 4.2 We shall write w(ρ) as shorthand for w(ρ(γ_{1}), …, ρ(γ_{r})). Note that if ρ(γ) is defined, this does not imply that w(ρ) is defined for every word w representing γ, only that w(ρ) is defined for some word w representing γ.

There is a canonical inclusion of R(F, G) in R_{Δ}(F, \overline{G}), given by ρ ↦ (F, ρ). We denote the image of the trivial representation by ρ_{0}^{F}.

Lemma 4.3 (a) Let (U_{1}, ρ_{1}), (U_{2}, ρ_{2}) ∈ R_{Δ}(F, \overline{G}). If ρ_{1}|_{Δ} = ρ_{2}|_{Δ} then (U_{1}, ρ_{1}) = (U_{2}, ρ_{2}).

(b) Let X_{1}, …, X_{r} ∈ \overline{G}. There exists an admissible representation (U, ρ) such that ρ(γ_{i}) = X_{i} for all 1 ≤ i ≤ r if and only if for any words w_{1}, w_{2} in γ_{1}, …, γ_{r} such that w_{1} and w_{2} represent the same element of F and both w_{1}(X_{1}, …, X_{r}), w_{2}(X_{1}, …, X_{r}) are defined, we have w_{1}(X_{1}, …, X_{r}) = w_{2}(X_{1}, …, X_{r}).

(c) Let (U, ρ) ∈ R_{Δ}(F, \overline{G}) and let γ, γ̃ ∈ U. If ρ(γ)ρ(γ̃) is defined then ρ(γγ̃) is defined and

ρ(γγ̃) = ρ(γ)ρ(γ̃).

(Note, however, that even if γ, γ̃ and γγ̃ belong to U, it does not necessarily follow that the product ρ(γ)ρ(γ̃) is defined.)

(d) Let (U, ρ) ∈ R_{Δ}(F, \overline{G}) and let γ ∈ F. Suppose there is a word w = γ_{m_{1}}^{η_{1}} … γ_{m_{s}}^{η_{s}} representing γ (with each η_{i} ∈ {±1}), not necessarily reduced, such that ρ(γ_{m_{i}}) is invertible whenever η_{i} = −1 and such that ρ(γ_{m_{1}})^{η_{1}} ⋯ ρ(γ_{m_{s}})^{η_{s}} is defined. Then ρ(γ) is defined and ρ(γ) = ρ(γ_{m_{1}})^{η_{1}} … ρ(γ_{m_{s}})^{η_{s}}.

Proof Parts (a) and (b) follow immediately from Definition 4.1, while (c) is a consequence of Lemma 2.5. Part (d) follows from (c) (compare the discussion following Lemma 2.5). □

Convention 4.4 In view of Lemma 4.3, we will regard R_{Δ}(F, \overline{G}) as a subset of \overline{G}^{r}, and R(F, G) as a subset of R_{Δ}(F, \overline{G}). Often we will simply write ρ instead of (U, ρ) for an admissible representation.

Remark 4.5 An admissible representation (U, ρ) is completely determined by its values ρ(γ_{1}), …, ρ(γ_{r}). However, the r-tuple (ρ(γ_{1}), …, ρ(γ_{r})) ∈ \overline{G}^{r} does not uniquely determine a subset Ũ of F and a function ρ̃: Ũ → F satisfying the homomorphism property of Lemma 4.3 (c). For example, there may exist Ũ ⊃ U and ρ_{1}, ρ_{2}: Ũ → \overline{G} such that ρ_{1}, ρ_{2} satisfy Lemma 4.3 (c) and ρ_{1}|_{U} = ρ_{2}|_{U}, yet ρ_{1} ≠ ρ_{2} — see Example 5.6 below. That is the motivation for Definition 4.1.

The next result shows that R_{Δ}(F, \overline{G}) is a compactification of R(F, G).

Proposition 4.6 (a) The subset R_{Δ}(F, \overline{G}) is closed in \overline{G}^{r}.

(b) The subset R(F, G) is open in R_{Δ}(F, \overline{G}).

Proof (a) Let (X_{1}, …, X_{r}) lie in the closure of R_{Δ}(F, \overline{G}), and suppose that (X_{1}, …, X_{r}) ∉ R_{Δ}(F, \overline{G}). By Lemma 4.3 (b), there exist words w_{1}, w_{2} that represent the same element of F such that w_{1}(X_{1}, …, X_{r}), w_{2}(X_{1}, …, X_{r}) are defined, but w_{1}(X_{1}, …, X_{r}) ≠ w_{2}(X_{1}, …, X_{r}). Let O be the set of r-tuples (Y_{1}, …, Y_{r}) ∈ \overline{G}^{r} such that w_{1}(Y_{1}, …, Y_{r}), w_{2}(Y_{1}, …, Y_{r}) are defined, but w_{1}(Y_{1}, …, Y_{r}) ≠ w_{2}(Y_{1}, …, Y_{r}). By Observation 2.4, O is an open neighbourhood of (X_{1}, …, X_{r}), hence some element (Y_{1}, …, Y_{r}) of R_{Δ}(F, \overline{G}) belongs to O. But then we have w_{1}(Y_{1}, …, Y_{r}) = w_{2}(Y_{1}, …, Y_{r}) by Definition 4.1 (b), a contradiction. We conclude that R_{Δ}(F, \overline{G}) is closed.

(b) We have R(F, G) = R_{Δ}(F, \overline{G}) ∩ G^{r}, and the result follows. □

Notation 4.7 We denote the closure of R(F, G) inside \overline{G}^{r} by R_{Δ}^{s}(F, \overline{G}). Example 4.8 (c) below shows that this may be a proper subset of R_{Δ}(F, \overline{G}).

Example 4.8 (a) Let F = F_{s} with the standard marking. Then R(F, G) = G^{s} is dense in \overline{G}^{s}, so R_{Δ}(F, \overline{G}) = R_{Δ}^{s}(F, \overline{G}) = \overline{G}^{s}.

(b) Let F = ⟨t, n| t^{3} = n^{2} = 1, ntn^{−1} = t^{2}⟩ = S_{3} (the symmetric group on three letters), and let Δ = {n, t}. Let G = PGL_{2}(k) with the standard embedding.

Let X = [( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )] ∈ PGL_{2}(k). Any reduced word w = w(n, t) such that

w([I], X) is defined cannot contain the letter t^{−1} and can contain the letter t at most once. Applying Lemma 4.3 (b), we see that ρ(n) = [I], ρ(t) = X defines an admissible representation ρ ∈ R_{Δ}(F, \overline{G}).

(c) Let F = ℤ⊕ ℤ = ⟨a, b| ab = ba⟩ and let Δ = {a, b}. Let G = GL_{3}(k) with the standard embedding. Define ρ ∈ R_{Δ}(F, \overline{G}) by

ρ(a) = [( \begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )], ρ(b) = [( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )]

— it is straightforward to check that this is well-defined. Now SL_{3}(k) acts on (M_{4}(k)\{0}) × (M_{4}(k)\{0}) by conjugation:

X.(A, B) = (ξ_{3}(X)Aξ_{3}(X^{−1}), ξ_{3}(X)Bξ_{3}(X^{−1})).

For A ∈ M_{4}(k), let Z(A) denote the centraliser of A in ξ_{3}(SL_{3}(k)). By [10], Lemma 3.7, the function ℵ: M_{4}(k)\{0}×M_{4}(k)\{0} → ℕ∪{0} given by ℵ(A, B) = dim (Z(A) ∩ Z(B)) is upper semicontinuous. The SL_{3}(k)-action commutes with scalar multiplication on each factor, so ℵ descends to a function \overline{ℵ}: ℙ(M_{4}(k)) × ℙ(M_{4}(k)) → ℕ ∪ \{0\}, and one can show that \overline{ℵ} is also upper semicontinuous.

Assume that k has characteristic zero. Then R(F, G) is irreducible by [11], §2, Proposition 3, so the open subset

O = {ρ̃ ∈ R(F, G)| ρ̃(a), ρ̃(b) are regular semisimple}

is dense. For ρ̃ ∈ O we have \overline{ℵ}(ρ̃) = 2, whence \overline{ℵ}(ρ̃) ≥ 2 for all ρ̃ ∈ R_{Δ}^{s}(F, \overline{G}). But \overline{ℵ}(ρ) = 1 by direct computation, so ρ ∉ R_{Δ}^{s}(F, \overline{G}). Thus R_{Δ}(F, \overline{G}) is not a strict compactification of R(F, G).

Let γ ∈ F. The set \{ρ ∈ R_{Δ}(F, \overline{G})| ρ(γ) is defined\} is open by Observation 2.4. Define a function ϵ_{γ} from this open set to \overline{G} by ϵ_{γ}(ρ) = ρ(γ). If ρ(γ) is defined then there is a reduced word w in γ_{1}, …, γ_{r} such that w represents γ and the function ρ̃ ↦ w(ρ̃(γ_{1}), …, ρ̃(γ_{r})) is well-defined and equal to ϵ_{γ} on an open neighbourhood of ρ. We deduce that ϵ_{γ} is a morphism of varieties. The morphisms ϵ_{γ_{i}} are defined everywhere on R_{Δ}(F, \overline{G}) and the canonical embedding of R_{Δ}(F, \overline{G}) in \overline{G}^{r} (see Convention 4.4) is just the product ϵ_{γ_{1}} × ⋯ × ϵ_{γ_{r}}.

Let X_{1}, …, X_{r} ∈ \overline{G}. Except for in simple cases, it is impractical to check whether the prescription ρ(γ_{i}) = X_{i} gives an admissible representation ρ ∈ R_{Δ}(F, \overline{G}) by directly verifying that the hypotheses of Lemma 4.3 (b) hold. An easier way to obtain admissible representations is by taking limits.

Let C be a smooth irreducible quasiprojective curve and let x_{0} ∈ C. If f: C\{x_{0}} → V is a morphism of varieties, we say that lim_{x→x_{0}} f(x) exists and equals v if f extends to a morphism f′: C → V such that f′(x_{0}) = v. If a limit exists then it is unique. The limit always exists if V is projective ([3], Chapter I, Proposition 6.8). Thus, given a morphism C\\{x_{0}\} → R_{Δ}(F, \overline{G}), x ↦ ρ_{x}, the limit lim_{x→x_{0}} ρ_{x} also belongs to R_{Δ}(F, \overline{G}). Moreover, if ρ_{x} ∈ R(F, G) for each x ∈ C\\{x_{0}\}, then lim_{x→x_{0}} ρ_{x} ∈ R_{Δ}^{s}(F, \overline{G}).

Example 4.9 Let F, Δ, G, \overline{G}, ρ be as in Example 4.8 (b). Assume that k has characteristic two. For x ∈ k^{∗}, define ρ_{x} ∈ R(F, G) by

ρ_{x}(n) = [( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} )], ρ_{x}(t) = [( \begin{matrix}ζ & 0 \\ x^{−1}(ζ + ζ^{2}) & ζ^{2} \end{matrix} )] = [( \begin{matrix}xζ & 0 \\ ζ + ζ^{2} & xζ^{2} \end{matrix} )],

where ζ is a primitive cube root of unity. Then lim_{x→0} ρ_{x} = ρ, and we deduce that ρ ∈ R_{Δ}^{s}(F, \overline{G}).

Remark 4.10 Let ρ = lim_{x→x_{0}} ρ_{x} for some map x ↦ ρ_{x} as above. Let γ ∈ F. If ρ_{y}(γ) is defined for some y ∈ C\{x_{0}} then the morphism x ↦ ρ_{x}(γ) is defined on an open neighbourhood C′ of y in C\{x_{0}}, and we can form the limit lim_{x→x_{0}} ρ_{x}(γ). In particular, if ρ(γ) is defined then lim_{x→x_{0}} ρ_{x}(γ) = lim_{x→x_{0}} ϵ_{γ}(ρ_{x}) exists because ϵ_{γ} is a morphism, and the limit is equal to ρ(γ). However, even when ρ(γ) is not defined, lim_{x→x_{0}} ρ_{x}(γ) will exist if ρ_{x}(γ) is defined for some x ∈ C\{x_{0}} (see Example 5.6).

5 Functoriality

Proposition 5.1 Let Φ: (F′, Δ′) → (F, Δ) be a homomorphism of marked groups, where Δ = \{γ_{1}, …, γ_{r}\} and Δ′ = \{γ′_{1}, …, γ′_{t}\}. Let (U, ρ) ∈ R_{Δ}(F, \overline{G}). The t-tuple (ρ′(γ′_{1}), …, ρ′(γ′_{t})) defined by ρ′(γ′_{i}) = ρ(|Φ|(γ′_{i})) determines a unique admissible representation (U′, ρ′) ∈ R_{Δ′}(F′, \overline{G}). We have |Φ|(U′) ⊂ U, with equality if Φ is a marked epimorphism, and

ρ′ = ρ ∘ |Φ|.  (1)

Proof Let w′, w̃′ be words in γ′_{1}, …, γ′_{t} such that w′, w̃′ both represent the same element γ′ ∈ F′ and w′(ρ′), w̃′(ρ′) are defined. Define a word w in γ_{1}, …, γ_{r} as follows: replace every occurrence of (γ′_{i})^{±1} in w′ with γ_{m_{i}}^{±1}, where γ_{m_{i}} = |Φ|(γ_{i}), and define w̃ from w̃′ similarly. Then w, w̃ both represent |Φ|(γ′) ∈ F, w(ρ) and w̃(ρ) are both defined, and we have w(ρ) = w′(ρ′) and w̃(ρ) = w̃′(ρ′). But w(ρ) = w̃(ρ) = ρ(|Φ|(γ′)), so w′(ρ′) = w̃′(ρ′). Therefore ρ′ is an admissible representation by Lemma 4.3 (b). Uniqueness follows from Lemma 4.3 (a), and Equation 1 is a consequence of Lemma 4.3 (d). Finally, if Φ is a marked epimorphism then every word w in γ_{1}, …, γ_{r} can be obtained from some word w′ in γ′_{1}, …, γ′_{t} in the way described above, and it follows easily that |Φ|(U′) = U. □

Definition 5.2 Define Φ^{#}: R_{Δ}(F, \overline{G}) → R_{Δ′}(F′, \overline{G}) by Φ^{#}((U, ρ)) = (U′, ρ′), where (U′, ρ′) is as given above.

In terms of our canonical embeddings in projective varieties (see Convention 4.4), Φ^{#} is the restriction to R_{Δ}(F, \overline{G}) of a product of projections onto factors of \overline{G}^{r}, so Φ^{#} is a morphism. It is easily verified that (F, Δ) ↦ R_{Δ}(F, \overline{G}), Φ ↦ Φ^{#} is a contravariant functor from 𝔐𝔊 to the category of projective varieties, and the restriction of Φ^{#} to R(F, G) is equal to ϕ^{#}: R(F, G) → R(F′, G) from §1.

Example 5.3 If Φ: (F′, Δ′) → (F, Δ) is a marked epimorphism then by considering the canonical embeddings of R_{Δ′}(F′, \overline{G}) and R_{Δ}(F, \overline{G}), we see that Φ^{#} is a closed embedding. In particular, if F′ is free on γ′_{1}, …, γ′_{t} and Δ′ = {γ′_{1}, …, γ′_{t}} then, identifying R_{Δ′}(F′, \overline{G}) with \overline{G}^{t} as in Example 4.8 (a), we have Φ^{#} = ϵ_{|Φ|(γ′_{1})} × ⋯ × ϵ_{|Φ|(γ′_{t})}.

When comparing different compactifications of R(F, G), it is helpful to consider inclusions J: (F, Δ′) → (F, Δ). If Δ′ is properly contained in Δ then J^{#} need not be an isomorphism (see Example 5.6 below). Here is a criterion for J^{#} to be an isomorphism near a point.

Lemma 5.4 Let Φ: (F, Δ′) → (F, Δ) be an inclusion. Set

O = \{ρ′ ∈ R_{Δ′}(F, \overline{G})| ρ′(γ) is defined for all γ ∈ Δ\Δ′\}.

Then O is open and Φ^{#}|_{(Φ^{#})^{−1}(O)}: (Φ^{#})^{−1}(O) → O is an isomorphism.

Proof Write Δ\Δ′ = {γ_{1}, …, γ_{s}}. Let ρ′ ∈ O. There are words w_{1}, …, w_{s} in the generators from Δ′ such that w_{i} represents γ_{i} and w_{i}(ρ′) is defined. Then w_{1}(ρ), …, w_{s}(ρ) are defined for ρ in an open neighbourhood O′ of ρ′. Define ψ: O′ → R_{Δ}(F, \overline{G}) by ψ(ρ)(γ′) = ρ(γ′) for γ′ ∈ Δ′, and ψ(ρ)(γ_{i}) = w_{i}(ρ). It is simple to check that ψ is well-defined, that ψ(O′) = (Φ^{#})^{−1}(O′), that O′ ⊂ O and that ψ: O′ → (Φ^{#})^{−1}(O′) is a morphism with inverse Φ^{#}|_{(Φ^{#})^{−1}(O′)}. The result follows. □

As an immediate corollary, we have

Corollary 5.5 Let (F, Δ) be a marked group, let γ ∈ F and let Φ: (F, Δ) → (F, Δ ∪ \{γ\}) be inclusion. Given ρ ∈ R_{Δ}(F, \overline{G}), if (Φ^{#})^{−1}(ρ) contains more than one point then ρ(γ) is not defined. □

Example 5.6 Let F = F_{2}, with free generators γ_{1}, γ_{2}, and let G = PGL_{2}(k) with the standard embedding. For any n ∈ ℕ, define Δ_{n} = {γ_{1}, γ_{2}, γ_{1}γ_{2}γ_{1}^{−1}, γ_{1}^{2}γ_{2}γ_{1}^{−2}, …, γ_{1}^{n}γ_{2}γ_{1}^{−n}}, and for N ∈ ℕ, x ∈ k^{∗}, define ρ_{x}^{(N)} ∈ R(F, G) by

ρ_{x}^{(N)}(γ_{1}) = [( \begin{matrix}x^{−1} & 0 \\ 0 & x \end{matrix} )], ρ_{x}^{(N)}(γ_{2}) = [( \begin{matrix}1 & x^{2N−1} \\ 0 & 1 \end{matrix} )].

Let Φ_{n}: (F, Δ_{n}) → (F, Δ_{n+1}) be inclusion.

Regarding each ρ_{x}^{(N)} as an element of R_{Δ_{n}}(F, \overline{G}), define σ_{n}, τ_{n} ∈ R_{Δ_{n}}^{s}(F, \overline{G}) to be lim_{x→0} ρ_{x}^{(n)} and lim_{x→0} ρ_{x}^{(n+1)} respectively. Then σ_{n} ≠ τ_{n} — for

σ_{n}(γ_{1}^{n}γ_{2}γ_{1}^{−n}) = [( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )] ≠ [I] = τ_{n}(γ_{1}^{n}γ_{2}γ_{1}^{−n})

— but Φ_{n}^{#}(σ_{n+1}) = Φ_{n}^{#}(τ_{n+1}) = τ_{n}. We conclude from Corollary 5.5 that τ_{n}(γ_{1}^{n+1}γ_{2}γ_{1}^{−(n+1)}) is undefined, even though lim_{x→0} ρ_{x}^{(n+1)}(γ_{1}^{n+1}γ_{2}γ_{1}^{−(n+1)}) =

[( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )] exists. Moreover, none of the compositions Φ_{n}^{#} ∘Φ_{n+1}^{#} ∘⋯∘Φ_{n+m−1}^{#}

gives an isomorphism from R_{Δ_{n+m}}^{s}(F, \overline{G}) to R_{Δ_{n}}^{s}(F, \overline{G}) for any n, m ∈ ℕ, so the strict compactifications R_{Δ_{n}}^{s}(F, \overline{G}) are pairwise nonisomorphic. This implies that the compactifications R_{Δ_{n}}(F, \overline{G}) are also pairwise nonisomorphic.

Example 5.6 shows that the compactifications arising from different generating sets may be distinct. Now we see that this problem cannot be avoided.

Theorem 5.7 There does not exist a contravariant functor F ↦ R̃(F, G), ϕ ↦ ϕ̃^{#} from the category 𝔉𝔊 to the category of projective varieties such that:

(i) for every finitely generated group F, R̃(F, G) is a compactification of R(F, G); and (ii) for every homomorphism ϕ: F′ → F of finitely generated groups, the restriction to R(F, G) of the morphism ϕ̃^{#}: R̃(F, G) → R̃(F′, G) equals ϕ^{#}.

Proof Suppose that such a functor exists. Let F, G, γ_{1}, γ_{2}, Δ_{n}, ρ_{x}^{(N)} be as in Example 5.6. Given γ ∈ F, let i_{γ}: ℤ → F be the homomorphism that takes 1 ∈ ℤ

to γ. By hypothesis, we have a morphism ĩ_{γ}^{#}: R̃(F, G) → R̃(ℤ, G) that extends

i_{γ}^{#}. Set D_{n} = ⋂_{γ∈Δ_{n}\{γ_{1}}} (ĩ_{γ}^{#})^{−1}(ρ_{0}^{ℤ}) ⊂ R̃(F, G). It is clear that D_{1} ⊃ D_{2} ⊃ ⋯ .

Define β_{n} = lim_{x→0} ρ_{x}^{(n+1)} ∈ R̃(F, G). Then

ĩ_{γ_{1}^{m}γ_{2}γ_{1}^{−m}}^{#}(β_{n}) = lim_{x→0} ĩ_{γ_{1}^{m}γ_{2}γ_{1}^{−m}}^{#}(ρ_{x}^{(n)}) = lim_{x→0} i_{γ_{1}^{m}γ_{2}γ_{1}^{−m}}^{#}(ρ_{x}^{(n)})

= lim_{x→0} ρ_{x}^{(n)}(γ_{1}^{m}γ_{2}γ_{1}^{−m}) = I

for all m such that 1 ≤ m ≤ n. However, we have ĩ_{γ_{1}^{n+1}γ_{2}γ_{1}^{−(n+1)}}^{#}(β_{n}) = lim_{x→0}

ρ_{x}^{(n)}(γ_{1}^{n+1}γ_{2}γ_{1}^{−(n+1)}) which does not equal I, so β_{n} ∈ D_{n}\D_{n+1}. But this implies that the sequence D_{1} ⊃ D_{2} ⊃ ⋯ is strictly decreasing, which contradicts the descending chain condition on closed subsets of R̃(F, G). □

Remark 5.8 If we restrict ourselves to the category of finite groups then we can find a functor with the properties given in Theorem 5.7. We set R̃(F, G) = R_{F} (F, \overline{G}). Then any homomorphism ϕ: F′ → F of finite groups gives rise to a marked homomorphism Φ: (F′, F′) → (F, F) such that |Φ| = ϕ, and we define ϕ̃^{#} = Φ^{#}.

Remark 5.9 Theorem 5.7 says that we cannot make a single uniform choice of compactification R̃(F, G) for every F and preserve functoriality. However, given a single pair of finitely generated groups F, F′ and a homomorphism ϕ: F′ → F, we can extend ϕ^{#} to a morphism Φ^{#}: R_{Δ}(F, \overline{G}) → R_{Δ′}(F′, \overline{G}) for an appropriate choice of markings Δ, Δ′.

6 Approximate Kernels

Let p: F → Q be an epimorphism of finitely generated groups, with kernel N.

Lemma 6.1 There exists an approximate G-kernel of p.

Proof Suppose not. Then we can find finitely generated subsets K_{1} ⊂ K_{2} ⊂ ⋯ of N such that the chain j_{1}^{−1}(ρ_{0}^{K_{1}}) ⊃ j_{2}^{−1}(ρ_{0}^{K_{2}}) ⊃ ⋯ is strictly decreasing, where j_{i} is the inclusion of K_{i} in F. But this contradicts the descending chain condition on closed subsets of R(F, G). □

Remark 6.2 If N is the normal closure of K in F, then for any G, K is an approximate G-kernel of p.

We want to describe the image of P^{#} for compactified representation varieties in a similar way.

Lemma 6.3 Let P: (F, Δ) → (Q, Ξ) be a marked epimorphism. Let (U, ρ) ∈ R_{Δ}(F, \overline{G}). Then ρ is in the image of P^{#} if and only if:

(∗) for all γ, γ̃ ∈ U such that |P |(γ) = |P |(γ̃), we have ρ(γ) = ρ(γ̃).

Proof If (∗) holds then we have a well-defined function σ: |P |(U) → \overline{G} given by σ(|P |(γ)) = ρ(γ) for γ ∈ U. It is easily checked that (|P |(U), σ) is an admissible representation and that ρ = P^{#}(σ) (compare the proof of Proposition 5.1). The converse follows from Equation 1. □

Definition 6.4 Let P: (F, Δ) → (Q, Ξ) be a marked epimorphism. Let K = (K, Θ) be a marked group with K ⊂ ker |P | and Θ ⊂ Δ, and let J: (K, Θ) → (F, Δ) be inclusion. We say that K is an approximate \overline{G}-kernel of P if P^{#}(R_{Ξ}(Q, \overline{G})) = (J^{#})^{−1}(ρ_{0}^{K}). (Note that P^{#}(R_{Ξ}(Q, \overline{G})) is always contained in (J^{#})^{−1}(ρ_{0}^{K}).)

If instead of P we are given an epimorphism p: F → Q then we take P to be the obvious marked epimorphism. If just a normal subgroup N of F is given then we take p to be the canonical epimorphism F → F/N. If just K is given then we take N to be the normal closure of K in F, and we say that K is an approximate kernel of (F, Δ).

Proposition 6.5 Let P: (F, Δ) → (Q, Ξ) be a marked epimorphism, and let K = (K, Θ) be a marked subgroup of (F, Δ) with K ⊂ ker |P |.

(a) If (K, Θ) is an approximate \overline{G}-kernel of P, then K is an approximate Gkernel of |P |.

(b) If K = ker |P | then (K, Θ) is an approximate \overline{G}-kernel of P.

Proof (a) Let ρ ∈ R_{Ξ}(Q, \overline{G}). Since |P | is surjective, ρ belongs to R(Q, G) if and only if P^{#}(ρ) belongs to R(F, G). Part (a) now follows.

(b) Let J: (K, Θ) → (F, Δ) be inclusion. Choose any (U, ρ) ∈ R_{Δ}(F, \overline{G}) such that J^{#}(ρ) = ρ_{0}^{K}. If γ, γ̃ ∈ U such that |P |(γ) = |P |(γ̃) then we have γ̃ = αγ for some α ∈ K. Since ρ(α) = [I], ρ(α)ρ(γ) is defined and ρ(γ̃) = [I] · ρ(γ) = ρ(γ) by Lemma 4.3 (c). Lemma 6.3 now implies that ρ lies in the image of P^{#}, as required. □

Example 6.6 (a) Let s ∈ ℕ and let Δ be the standard marking for F_{s}. Given Θ ⊂ Δ, set K = (⟨Θ⟩, Θ). It is clear that K is an approximate \overline{G}-kernel of (F_{s}, Δ).

(b) Let F, Δ, G, \overline{G}, ρ be as in Example 4.8 (b). Let P be the marked epimorphism from (F, Δ) onto the trivial marked group (1, {1}). Let K = (⟨n⟩, {n}) and let N be the normal closure of K in F. Then N = F = ker |P |; but K is not an approximate \overline{G}-kernel of P, for ρ is nontrivial yet ρ(n) = [I].

Given an extension 1 → N → F → Q → 1 of groups with F finitely generated and given a marking Δ for F, one shouldn’t expect an approximate \overline{G}-kernel to exist, for the reasons given at the end of §3. This leads us to broaden our definition of approximate kernel.

Lemma 6.7 Let p: F → Q be an epimorphism, let Δ_{1}, Δ_{2} be markings for F and let (K, Θ_{1}), (K, Θ_{2}) be marked subgroups of (F, Δ_{1}), (F, Δ_{2}) respectively, with K ⊂ ker p. Suppose that Δ_{1}\Θ_{1} = Δ_{2}\Θ_{2}. Then (K, Θ_{1}) is an approximate \overline{G}-kernel of (F, Δ_{1}) if and only if (K, Θ_{2}) is an approximate \overline{G}-kernel of (F, Δ_{2}).

Proof Let P_{1}: (F, Δ_{1}) → (Q, p(Δ_{1})) and P_{2}: (F, Δ_{2}) → (Q, p(Δ_{2})) be the marked epimorphisms induced by p. Set S = Δ_{1}\Θ_{1} = Δ_{2}\Θ_{2}. Let ρ_{1} ∈ R_{Δ_{1}}(F, \overline{G}) such that ρ_{1}(γ) = [I] for all γ ∈ Θ_{1}. For each γ ∈ Θ_{1}, choose a word v_{γ} in the generators from Θ_{2} such that v_{γ} represents γ. Given a word w_{1} in the generators from Δ_{1}, we obtain a word w_{2} in the generators from Δ_{2} by replacing every occurrence of γ^{±1} in w_{1} by v_{γ}^{±1}, for all γ ∈ Θ_{1}. Then w_{1}, w_{2} represent the same element of F and (w_{1})_{S} = (w_{2})_{S}, where (w_{1})_{S}, (w_{2})_{S} are defined as in Example 2.6.

Define ρ_{2} by ρ_{2}(γ) = [I] for γ ∈ Θ_{2} and ρ_{2}(γ) = ρ_{1}(γ) for γ ∈ S. If w_{1}(ρ_{1}) is defined then w_{2}(ρ_{2}) is defined and w_{2}(ρ_{2}) = (w_{2})_{S}(ρ_{2}) = (w_{1})_{S}(ρ_{1}) = w_{1}(ρ_{1}). It is now straightforward to verify that ρ_{2} belongs to R_{Δ_{2}}(F, \overline{G}) and that moreover, ρ_{1} ∈ Im P_{1}^{#} if and only if ρ_{2} ∈ Im P_{2}^{#}.

By symmetry, we can interchange the roles of (F, Δ_{1}) and (F, Δ_{2}), and the result follows. □

Definition 6.8 We say that a subgroup K of ker p is an approximate \overline{G}- kernel of (F, Δ) if

(∗) (K, Θ) is an approximate \overline{G}-kernel of (F, Δ ∪ Θ) in the sense of Definition 6.4,

for some marking Θ. In view of Lemma 6.7, the question of whether (∗) holds is independent of the choice of Θ.

Theorem 6.9 Let P: (F, Δ) → (Q, Ξ) be a marked epimorphism. There exists a finitely generated subgroup K of ker |P | such that K is an approximate kernel of P.

Proof Given a marked group K̃ = (K̃, Θ̃) with K̃ ⊂ ker |P |, let J_{K}: (K̃, Θ̃) → (F, Δ ∪ Θ̃) and L_{K}: (F, Δ) → (F, Δ ∪ Θ̃) be the inclusions. Define a closed

subset V_{K} of R_{Δ}(F, \overline{G}) by V_{K} = L# ; _{K}((J # ; _{K})^{−1}(ρ_{K} ; 0)). Since the variety R_{Δ}(F, \overline{G}) is a noetherian topological space, we can choose K = (K, Θ) such that V_{K} is minimal. We show that K is an approximate \overline{G}-kernel of (F, Δ ∪ Θ).

Let (U, ρ) ∈ R_{Δ∪Θ}(F, \overline{G}) such that (J_{K}^{#})^{−1}(ρ) = ρ_{0}^{K}. Choose any γ, δ ∈ U such that |P |(γ) = |P |(δ). Then δ = αγ for some α ∈ ker |P |. Set K̃ = (K̃, Θ̃), where K̃ = ⟨K ∪ {α}⟩ and Θ̃ = Θ ∪ {α}, and let M: (F, Δ ∪ Θ) → (F, Δ ∪ Θ̃) be inclusion. Then V_{K} ⊂ V_{K}. Thus V_{K} = V_{K} by choice of K, so we can find

(Ũ, ρ̃) ∈ R_{Δ∪_{Θ}}(F, \overline{G}) such that L# ; _{K}(ρ̃) = L_{K}^{#}(ρ) and (J # ; _{K})^{−1}(ρ̃) = ρ_{K} ; 0. It is easily

checked that M^{#}(ρ̃) = ρ, whence γ, δ ∈ Ũ and ρ̃(γ) = ρ(γ), ρ̃(δ) = ρ(δ). But ρ̃(α) = [I], so

ρ(δ) = ρ̃(δ) = [I] · ρ̃(γ) = ρ̃(γ) = ρ(γ).

By Lemma 6.3, we are done. □

The next example illustrates that we cannot choose a K that works for all choices of marking Δ.

Example 6.10 Let F = F_{2} with free generators γ_{1}, γ_{2} and let Δ = {γ_{1}, γ_{2}}. Let Q = ℤ with generator γ_{1} and let Ξ = {γ_{1}}. Define P: (F, Δ) → (Q, Ξ) by |P |(γ_{1}) = γ_{1}, |P |(γ_{2}) = 1. Let G = PGL_{2}(k) with the standard embedding.

Now choose any finitely generated subgroup K of ker |P | and any marking Θ for K. Since ker |P | is not finitely generated ([7], Theorem 2), we can pick δ ∈ ker |P | such that δ ∉ ⟨K ∪ {γ_{2}}⟩. Let Δ_{δ} = {γ_{1}, γ_{2}, δ}. We show that (K, Θ) is not an approximate \overline{G}-kernel of (F, Δ_{δ} ∪ Θ).

Define ρ(γ) for γ ∈ Δ_{δ} ∪ Θ by

ρ(γ) = [I] for γ ∈ Θ ∪ \{γ_{2}\}, ρ(γ_{1}) = ρ(δ) = [( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )].

We check that ρ gives an admissible representation. If w is any reduced word in the generators from Δ_{δ} ∪ Θ such that w(ρ) is defined, then neither of the letters γ_{1}^{−1} and δ^{−1} can appear in w. There are three possibilities:

(i) neither of γ_{1}, δ appears in w (in which case w(ρ) = [I]); (ii) γ_{1} appears exactly once in w and δ does not appear (in which case w(ρ) =

[( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )]);

(iii) δ appears exactly once in w and γ_{1} does not appear (in which case w(ρ) =

[( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )]).

Let w, w̃ be two reduced words in the generators from Δ_{δ}∪Θ such that w(ρ), w̃(ρ) are defined and w, w̃ represent the same element of F. We must show that w(ρ) = w̃(ρ). Clearly we need only consider the case when (i) holds for w and (iii) holds for w̃: say w̃ = uδv, for some words u, v in the generators from Θ∪{γ_{2}}. But then δ and u^{−1}wv^{−1} represent the same element of F, which is impossible because u^{−1}wv^{−1} represents an element of ⟨K ∪ {γ_{2}}⟩. Thus ρ is well-defined.

We see that ρ(γ) = [I] for γ ∈ Θ, but ρ(δ) ≠ [I], so (K, Θ) is not an approximate \overline{G}-kernel of (F, Δ_{δ} ∪Θ), as claimed. It follows that K is not an approximate \overline{G}-kernel of (F, Δ_{δ}).

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School of Mathematics and Statistics F07 University of Sydney Sydney NSW 2006 Australia