On the profinite distinguishability of hyperbolic Dehn fillings of finite-volume 3-manifolds
aa r X i v : . [ m a t h . A T ] F e b ON THE PROFINITE DISTINGUISHABILITY OF HYPERBOLIC DEHNFILLINGS OF FINITE-VOLUME 3-MANIFOLDS
PAUL RAPOPORT
Abstract.
We use model theory to study relative profinite rigidity of 3-manifold groups andshow that given any residually finite group Γ with finite character variety and single-cusped finitevolume hyperbolic 3-manifold M , cofinitely many Dehn fillings M p/q are profinitely distinguish-able from Γ. Introduction
One of the foundational lines of research in modern geometry has been the study of 3-manifoldtopology; some relevant recent progress has been made primarily through teasing out differencesbetween different manifolds through appeal to their fundamental groups. For some examples, see[19], [1], [17], and [13].It is well known that fundamental groups of finite-volume hyperbolic 3-manifolds are finitely pre-sented, and that the fundamental groups of finite-volume hyperbolic 3-manifolds are residually finite(see [10]). Thus it seems like a natural line of inquiry to try to distinguish the fundamental groupsof 3-manifolds by looking at their finite quotients, since finite quotients of fundamental groups cor-respond to finite-sheeted regular coverings, so that commensurability of 3-manifolds coincides withcommensurability of their fundamental groups.Tying together all of the information about finite quotients of fundamental groups through an in-verse limit leads us to the concept of the profinite completion of the group (see Definition 2.1), andto profinite rigidity (see Definition 2.6); this thus follows in the path laid down by [12], [14], and[18], among many others. In particular, Wilton and Zaleskii in [17, Thm 8.4] show that the profinitecompletion of a geometric 3-manifold group determines its geometry, and further, in [18], they showthat the profinite completion of a 3-manifold group also determines the JSJ decomposition of themanifold.There are two natural notions of profinite rigidity: relative and absolute. Relative profinite rigiditydeals with whether within some family of groups, a given pair of distinct groups can be distin-guished by their profinite completions, while absolute profinite rigidity is more general, dealingwith whether a given group is profinitely distinguishable from every other residually finite group.(For more precise definitions, see Definition 2.6 later on.) Since it depends on a more general no-tion, we might expect that there have been some attempts to construct a few 3-manifold groupswhich are provably profinitely rigid in the absolute sense. In Bridson, MacReynolds, Reid, andSpitler’s breakthrough paper [2], the authors do just that, producing the only known examples ofabsolutely profinitely rigid 3-manifold groups, through careful examination of arithmetic latticesin PSL(2 , C ) ∼ = Isom + ( H ) and the employment of strong assumptions, though thus far there areno infinite families of absolutely profinitely rigid 3-manifold groups, and very little else is known Date : February 23, 2021. about the absolute case. Accordingly, I have restricted my work to the realm of relative profiniterigidity within the set of 3-manifold groups, since its study has more work done for me to build onand more tools available for me to use. On the other hand, the powerful work of Agol in [1] andWise in [19] in particular provided many tools for the study of finite covers of 3-manifolds, openingup in turn the ability to make more progress on the study of profinite rigidity within the realm of3-manifold groups, and my work follows in this vein: my use of the non-Haken assumption herecomes from the foundational work of [3].The study of finite-volume hyperbolic 3-manifolds is a central area within 3-manifold topology:among 3-manifolds, the possession of a hyperbolic structure is the “generic” case, as we see byThurston’s Hyperbolic Dehn Surgery Theorem in [16, Theorem 2.6] and which is explored in moreformal and precise detail by Maher in [9]. We recall further that the study of Dehn fillings isfundamental to 3-manifold topology, and that the volumes of the Dehn fillings M p/q of a hyper-bolic 3-manifold M are strictly less than and converge to vol ( M ) , by [4, Thurston’s Theorem].Additionally, Mostow’s rigidity theorem, which says that geometric invariants of a complete andfinite-volume hyperbolic 3-manifold M , such as volume, are topological invariants, and thus giveinvariants of their fundamental groups. Accordingly, we have even further reason to suspect thatthe study of finite-volume hyperbolic 3-manifolds might be fruitful. Thus, in being different, thisfact gives us (again by Mostow rigidity) that the fundamental groups π ( M p/q ) are different, and itbecomes a natural question whether the profinite completions of these groups are also distinguish-able. As below, this is a question that I answer in the affirmative in cofinitely many cases.It is an open question as to whether all finite-volume hyperbolic 3-manifold groups are absolutelyprofinitely rigid, and furthermore, it is also currently unknown whether there exists some pair ofnon-isometric hyperbolic manifolds that are not profinitely distinguishable. However, my work pre-sented here has helped to further explore this line of inquiry by ruling out a class of possibilities.In particular, we have the following theorem, proved in Section 3: Theorem A.
Let Γ be any finitely generated residually finite group with | χ I C (Γ) | < ∞ , and let M be an oriented finite-volume hyperbolic 3-manifold with a single cusp. Then for hyperbolic Dehnfillings with all but finitely many choices of surgery coefficient M m/n with Λ = π ( M m/n ) , we getthat b Γ = b Λ . My journey takes us through representation theory, as well; in Section 2.2, I define | χ I C (Γ) | inthe above theorem. It turns out to be easier to look at the SL(2 , k )-representation of 3-manifoldgroups over careful choices of field k rather than at the groups themselves, and in turn at thecharacter variety of a given representation than at the representation itself; by a theorem of Cullerand Shalen, [3, Proposition 1.5.2], which here is Theorem 2.8, a point in the character variety of a3-manifold group picks out an irreducible representation up to conjugacy, and by a result of [3] -Theorem 5.1 here - the character variety of a finite-volume non-Haken 3-manifold group is in factfinite. I owe a debt of gratitude to [2], who have used representation theory to think about profinitedistinguishability of specific 3-manifold groups in a different and in some sense less restrictive way.The “special sauce” here is my use of model theory after [11], rather than algebraic geometry:Marker’s work ensures that I can pass back and forth between SL(2 , C ) and SL(2 , F p )-representationsas needed, since both have desirable properties - SL(2 , F p ) is locally finite, while representations intoSL(2 , C ) are both much better understood and more directly connected to more concrete geometricapplications: in particular, any finite-volume hyperbolic manifold corresponds naturally to a finite-covolume lattice within (P)SL(2 , C ), and this gives us a canonical representation. In fact, the order type of the set of all volumes of Dehn fillings of all 3-manifolds is ω ω ! ROFINITE DISTINGUISHABILITY OF HYPERBOLIC 3-MANIFOLDS 3
While this paper was in preparation, [6] was published, which proves a more general version ofCorollary 5.2.1 using totally different methods.2.
Preliminaries
Before we can begin in earnest, we need a few preliminaries on the basics of profinite comple-tions of groups and profinite distinguishability and rigidity, as well as some of the finer aspects ofrepresentation theory we use frequently for the rest of the paper.2.1.
Profinite Completions.
We start by recalling some basic properties of profinite completionsof groups. For a more complete and formal treatment of the elementary characterizing propertiesof profinite completions of groups, I recommend reading through [14].
Definition 2.1.
Let G be a group. The profinite completion of G , denoted by b G , is the inverselimit of the groups G/N , where N varies over all normal subgroups of G of finite index. Proposition 2.2.
The natural homomorphism ι : G → b G has dense image, that is, ι ( G ) = b G . Definition 2.3.
Let G be a group. We call G residually finite if for all = g ∈ G , there exists afinite group H and a homomorphism φ : G → H such that φ ( g ) = 1 . Remark.
We may think of residual finiteness as the capacity for at least one of the finite-index(normal) subgroups of G to tell an arbitrary g ∈ G \{ } apart from the identity, and we may thinkof the profinite completion of a group b G to be the packaging together of all of this finite-indexinformation. Proposition 2.4.
Given a group G , denote by N ( G ) the set of all finite-index normal subgroupsof G . Then the following are equivalent: • G is residually finite; • \ N ∈ N ( G ) N = { } ; and • The natural homomorphism ι : G → b G is injective. Definition 2.5.
We say that two groups
G, H are profinitely equivalent if b G ∼ = b H . Definition 2.6.
Let G be a residually finite group. We say that a residually finite group G is profinitely rigid if for all residually finite groups H , whenever b G ∼ = b H , we have G ∼ = H . We saythat a property is profinitely rigid if any group possessing that property is profinitely rigid. We calltwo groups G, G ′ profinitely distinguishable if b G = c G ′ . Remark.
The central focus of this paper is the profinite distinguishability of -manifold groups. Inthe above definitions, we say that the rigidity or distinguishability is absolute if we consider them inthe general case, allowing H to range over all residually finite groups, and we say that the rigidityor distinguishability is relative to a specific family if we restrict our attention to a specific familyof groups. In that case, we call the set of all groups profinitely equivalent to some group G withinthat family the genus of G in that family. In this paper, we are concerned with 3-manifold groups. The terminology here is not accidental, but borrowed from taxonomy, where family, genus, and species representincreasing levels of specificity of organism.
PAUL RAPOPORT
Representation Theory.Definition 2.7.
Let G be a group, k a field, and H a matrix group over k . Let Hom
Irr ( G, H ) bethe set of irreducible representations of G in H . We define χ Ik (Γ) := Hom Irr (Γ , SL(2 , k )) / ∼ , where ∼ denotes the conjugacy relation. In particular, χ I C (Γ) := Hom Irr (Γ , SL(2 , C )) / ∼ ,χ I F p (Γ) := Hom Irr (Γ , SL(2 , F p )) / ∼ . This last we write as χ Ip (Γ) for the sake of concision. By [3, Proposition 1.5.2], for irreduciblerepresentations into SL(2 , C ), irreducible representations with the same character are the same upto conjugacy, so it suffices to talk about the characters: Theorem 2.8. [3, Proposition 1.5.2]
Let ρ, σ be representations of Π in SL(2 , C ) with χ ρ = χ σ . If ρ is irreducible, then ρ, σ are conjugate. Remark.
This gives us a bijection between the set of irreducible characters, and the set of irreduciblerepresentations up to conjugacy. Accordingly, we call χ I C (Γ) the C -character variety of Γ . We notethat character varieties truly are (reducible) varieties in their own right - for a detailed treatment,see [3] and [7] . Model Theory And Its Uses
What could model theory be doing in a paper on geometric group theory and representationtheory? Well, we use it as a means to prove the following theorem, whose proof is at the end ofthis section:
Theorem 3.1.
Suppose | χ I C (Γ) | = n . Then for cofinitely many p , | χ Ip (Γ) | = n as well. Where model theory comes in is its ability to permit us to pass back and forth between statementsabout C and the same single statement about cofinitely many F p . In any case, if for whatever reasonyou want to blackbox this, you can simply use the above Theorem 3.1.The remainder of this section is devoted to the proof of this Theorem 3.1, but first we need afew closely related definitions and a pair of results from model theory; for a background reference,refer to [11]. Definition 3.2.
The first-order theory of fields has signature given by the constants , , andthe binary functions + , × . It has the axioms that addition makes the set into an abelian group,multiplication is associative, commutative, and distributive with identity , ¬ , and ∀ x : ¬ x =0 → ∃ y : xy = 1 . Definition 3.3.
The first-order theory of algebraically closed fields,
ACF , extends the first-ordertheory of fields by appending countably many axioms, one for each natural number, each of the formthat every polynomial of degree n has at least one root. Definition 3.4.
The first-order theory of algebraically closed fields of characteristic p,
ACF p ,extends ACF by appending the additional axiom that p copies of 1 z }| { · · · + 1 = 0 . ROFINITE DISTINGUISHABILITY OF HYPERBOLIC 3-MANIFOLDS 5
Definition 3.5.
The first-order theory of algebraically closed fields of characteristic 0,
ACF ,extends ACF by appending countably many axioms, one for each prime p , each of the form that ¬ p copies of 1 z }| { · · · + 1 = 0 . Lemma 3.6. [11, Corollary 1.2]
Let
ACF be the first-order theory of algebraically closed fields of characteristic 0, and for rationalprime p , let ACF p be the first-order theory of algebraically complete fields of characteristic p . Let Σ be an L -sentence. Then the following are equivalent:(i) ACF | = Σ ;(ii) ACF p | = Σ for cofinitely many choices of p ;(iii) ACF p | = Σ for infinitely many choices of p ; and(iv) C | = Σ . Lemma 3.7.
Let S ( k ) be a first-order statement in the theory of a field k . Then S ( F p ) holds forinfinitely (in fact, cofinitely) many choices of p if and only if S ( C ) holds.Proof. Assume that S ( C ) holds. By Lemma 3.6, this means that ACF | = S , and thus also that ACF p | = S for cofinitely many p , that is, S ( F p ) holds for those p . On the other hand, assume that S ( C ) does not hold. Then by Lemma 3.6 again we must have ACF | = ¬ S , and thus also that ACF p | = ¬ S for cofinitely many p , that is, S ( F p ) does not hold for those p . (cid:3) We start by looking at how we can use
ACF to talk about matrices in SL(2 , k ). Let x , x , x , x be variables in ACF . Then we define the predicate M ( x , x , x , x )to be x · x − x · x = 1 . The attentive reader may notice that this is exactly the defining relation for the determinant ofa matrix in M(2 , k ) to be 1 in terms of its elements. More subtly, and perhaps more powerfully,one may note a tactic that will be used throughout this section: namely, that we will make ourpredicates complex and full of equations, so that they can do the heavy lifting that a mere abstracttuple cannot do. More simply, though, we bundle 4-tuples of variables that satisfy M and notatethem as matrices A ∈ SL(2 , C ) unless we really do need access to the entries.To write that a given matrix is the identity is actually even easier. We define the predicate Id ( x , x , x , x )to be x = 1 ∧ x = 0 ∧ x = 0 ∧ x = 1 . We can also just write Id ( A ), when we have a 4-tuple as mentioned above.Before we can look at how to extend our method for talking about matrices of SL(2 , k ) in ACF to a method for talking about representations Γ → SL(2 , k ), we are going to need to be able to writepredicates that verify that each relation of Γ is satisfied. Consider how, for some finitely presentedgroup Γ = h L | R i , one might describe a new representation ρ : Γ → SL (2 , k ). It suffices to specify what the mapdoes to a given choice of generators of G , ( l i ) ρ ( l i ). It is worth noting that this gives us PAUL RAPOPORT a natural map Hom(Γ , SL (2 , k )) → k l determined by which element of SL(2 , k ) that particular ρ ∈ Hom(Γ , SL(2 , k )) sends the ordered l -tuple of generators of Γ to, interpreted by reading offthe matrix entries. This is certainly injective - different elements of Hom(Γ , SL(2 , k )) send at leastone generator of Γ to different matrices. What [3] gives us is the conceptualization of the imageas also some vanishing set V Γ ( k ), and then additionally that using the next few results (when wehave proven them) that we can also can recognize which points of k l are in V Γ ( k ) and are thustrue representations by understanding V Γ ( k ) as the vanishing set of the polynomial relations in R ,along with the polynomials ensuring that the generators map to elements of SL(2 , k ). But to makeuse of all thus, we have to tackle the challenge of how to communicate all of the machinery usedhere in ACF first. Bringing this to
ACF , let ( A i ) := A , · · · , A l be 4-tuples such that l ^ i =1 M ( A i ) . That is, the vector in k l can be thought of more helpfully as a l -tuple of 4-tuples, each of whichwe conceptualize as a matrix. As such, we write the l -tuple ( A i ) li =1 as ~A . A couple of lemmata onthe relation between the entries of matrices and those of their products and inverses mean we canuse T to talk about the satisfaction of relations: Lemma 3.8.
Let
A, B ∈ SL(2 , k ) . Then the entries of AB, A − are polynomial in the entries of A, B . Corollary 3.8.1.
Let F h L i be the set of freely reduced words on some finite set of letters and theirinverses, l = | L | , r ∈ F h L i . Let f r : k l → k be the map treating successive -tuples of the argumentas matrix elements of a generating set, interpreting concatenation as matrix multiplication, andinverses of letters as inverses of generators, to take r to its image under this choice of assignment.Then for all ~z ∈ k l , f r ( ~z ) is polynomial in the z i . In particular, we use this in the case where the z i represent elements of SL(2 , C ).Now that we have ensured that nothing goes horribly wrong when we talk about matrix inver-sions, matrix products, and free reduction in SL(2 , k ), we can now figure out how to use ACF towrite that a relationship r ∈ R is satisfied by some l -tuple of matrices. For r ∈ R , we define thepredicate SAT r ( ~A )to be Id ( f r ( ~A )) . That is, the f r map reads off the word r and takes the l -tuple it spells out from the A i , to theidentity, as is required by the fact that the r ∈ R are the relations of G .Perhaps the most important immediate idea of this section is that we can use these previouspredicates to write a statement in first-order logic that says whether or not a given 4 l -tuple is arepresentation of G into SL(2 , k ). In particular, we define REP G ( ~A ) ROFINITE DISTINGUISHABILITY OF HYPERBOLIC 3-MANIFOLDS 7 to be l ^ i =1 M ( A i ) ! ∧ ^ r ∈ R SAT r ( ~A ) ! , where the G subscript reminds us that we started by fixing G and a presentation for G once andfor all. Half-translating towards prose, this says that a 4 l -tuple A corresponds to a representationof G if all l of the 4-tuples are matrices of SL(2 , k ), and that all of the relations r ∈ R of G aresatisfied. Later on we use this for discrete finite-covolume subgroups of SL(2 , C ), designated bycapital Greek letters such as Γ. Proposition 3.9. [3, Corollary to Proposition 1.4.1]
Let Γ be a finitely presented group, with finite presentation Γ = h L | R i , and let V Γ ( k ) = { ~A | REP Γ ( ~A ) } = { ~A | V ( f r ( ~A ) − I ) , M ( A i ) li =1 } ⊆ k l . Then the map Φ Γ : V Γ → Hom(Γ , SL(2 , k )) taking points in V Γ to the representations that they determine is well-defined and is a bijection. Remark.
The Hilbert basis theorem permits us to assume finite generation without loss of gener-ality, rather than the stronger condition of finite presentation, and, in proofs, to pass to the case offinite presentation rather than finite generation.Proof.
We start by noting that the first part of
REP Γ uses the M predicate to check that the4 l -tuple genuinely is an l -tuple of matrices in SL(2 , k ). Thinking once more of ~A as a 4 l -tuple v ,this defines a map ψ v : L → SL(2 , k ) , a i A i . Then by the universal property of free groups, thisextends to a map ˇ ψ v : F ( L ) → SL(2 , k ) under multiplication and inverses (see the diagram below).Subsequently, the second part of
REP Γ uses each SAT ( r ) to check that each relation of Γ issatisfied, and by the universal property of groups defined by presentations, this uniquely determinesa representation ρ v : Γ → SL(2 , k ). But then that means that Φ Γ ( v ) = ρ v . (cid:3) L Γ F ( L Γ ) ΓSL(2 , k ) ψ v ˇ ψ v ρ v Remark.
After [3] , we call V Γ ( k ) the SL(2 , k )-representation variety of Γ . Now that we have established that we can talk about whether a given 4 l -tuple corresponds to arepresentation of G , we can talk about whether that representation is irreducible. As it turns out,though, it is much easier to start with reducibility. We recall that a representation into SL(2 , k ) is reducible if the action by all of the images of the generators on k fix some line through the origin:more formally, for some generator-dependent λ ∈ k , (cid:18) a bc d (cid:19) (cid:18) v v (cid:19) = (cid:18) λv λv (cid:19) , where (cid:18) a bc d (cid:19) ranges over the images of all generators of Γ. We can therefore define thepredicate RED ( ~A ) PAUL RAPOPORT to be
REP G ( ~A ) ∧ ∃ a ∃ b n ^ i =1 ∃ λ i : (( a, b ) = (0 , ∧ A i · h a, b i = λ i h a, b i , where we treat h a, b i as a column vector, and matrix and scalar multiplication are accordinglyappropriately defined. Proposition 3.10.
For all l -tuples ( A i ) , ACF | = RED ( ~A ) if and only if Φ( ~A ) is a reduciblerepresentation. We can then talk about irreducibility, defining the predicate
IRREP ( ~A )to be REP G ( ~A ) ∧ ¬ RED ( ~A ) . With a little more work, we can also talk about whether two representations are conjugate. Werecall that two representations ρ, σ : G → SL(2 , k ) are conjugate if there exists some matrix M suchthat for all i , M ρ ( x i ) M − = σ ( x i ), where x i is the i th generator of G under some fixed choice ofordering. We can represent this in model theory by defining the predicate CON J ( ~A, ~B )to be REP G ( ~A ) ∧ REP G ( ~B ) ∧ ∃ C : M ( C ) ∧ l ^ j =1 ( CA j C − = B j ) . Proposition 3.11.
For all pairs of l -tuples ~A, ~B , ACF | = CON J ( ~A, ~B ) if and only if Φ( ~A ) ∼ Φ( ~B ) , that is, there exists some C ∈ SL(2 , k ) with τ C ◦ Φ( ~A ) = Φ( ~B ) , where τ C is the innerautomorphism of SL(2 , k ) that C defines. The whole point of this section, of course, was to be able to talk about the number of irreduciblerepresentations of a given finitely generated group G into SL(2 , k ), up to conjugacy. However,this certainly is not a proper sentence in first-order logic. We might think of the “half-translated”version of Σ G,n as the following:There exist n irreducible representations up to conjugacy of G into SL(2 , k ), they are notconjugate to each other, and any other irreducible representation of G into SL(2 , k ) must beconjugate to one of the n representations previously mentioned.We now have all the tools we need to write the previously mentioned sentence Σ G,n ; this will bealmost exactly a predicate-by-predicate, symbol-by-symbol calque of what was written above as thehalf-translation, making use of the predicates we have constructed here. We write the first-ordersentence Σ
G,n as n ^ j =1 ∃ ~A ( j ) : IRREP ( ~A ( j ) ) ∧ n ^ j,j ′ =1 j = j ′ ¬ CON J ( ~A ( j ) , ~A ( j ′ ) ) ∧ ∀ ~B : IRREP ( ~B ) ⇒ n _ j =1 CON J ( ~B, ~A ( j ) ) . Theorem 3.12.
Let G be a finitely generated group. Then there exists a sentence Σ G,n in ACF such that
ACF | = Σ G,n if and only if | χ I C ( G ) | = n . ROFINITE DISTINGUISHABILITY OF HYPERBOLIC 3-MANIFOLDS 9
Proof.
Since G is merely finitely generated and not finitely presented, we must invoke the HilbertBasis Theorem: using it, we have that there exists a finitely presented ˜ G, η : ˜ G → G such thatthe induced map η ∗ : χ I C ( G ) → χ I C ( ˜ G ) is a bijection. Then the sentence Σ ˜ G,n for ˜ G also works for G . (cid:3) Lemma 3.13.
Let G be a finitely generated subgroup of SL(2 , F p ) . Then | G | is finite.Proof. To see this, we note that some generator will have an entry belonging to the largest F p k among the set of generators, and neither addition nor matrix multiplication can increase the size;finally, any given SL(2 , F p k ) is finite. (cid:3) Theorem 3.1.
Suppose | χ I C (Γ) | = n . Then for cofinitely many p , | χ Ip (Γ) | = n as well.Proof. This follows immediately from Lemma 3.6 and Theorem 3.12; we recall that in particular,since | χ I C (Γ) | = n , | χ Ip (Γ) | = n for infinitely many p , and thus by Lemma 3.6, for cofinitely many p . (cid:3) Remark.
This will be the key tool allowing us to pass between representations into
SL(2 , C ) , whichwe understand well, and SL(2 , F p ) , which we prize for its local finiteness. The goal will now be touse the theory of profinite distinguishability to prove the Main Theorem. Constraints on Profinite Completions
Now that we have established that we can pass (almost) freely between representations intoSL(2 , C ) and into SL(2 , F p ), we can start to get a sense of what this means for the profinite distin-guishability of groups. Lemma 4.1.
Let Γ , Λ be two finitely generated groups such that b Γ ∼ = b Λ . Suppose that | χ Ip (Γ) | = n for cofinitely many p . Then | χ Ip (Γ) | = | χ Ip (Λ) | = n for those p .Proof. Consider the following commutative diagram.Γ SL(2 , F p ) b Γ = b Λ Λ i Γ i Λ ˆ ρ =ˆ σρ σ We note that by the universal property of profinite completions, any representation ρ : Γ → SL(2 , F p ) extends profinitely to a representation ˆ ρ : b Γ → SL(2 , F p ) by the local finiteness ofSL(2 , F p ), as in Lemma 3.13. We may note that by the commutativity of the diagram, any represen-tation from χ Ip (Γ) must factor as a composition of the canonical injection into b Γ and a representationfrom χ Ip ( b Γ), so that | χ Ip (Γ) | = | χ Ip ( b Γ) | . Looking at the other half of the diagram, we note that since b Γ = b Λ, the same argument applies in reverse: compositions of the canonical injection of Λ into b Λwith representations from χ Ip ( b Λ) must yield all of χ Ip (Λ), so that | χ Ip (Λ) | = | χ Ip ( b Λ) | . (cid:3) Theorem 4.2.
Let Γ , Λ be two finitely generated groups with b Γ ∼ = b Λ and | χ I C (Γ) | = n for some n ∈ N . Then | χ I C (Γ) | = | χ I C (Λ) | . Proof.
Since | χ I C (Γ) | = n , by Lemma 3.7 and the statement of Σ Γ ,n , we know that | χ Ip (Γ) | = n forcofinitely many p . By profinite equivalence and Lemma 4.1, we know that | χ Ip (Λ) | = | χ Ip (Γ) | = n .Finally, by another application of Lemma 3.7 and the statement of Σ Λ ,n , | χ I C (Λ) | = n . (cid:3) Theorem 4.3.
Let
Γ = π ( M ) , where M is a compact hyperbolic -manifold. If deg ( T F (Γ)) ≥ d for some d ∈ N , then | χ I C (Γ) | ≥ d .Proof. Let Γ be the fundamental group of a finite-volume hyperbolic 3-manifold, and let k = T F (Γ)be its trace field. Let { σ i } : k ֒ → C be a family of distinct embeddings, and let θ : Γ ֒ → SL(2 , k )be the map realizing elements of Γ as matrices. Let ˆ σ i : SL(2 , k ) ֒ → SL(2 , C ) be the map byinterpretation of matrix entries induced by the { σ i } . Then the result will follow if ρ i = ˆ σ i ◦ θ is arepresentation ρ i : Γ → SL(2 , C ), and if for i = j , ρ i , ρ j are nonconjugate.To see this, recall that k is a number field by [8, Theorem 3.1.2]. Denote its degree over Q as d = deg ( k ), so that { σ i } di =1 : k ֒ → C are its d -many embeddings. Then since the maps { σ i } : k → C are all different, and are all embeddings, they cannot agree on every γ ∈ Γ: there must exist some γ ∈ Γ such that ˆ σ i ◦ θ ( γ ) = ˆ σ j ◦ θ ( γ ).But then given that ρ i = ˆ σ i ◦ θ , ρ j = ˆ σ j ◦ θ , for tr : SL(2 , C ) → C the trace map, tr ◦ ρ i ( γ ) = tr ◦ ρ j ( γ ).Thus γ represents a witnessing element of Γ on which the representations ρ i , ρ j have different traces,which by Proposition 2.8 tells us that the two representations cannot be conjugate. (cid:3) Having shown that profinite equivalence means that the number of representations up to conju-gacy into SL(2 , k ) (if finite) are the same between the profinitely equivalent groups, the goal is nowto attack the main theorem. 5.
Main Theorem
Before we get to the main theorem, we need to call on a pair of results. First, a consequence ofthe results in [3]; for the sake of precision, we cite the result as stated in [15]:
Theorem 5.1. [15, Lemma 2.2]
Let M be a non-Haken orientable hyperbolic -manifold of finite volume. Then its character varietyis finite. And now this, from [7]:
Theorem 5.2. [7, Theorem 3.2]
Let M be an orientable hyperbolic -manifold of finite volume, and d ∈ N . Then there are finitelymany pairs ( m, n ) such that tr ( ρ ( π ( M m/n )) ∈ k where deg ( k/ Q ) ≤ d as an extension. Theorem A.
Let Γ be any finitely generated residually finite group with | χ I C (Γ) | < ∞ , and let M be an oriented finite-volume hyperbolic 3-manifold with a single cusp. Then for hyperbolic Dehnfillings with all but finitely many choices of surgery coefficient M m/n with Λ = π ( M m/n ) , we getthat b Γ = b Λ .Proof. It suffices to show that b Γ ∼ = b Λ only for finitely many Λ; we thus assume that b Γ ∼ = b Λ. Byassumption, | χ I C (Γ) | < ∞ ; let | χ I C (Γ) | = d . However, by Lemma 4.2, we know that since b Γ ∼ = b Λ, | χ I C (Γ) | = | χ I C (Λ) | . Now, by Theorem 5.2, we know that there are at most finitely many choices ofsurgery coefficient resulting in a manifold with degree of trace field of fundamental group with atmost a given degree d + 1, and by Theorem 4.3, we know that if deg ( T F (Λ)) > d , then | χ I C (Λ) | > d as well, so that it is exactly these finitely many choices of surgery coefficient where it is even possible ROFINITE DISTINGUISHABILITY OF HYPERBOLIC 3-MANIFOLDS 11 for us to have | χ I C (Λ) | = d . Finally, by Proposition 2.8, we know that it suffices to check on thelevel of traces, since irreducible representations with the same trace are conjugate. (cid:3) The above result thus lends itself to the following more geometrically focused corollary:
Corollary 5.2.1.
Let M be a one-cusped, finite-volume, hyperbolic 3-manifold. Suppose M m/n isa hyperbolic Dehn filling of M with surgery coefficients m/n and with finite character variety (forinstance, a non-Haken such filling), and let Γ = π ( M m/n ) . Then for hyperbolic Dehn fillings withall but finitely many other choices of surgery coefficient M m ′ /n ′ with Λ = π ( M m ′ /n ′ ) , we get that b Γ = b Λ .Proof. We may start by passing without loss of generality to the case where M m ′ /n ′ has hyperbolicstructure, thanks to Theorems A and 8.4 from [17]. By assumption, | χ I C (Γ) | < ∞ , so that TheoremA applies. (cid:3) Remark.
Liu in [6] proves a more general version of this result using completely different methods.
We can extend this to the question that actually motivated this entire line of inquiry with a littlehelp from [5]:
Definition 5.3.
A knot K is small if its complement S \ K contains no closed incompressiblesurface. Theorem 5.4. [5, Unnumbered Theorem from p. 373-374]
Let K be a small knot. Then all but finitely many of its Dehn fillings M m/n = { S \ K } m/n arenon-Haken. This allows us to narrow in on the trailhead for this line of thought: that we can use all of thisto say something interesting about hyperbolic Dehn fillings of small knots.
Corollary 5.4.1.
Let K be a small knot such that S \ K = M is a one-cusped, finite-volume,hyperbolic 3-manifold. Let Γ = π ( M m/n ) be the fundamental group of a non-Haken hyperbolicDehn filling of M with surgery coefficients m/n . Then for Dehn fillings with all but finitely manyother choices of surgery coefficient M m ′ /n ′ with Λ = π ( M m ′ /n ′ ) , we get that b Γ = b Λ .Proof. This follows from Theorems A and 5.4, given the fact that knot complements have infinitecyclic first homology and thus that fillings whose slopes have different numerators have differentfirst homology, which profinite completion detects. (cid:3)
Acknowledgments
My deepest thanks to Daniel Groves, without whose generosity of time, attention, and expertise,none of this would have been possible from the start and to Alan Reid, who provided the initialline of pursuit, as well as some improvements to the main theorem.
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