DOCSLIB.ORG

Algebraic Completions of Discrete and Profinite Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebraic Completions of Discrete and Profinite Groups Motivation Unipotent Completion Relative Unipotent Completion The Profinite Case Weighted Completion Algebraic Completions of Discrete and Profinite Groups Richard Hain Duke University November 6, 2008 Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Unipotent Completion What? Relative Unipotent Completion Why? The Profinite Case How? Weighted Completion The goal The problem is to “linearlize” a given a discrete (or profinite) group Γ. That is, we want to turn Γ into a vector space over a given field F. Naive Example ab We can replace Γ by H1(Γ, F) := Γ ⊗Z F. This is not very enlightening as, in general, we lose too much information — such as all information about the non-abelian properties of Γ. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Unipotent Completion What? Relative Unipotent Completion Why? The Profinite Case How? Weighted Completion Why? 1 To understand Γ by “simplifying” it. In general, finitely presented groups are difficult to understand, even if you have a presentation. If you don’t believe this, I will give you a nice presentation of the mapping class group Γg in any genus ≥ 2, and you can tell 2 me what H (Γg, V ) is, where V is any non-trivial representation of Spg, pulled back along the natural homomorphism Γg → Spg. 2 To accomodate and study “motivic” structures on Γ, such as Hodge structures, (linear) Galois actions, etc. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Unipotent Completion What? Relative Unipotent Completion Why? The Profinite Case How? Weighted Completion How? We will replace Γ by an affine proalgebraic group G over F. An affine proalgebraic group is just an inverse limit G = ←−lim Gα α of affine F-groups G. Knowing G is equivalent to knowning its coordinate ring O(G) := O( ), −→lim Gα α which is a Hopf algebra — and so a vector space. G g := g We can also replace by its Lie algebra −→lim α. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Constructions Unipotent Completion Examples Relative Unipotent Completion Presentations The Profinite Case More structure Weighted Completion More Examples Definition Unipotent completion, also called Malcev completion, has its origins in a theorem of Malcev and is probably due to Quillen (1969). Fix a field F of characteristic zero, such as Q. The unipotent completion of Γ over F is defined by Γun = Γun := lim U /F ←− ρ ρ where Uρ is a unipotent F-group and ρ :Γ → Uρ(F) is a Zariski dense homomorphism. There is a natural homomorphism ρ˜ :Γ → Γun(F). Every homomorphism from Γ to a unipotent F-group factors through ρ˜. Consequently, the category of unipotent representations of Γ over F is the category of finite dimensional representations of Γun. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Constructions Unipotent Completion Examples Relative Unipotent Completion Presentations The Profinite Case More structure Weighted Completion More Examples Alternative construction Let J be the kernel of the augmentation : FΓ → F. The J-adic completion of FΓ is Γ∧ : Γ/ n. F ←−lim F J n This is a complete Hopf algebra. Then Γun(F) = group-like elets of FΓ∧ = {x ∈ 1 + J∧ : ∆x = x ⊗ x}. Its Lie algebra is the set of primitive elements {x ∈ FΓ∧ : ∆(x) = 1 ⊗ x + x ⊗ 1} of FΓ∧ and is pronilpotent. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Constructions Unipotent Completion Examples Relative Unipotent Completion Presentations The Profinite Case More structure Weighted Completion More Examples Basic examples 1 The completion of a finitely generated abelian group Γ is n un n Γ ⊗Z F. For example (Z ) = (Ga) : 1 n 1 F n n =∼ Z → =∼ (F)n. Z 0 1 0 1 Ga 2 The unipotent completion of the integral Heisenberg group Γ is the Heisenberg group H: 1 ZZ 1 FF Γ = 0 1 Z → 0 1 F = H(F). 0 0 1 0 0 1 Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Constructions Unipotent Completion Examples Relative Unipotent Completion Presentations The Profinite Case More structure Weighted Completion More Examples The unipotent completion of a free group If Γ is the free group Γ = hx1,..., xni, then ∧ ' θ : FΓ −→ FhhX1,..., Xnii is a (complete) Hopf algebra isomorphism where each Xj is defined to be primitive and θ(xj ) = exp Xj j = 1,..., n. un ' ∧ θ induces an isomorphism Γ (F) −→ exp L(X1,..., Xn) . The un ∧ Lie algebra of Γ is L(X1,..., Xn) . Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Constructions Unipotent Completion Examples Relative Unipotent Completion Presentations The Profinite Case More structure Weighted Completion More Examples The unipotent completion of a finitely presented group un If Γ = hx1,..., xn : r1,..., rmi, then the Lie algebra of Γ is ∧ L(X1,..., Xn) /(log θ(rj ): j = 1,..., m). This works in simple cases, but is not useful in general. Examples 1 If H1(Γ, F) = F (e.g., Γ is a knot group or a braid group Bn), un ∼ then Γ = Ga. 2 If H1(Γ, F) = 0 (e.g. Γ is a mapping class group or an irreducible lattice in a semi-simple Lie group of real rank ≥ 2), then Γun is trivial. These illustrate some limitations of unipotent completion. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Constructions Unipotent Completion Examples Relative Unipotent Completion Presentations The Profinite Case More structure Weighted Completion More Examples Minimal presentations of pronilpotent Lie algebras Suppose that u is a pronilpotent Lie algebra over F. A section s of u → H1(u) induces a continuous surjection ∧ f := L(H1(u)) → u, so that u =∼ f/r where r ⊆ [f, f]. The image of any section of r → r/[r, f] = H2(u) will be a minimal set of relations. Thus u has a minimal presentation of the form ∧ u = L(H1(u)) /(H2(u)) ∧ ∧ for a suitable embedding H2(u) ,→ [L , L ]. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Constructions Unipotent Completion Examples Relative Unipotent Completion Presentations The Profinite Case More structure Weighted Completion More Examples Basic structure theorem un The Lie algebra u of Γ is controlled by H1(u) and H2(u). Theorem There are natural homomorphisms Hk (Γ, F) → Hk (u) ∼ If Γ is a finitely presented group, then H1(u) = H1(Γ, F) and H2(Γ, F) → H2(u) is surjective. The homomorphism ∧ ∧ ∧ ∧ ∧ ∼ 2 H2(u) → [L , L ]/[L , L , L ] = Λ H1(u) is dual to the cup product Λ2H1(u) → H2(u). There is also a version for Γ that are not finitely presented. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Constructions Unipotent Completion Examples Relative Unipotent Completion Presentations The Profinite Case More structure Weighted Completion More Examples Hodge and Galois structures Theorem If X is an algebraic variety over a subfield K of C, then un 1 (Morgan, Hain) π1(X( ), x) has a natural mixed Hodge C /Q structure for all x ∈ X(C); un 2 (Deligne) π1(X( ), x) has a natural GK -action for all C /Q` K -rational points x of X. Corollary (Morgan (1978)) If X is smooth complex algebraic variety, then the Lie algebra of un π1(X, x) has a minimal presentation with generators of weights in {−1, −2} and relations of weights in {−2, −3, −4}. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Constructions Unipotent Completion Examples Relative Unipotent Completion Presentations The Profinite Case More structure Weighted Completion More Examples Serre’s question In the 1960’s Serre asked if every finitely presented group could occur at the fundamental group of a smooth quasi projective variety X over C. un Morgan’s result implies that the Lie algebra u of π1(X, x) is “weighted homogeneous” with at at worst quadratic, cubic and quartic relations. Example (Morgan) The group Γ = hx, y :[x, [x, [x, [x, y]]]]i is not the fundamental group of any smooth quasi-projective variety. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Constructions Unipotent Completion Examples Relative Unipotent Completion Presentations The Profinite Case More structure Weighted Completion More Examples Pure braid groups Example (Kohno) The pure braid group Pn is the fundamental group of a hyperplane complement Xn whose H1 is pure of weight −2. un This implies that the Lie algebra of Pn is quadratically presented: ∧ pn = L(Xjk ) /([Xij , Xkm], [Xij , Xik + Xjk ]: i, j, k, m distinct) 2 1 2 The quadratic relations are dual to Λ H (Xn) → H (Xn); they are easily computed from Brieskorn’s description of the cohomology ring of Xn. Similarly, the Lie algebra of the unipotent completion of π1 of any compact Kähler manifold X is quadratically presented. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Generalities Unipotent Completion Structure Relative Unipotent Completion Examples The Profinite Case More structure Weighted Completion Definition Suppose that Γ is a discrete group, S is a reductive F-group and that ρ :Γ → S(F) is a Zariski dense representation. The completion of Γ relative to ρ is G := ←−lim Gφ φ where φ ranges over all Zariski dense representations Γ → Gφ(F) such that: (1) Gφ is an affine F group that is an extension of S by a unipotent group: 1 → Uφ → Gφ → S → 1 and; (2) the composite Γ → Gφ(F) → S(F) is ρ. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Generalities Unipotent Completion Structure Relative Unipotent Completion Examples The Profinite Case More structure Weighted Completion Observations Unipotent completion is the case where S is trivial. Note that G is an extension 1 → U → G → S → 1 U = of S by the prounipotent group ←−lim Uφ.
Load more

Recommended publications
  • Fundamental Groups of Schemes
    Fundamental Groups of Schemes Master thesis under the supervision of Jilong Tong Lei Yang Universite Bordeaux 1 E-mail address: [email protected] Chapter 1. Introduction 3 Chapter 2. Galois categories 5 1. Galois categories 5 §1. Definition and elementary properties. 5 §2. Examples and the main theorem 7 §2.1. The topological covers 7 §2.2. The category C(Π) and the main theorem 7 2. Galois objects. 8 3. Proof of the main theorem 12 4. Functoriality of Galois categories 15 Chapter 3. Etale covers 19 1. Some results in scheme theory. 19 2. The category of étale covers of a connected scheme 20 3. Reformulation of functoriality 22 Chapter 4. Properties and examples of the étale fundamental group 25 1. Spectrum of a field 25 2. The first homotopy sequence. 25 3. More examples 30 §1. Normal base scheme 30 §2. Abelian varieties 33 §2.1. Group schemes 33 §2.2. Abelian Varieties 35 §3. Geometrically connected schemes of finite type 39 4. G.A.G.A. theorems 39 Chapter 5. Structure of geometric fundamental groups of smooth curves 41 1. Introduction 41 2. Case of characteristic zero 42 §1. The case k = C 43 §2. General case 43 3. Case of positive characteristic 44 (p0) §1. π1(X) 44 §1.1. Lifting of curves to characteristic 0 44 §1.2. the specialization theory of Grothendieck 45 §1.3. Conclusion 45 ab §2. π1 46 §3. Some words about open curves. 47 Bibliography 49 Contents CHAPTER 1 Introduction The topological fundamental group can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the asso- ciated universal covering space.
    [Show full text]
  • Abstract Quotients of Profinite Groups, After Nikolov and Segal
    ABSTRACT QUOTIENTS OF PROFINITE GROUPS, AFTER NIKOLOV AND SEGAL BENJAMIN KLOPSCH Abstract. In this expanded account of a talk given at the Oberwolfach Ar- beitsgemeinschaft “Totally Disconnected Groups”, October 2014, we discuss results of Nikolay Nikolov and Dan Segal on abstract quotients of compact Hausdorff topological groups, paying special attention to the class of finitely generated profinite groups. Our primary source is [17]. Sidestepping all difficult and technical proofs, we present a selection of accessible arguments to illuminate key ideas in the subject. 1. Introduction §1.1. Many concepts and techniques in the theory of finite groups depend in- trinsically on the assumption that the groups considered are a priori finite. The theoretical framework based on such methods has led to marvellous achievements, including – as a particular highlight – the classification of all finite simple groups. Notwithstanding, the same methods are only of limited use in the study of infinite groups: it remains mysterious how one could possibly pin down the structure of a general infinite group in a systematic way. Significantly more can be said if such a group comes equipped with additional information, such as a structure-preserving action on a notable geometric object. A coherent approach to studying restricted classes of infinite groups is found by imposing suitable ‘finiteness conditions’, i.e., conditions that generalise the notion of being finite but are significantly more flexible, such as the group being finitely generated or compact with respect to a natural topology. One rather fruitful theme, fusing methods from finite and infinite group theory, consists in studying the interplay between an infinite group Γ and the collection of all its finite quotients.
    [Show full text]
  • 24 the Idele Group, Profinite Groups, Infinite Galois Theory
    18.785 Number theory I Fall 2016 Lecture #24 12/06/2015 24 The idele group, profinite groups, infinite Galois theory 24.1 The idele group Let K be a global field. Having introduced the ring of adeles AK in the previous lecture, it is natural to ask about its unit group. As a group we have × × × AK = f(av) 2 AK : av 2 Kv for all v 2 MK ; and av 2 Ov for almost all v 2 MK g; however, as a subspace of AK , this is not a topological group because the inversion map a 7! a−1 need not be continuous. Example 24.1. Consider K = Q and for each prime p let xp = (1; 1;:::; 1; 1; p; 1; 1;:::) 2 AQ be the adele with a p in its pth component and 1's elsewhere. Every basic open set U about 1 in A has the form Q Y Y U = Uv × Ov; v2S V 62S with S ⊆ MQ finite and 1v 2 Uv, and it is clear that U contains xp for all sufficiently large p. It follows that the sequence x2; x3; x5;::: converges to 1 in the topology of AQ. But notice −1 −1 −1 −1 that U does not contain xp for all sufficiently large p, so the sequence x2 ; x3 ; x5 ;::: −1 −1 cannot possibly converge to 1 = 1 in AQ. Thus the function x ! x is not continuous × in the subspace topology for AK . This problem is not specific to rings of adeles. In general, given a topological ring R, there is no reason to expect its unit group R× ⊆ R to be a topological group in the subspace topology unless R happens to be a subring of a topological field (the definition of which × requires inversion to be continuous), as is the case for the unit group OK ; this explains why we have not encountered this problem up to now.
    [Show full text]
  • Subgroups of Finite Index in Profinite Groups
    Subgroups of Finite Index in Profinite Groups Sara Jensen May 14, 2013 1 Introduction In addition to having a group structure, profinite groups have a nontrivial topological structure. Many results pertaining to profinite groups exploit both structures, and there- fore both structures are important in understanding profinite groups. An amazing result due to Nikolov and Segal is the following theorem. Theorem 1.1. Suppose that G is a topologically finitely generated profinite group. Then every subgroup of G of finite index is open. One way to view Theorem 1.1 is as a statement that the algebraic structure of a finitely generated profinite group somehow also encodes the topological structure. That is, if one wishes to know the open subgroups of a profinite group G, a topological property, one must only consider the subgroups of G of finite index, an algebraic property. As profinite groups are compact topological spaces, an open subgroup of G necessarily has finite index. Thus it is also possible to begin with a subgroup of G having a particular topological property (the subgroup is open) and deduce that this subgroup must also have a particular algebraic property (the subgroup has finite index). The proof of Theorem 1.1 is quite extensive, and requires the classification of finite simple groups. However, if one restricts attention to a smaller class of groups, the result can be done in a fairly straightforward manner. Suppose that G is a finite group having a normal series 1 = Gl ⊆ Gl−1 ⊆ ::: ⊆ G1 ⊆ G0 = G such that Gi=Gi+1 is nilpotent for all 0 ≤ i < l.
    [Show full text]
  • The Equivalence Between Rational G-Sheaves and Rational G-Mackey
    THE EQUIVALENCE BETWEEN RATIONAL G-SHEAVES AND RATIONAL G-MACKEY FUNCTORS FOR PROFINITE G DAVID BARNES AND DANNY SUGRUE Abstract. For G a profinite group, we construct an equivalence between rational G-Mackey functors and a certain full subcategory of G-sheaves over the space of closed subgroups of G called Weyl-G-sheaves. This subcategory consists of those sheaves whose stalk over a subgroup K is K-fixed. This extends the classification of rational G-Mackey functors for finite G of Thévenaz and Webb, and Greenlees and May to a new class of examples. Moreover, this equivalence is instru- mental in the classification of rational G-spectra for profinite G, as given in the second author’s thesis. Contents 1. Introduction 2 1.1. Organisation 3 2. Mackey functors and sheaves 3 2.1. Basic facts on profinite groups 3 2.2. Mackey functors for profinite groups 4 2.3. Burnside ring idempotents 6 2.4. Equivariant sheaves 10 3. The functors 11 3.1. Weyl-G-sheaves determine Mackey Functors 11 3.2. Mackey Functors determine Weyl-G-sheaves 13 4. The equivalence 16 4.1. The equivalence on sheaves 17 4.2. The equivalence on Mackey functors 19 5. Consequences 20 5.1. Examples 20 arXiv:2002.11745v1 [math.AT] 26 Feb 2020 5.2. Weyl Sheaves from equivariant sheaves 21 References 22 1 WEYL SHEAVES AND MACKEY FUNCTORS 2 1. Introduction The classification of rational Mackey functors for finite groups is well-known and highly useful. An algebraic version of this result is given by Thévenaz and Webb [TW95].
    [Show full text]
  • The Étale Fundamental Group Wouter Zomervrucht, December 9, 2014
    The étale fundamental group Wouter Zomervrucht, December 9, 2014 1. Topology Let X be a connected topological space. Let x 2 X be a point. An important invariant of (X, x) is the (topological) fundamental group p(X, x) := loops x x in X /'. It can also be described in terms of covers. A cover of X is a map p : Y ! X such that every point x 2 X has an open neighborhood U ⊆ X with p−1(U) =∼ U × p−1(x) as spaces over U (endowing p−1(x) with the discrete topology). A cover Y ! X is universal if Y is simply connected. In this case p(X, x) = AutX Y. Theorem 1.1. Suppose X admits a universal cover. Then the functor Cov X ! p(X, x)-Set, p 7! p−1(x) is an equivalence. Theorem 1.2. There is a profinite group p, unique up to isomorphism, such that FCov X ≈ p-FSet. If X admits a universal cover, then p is isomorphic to the profinite completion pˆ (X, x). All data in this theorem can be made functorial in (X, x). Example 1.3. The circle S1 has fundamental group p(S1, x) = Z. It has the universal cover R ! S1, t 7! exp 2pit, with automorphism group generated by the shift t 7! t + 1. In the set- ting of theorem 1.2, suppose A is a transitive finite Zˆ -set. Then A =∼ Z/nZ, and it corresponds to the finite cover R/nZ ! S1, t 7! exp 2pit. 2. Algebraic geometry Let X be a connected scheme.
    [Show full text]
  • Abstract Commensurators of Profinite Groups 1
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 10, October 2011, Pages 5381–5417 S 0002-9947(2011)05295-5 Article electronically published on March 28, 2011 ABSTRACT COMMENSURATORS OF PROFINITE GROUPS YIFTACH BARNEA, MIKHAIL ERSHOV, AND THOMAS WEIGEL Abstract. In this paper we initiate a systematic study of the abstract com- mensurators of profinite groups. The abstract commensurator of a profinite group G is a group Comm(G) which depends only on the commensurability class of G. We study various properties of Comm(G); in particular, we find two natural ways to turn it into a topological group. We also use Comm(G) to study topological groups which contain G as an open subgroup (all such groups are totally disconnected and locally compact). For instance, we con- struct a topologically simple group which contains the pro-2 completion of the Grigorchuk group as an open subgroup. On the other hand, we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups. Several celebrated rigidity theorems, such as Pink’s analogue of Mostow’s strong rigidity theorem for simple alge- braic groups defined over local fields and the Neukirch-Uchida theorem, can be reformulated as structure theorems for the commensurators of certain profinite groups. 1. Introduction Let G be a group and let H be a subgroup G. The (relative) commensurator of H in G, denoted CommG(H), is defined as the set of all g ∈ G such that the group gHg−1 ∩ H has finite index in both H and gHg−1.
    [Show full text]
  • The Complexity of Topological Group Isomorphism
    THE COMPLEXITY OF TOPOLOGICAL GROUP ISOMORPHISM ALEXANDER S. KECHRIS, ANDRE´ NIES AND KATRIN TENT Abstract. We study the complexity of the topological isomorphism relation for various classes of closed subgroups of the group of permu- tations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Borel spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound. 1. Introduction Let S1 denote the Polish group of permutations of !. It is well-known that the closed subgroups of S1 (or equivalently, the non-Archimedean Pol- ish groups) are, up to topological group isomorphism, the automorphism groups of countable structures. Algebra or model theory can sometimes be used to understand natural classes of closed subgroups of S1. Firstly, the separable profinite groups are precisely the Galois groups of Galois exten- sions of countable fields. For a second example, consider the oligomorphic groups, namely the closed subgroups of S1 such that for each n there are only finitely many n-orbits. These groups are precisely the automorphism groups of !-categorical structures with domain the natural numbers. Under this correspondence, topological isomorphism turns into bi{interpretability of the structures by a result in Ahlbrandt and Ziegler [1] going back to unpublished work of Coquand. The closed subgroups of S1 form the points of a standard Borel space. Our main goal is to determine the complexity of the topological isomorphism relation for various classes of closed subgroups of S1 within the setting of Borel reducibility between equivalence relations.
    [Show full text]
  • Continuous Cohomology and Homology of Profinite Groups
    1 Continuous cohomology and homology of profinite groups Marco Boggi and Ged Corob Cook Abstract. We develop cohomological and homological theories for a profinite group G with coefficients in the Pontryagin dual categories of pro-discrete and ind-profinite G-modules, respectively. The standard results of group (co)homology hold for this theory: we prove versions of the Universal Coefficient Theorem, the Lyndon-Hochschild-Serre spectral sequence and Shapiro's Lemma. 2000 Mathematics Subject Classification: Primary 20J06; Secondary 20E18, 20J05, 13J10. Keywords and Phrases: Continuous cohomology, profinite groups, quasi-abelian categories. Introduction Cohomology groups Hn(G; M) can be studied for profinite groups G in much the same way as abstract groups. The coefficients M will lie in some category of topological modules, but it is not clear what the right category is. The classical solution is to allow only discrete modules, in which case Hn(G; M) is discrete: see [9] for this approach. For many applications, it is useful to take M to be a profinite G-module. A cohomology theory allowing discrete and profinite coefficients is developed in [12] when G is of type FP1, but for arbitrary profinite groups there has not previously been a satisfactory definition of cohomology with profinite coefficients. A difficulty is that the category of profinite G-modules does not have enough injectives. We define the cohomology of a profinite group with coefficients in the category of pro-discrete Z^ G -modules, PD(Z^ G ). This category contains the discrete J K J K Z^ G -modules and the second-countable profinite Z^ G -modules; when G itself isJ second-countable,K this is sufficient for many applications.J K PD(Z^ G ) is not an abelian category: instead it is quasi-abelian { homological algebraJ overK this generalisation is treated in detail in [8] and [10], and we give 2 an overview of the results we will need in Section 5.
    [Show full text]
  • INVERSE LIMITS and PROFINITE GROUPS We Discuss the Inverse Limit Construction, and Consider the Special Case of Inverse Limits O
    INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological groups, and can be characterized by their topological properties. These are profinite groups, which arise naturally in infinite Galois theory. No arguments in these notes are original in any way. 1. Inverse limits Standard examples of inverse limits arise from sequences of groups, with maps n between them: for instance, if we have the sequence Gn = Z=p Z for n ≥ 0, with the natural quotient maps πn+1 : Gn+1 ! Gn, the inverse limit consists of Q tuples (g0; g1;::: ) 2 n≥0 Gn such that πn+1(gn+1) = gn for all n ≥ 0. This is a description of the p-adic integers Zp. It is clear that more generally if the Gn are any groups and πn+1 : Gn+1 ! Gn any homomorphisms, we can define a notion of the inverse limit group in the same way. However, we will make a more general definition. Definition 1.1. Suppose we have a set S, with a partial order on it. We say that S (with the partial order) is directed if given any s1; s2 2 S, there exists s3 2 S such that s1 ≤ s3 and s2 ≤ s3. An inverse system of groups is a directed set S, together with groups Gs for every s 2 S, as well as for all s1; s2 satisfying s1 ≤ s2, a homomorphism f s2 : G ! G . These homomorphisms must satisfy the conditions that f s = id s1 s2 s1 s for all s 2 S, and that for any s1 ≤ s2 ≤ s3, we have f s2 ◦ f s3 = f s3 : s1 s2 s1 Given an inverse system of groups, the inverse limit lim Gs is the subgroup of Q −s s2S Gs consisting of elements (gs)s2S satisfying the condition that for all s1 ≤ s2, we have f s2 (s ) = s .
    [Show full text]
  • Profinite Properties of Discrete Groups
    PROFINITE PROPERTIES OF DISCRETE GROUPS ALAN W. REID 1. Introduction This paper is based on a series of 4 lectures delivered at Groups St Andrews 2013. The main theme of the lectures was distinguishing finitely generated residually finite groups by their finite quotients. The purpose of this paper is to expand and develop the lectures. The paper is organized as follows. In x2 we collect some questions that motivated the lectures and this article, and in x3 discuss some examples related to these questions. In x4 we recall profinite groups, profinite completions and the formulation of the questions in the language of the profinite completion. In x5, we recall a particular case of the question of when groups have the same profinite completion, namely Grothendeick's question. In x6 we discuss how the methods of L2-cohomology can be brought to bear on the questions in x2, and in x7, we give a similar discussion using the methods of the cohomology of profinite groups. In x8 we discuss the questions in x2 in the context of groups arising naturally in low-dimensional topology and geometry, and in x9 discuss parafree groups. Finally in x10 we collect a list of open problems that may be of interest. Acknoweldgement: The material in this paper is based largely on joint work with M. R. Bridson, and with M. R. Bridson and M. Conder and I would like to thank them for their collaborations. I would also like to thank the organizers of Groups St Andrews 2013 for their invitation to deliver the lectures, for their hopsitality at the conference, and for their patience whilst this article was completed.
    [Show full text]
  • Arxiv:1907.02262V5 [Math.GR] 22 May 2021
    FINITE AXIOMATIZABILITY FOR PROFINITE GROUPS ANDRE NIES, DAN SEGAL AND KATRIN TENT Abstract. A group is finitely axiomatizable (FA) in a class C if it can be de- termined up to isomorphism within C by a sentence in the first-order language of group theory. We show that profinite groups of various kinds are FA in the class of profinite groups, or in the class of pro-p groups for some prime p. Both algebraic and model-theoretic methods are developed for the purpose. Reasons why certain groups cannot be FA are also discussed. Contents 1. Introduction 1 2. Definable subgroups 7 3. Finite extensions 9 4. Bi-interpretation 12 5. Profinite groups of finite rank 22 6. Special linear groups 35 7. Some negative results 39 8. List of formulae 40 References 40 1. Introduction Some properties of a group can be expressed by a sentence in the first-order language Lgp of group theory, and some cannot. If the group is assumed to be finite, a lot more can be said about it in first-order language than in the general case. We mention examples of these phenomena below. The strongest property of a group G is that of ‘being isomorphic to G’. If this arXiv:1907.02262v5 [math.GR] 22 May 2021 can be expressed by a first-order sentence, G is said to be finitely axiomatizable, henceforth abbreviated to FA. It is obvious that every finite group is FA: if |G| = n, the fact that G has exactly n elements and that they satisfy the multiplication table of G is clearly a first-order property.
    [Show full text]