WHY WORD PROBLEMS ARE HARD 1. Introduction the Title Above Is A
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On Abelian Subgroups of Finitely Generated Metabelian
J. Group Theory 16 (2013), 695–705 DOI 10.1515/jgt-2013-0011 © de Gruyter 2013 On abelian subgroups of finitely generated metabelian groups Vahagn H. Mikaelian and Alexander Y. Olshanskii Communicated by John S. Wilson To Professor Gilbert Baumslag to his 80th birthday Abstract. In this note we introduce the class of H-groups (or Hall groups) related to the class of B-groups defined by P. Hall in the 1950s. Establishing some basic properties of Hall groups we use them to obtain results concerning embeddings of abelian groups. In particular, we give an explicit classification of all abelian groups that can occur as subgroups in finitely generated metabelian groups. Hall groups allow us to give a negative answer to G. Baumslag’s conjecture of 1990 on the cardinality of the set of isomorphism classes for abelian subgroups in finitely generated metabelian groups. 1 Introduction The subject of our note goes back to the paper of P. Hall [7], which established the properties of abelian normal subgroups in finitely generated metabelian and abelian-by-polycyclic groups. Let B be the class of all abelian groups B, where B is an abelian normal subgroup of some finitely generated group G with polycyclic quotient G=B. It is proved in [7, Lemmas 8 and 5.2] that B H, where the class H of countable abelian groups can be defined as follows (in the present paper, we will call the groups from H Hall groups). By definition, H H if 2 (1) H is a (finite or) countable abelian group, (2) H T K; where T is a bounded torsion group (i.e., the orders of all ele- D ˚ ments in T are bounded), K is torsion-free, (3) K has a free abelian subgroup F such that K=F is a torsion group with trivial p-subgroups for all primes except for the members of a finite set .K/. -
Filling Functions Notes for an Advanced Course on the Geometry of the Word Problem for Finitely Generated Groups Centre De Recer
Filling Functions Notes for an advanced course on The Geometry of the Word Problem for Finitely Generated Groups Centre de Recerca Mathematica` Barcelona T.R.Riley July 2005 Revised February 2006 Contents Notation vi 1Introduction 1 2Fillingfunctions 5 2.1 Van Kampen diagrams . 5 2.2 Filling functions via van Kampen diagrams . .... 6 2.3 Example: combable groups . 10 2.4 Filling functions interpreted algebraically . ......... 15 2.5 Filling functions interpreted computationally . ......... 16 2.6 Filling functions for Riemannian manifolds . ...... 21 2.7 Quasi-isometry invariance . .22 3Relationshipsbetweenfillingfunctions 25 3.1 The Double Exponential Theorem . 26 3.2 Filling length and duality of spanning trees in planar graphs . 31 3.3 Extrinsic diameter versus intrinsic diameter . ........ 35 3.4 Free filling length . 35 4Example:nilpotentgroups 39 4.1 The Dehn and filling length functions . .. 39 4.2 Open questions . 42 5Asymptoticcones 45 5.1 The definition . 45 5.2 Hyperbolic groups . 47 5.3 Groups with simply connected asymptotic cones . ...... 53 5.4 Higher dimensions . 57 Bibliography 68 v Notation f, g :[0, ∞) → [0, ∞)satisfy f ≼ g when there exists C > 0 such that f (n) ≤ Cg(Cn+ C) + Cn+ C for all n,satisfy f ≽ g ≼, ≽, ≃ when g ≼ f ,andsatisfy f ≃ g when f ≼ g and g ≼ f .These relations are extended to functions f : N → N by considering such f to be constant on the intervals [n, n + 1). ab, a−b,[a, b] b−1ab, b−1a−1b, a−1b−1ab Cay1(G, X) the Cayley graph of G with respect to a generating set X Cay2(P) the Cayley 2-complex of a -
Finitely Generated Groups Are Universal
Finitely Generated Groups Are Universal Matthew Harrison-Trainor Meng-Che Ho December 1, 2017 Abstract Universality has been an important concept in computable structure theory. A class C of structures is universal if, informally, for any structure, of any kind, there is a structure in C with the same computability-theoretic properties as the given structure. Many classes such as graphs, groups, and fields are known to be universal. This paper is about the class of finitely generated groups. Because finitely generated structures are relatively simple, the class of finitely generated groups has no hope of being universal. We show that finitely generated groups are as universal as possible, given that they are finitely generated: for every finitely generated structure, there is a finitely generated group which has the same computability-theoretic properties. The same is not true for finitely generated fields. We apply the results of this investigation to quasi Scott sentences. 1 Introduction Whenever we have a structure with interesting computability-theoretic properties, it is natural to ask whether such examples can be found within particular classes. While one could try to adapt the proof within the new class, it is often simpler to try and code the original structure into a structure in the given class. It has long been known that for certain classes, such as graphs, this is always possible. Hirschfeldt, Khoussainov, Shore, and Slinko [HKSS02] proved that classes of graphs, partial orderings, lattices, integral domains, and 2-step nilpotent groups are \complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations". -
Decomposing Some Finitely Generated Groups Into Free Products With
Decomposing some finitely generated groups into free products with amalgamation 1 V.V. Benyash-Krivets Introduction . We shall say that a group G is a non-trivial free product with amalgamation if ¡ ¡ G = G1 A G2, where G1 = A = G2 (see [1]). Wall [2] has posed the following question: Which one-relator groups are non-trivial free products with amalgamation? £ Let G = ¢ g , . , g R = = R = 1 be a group with m generators and n 1 m | 1 · · · n relations such that def G = m n ¤ 2. It is proved in [4] that G is a non-trivial − free product with amalgamation. In particular, if G is a group with m ¤ 3 generators and one relation, then G is a non-trivial free product with amalgamation. A case of groups with two generators and one relation is more complicated. For example, the free −1 −1 £ abelian group G = ¢ a, b [a, b] = 1 of rank 2, where [a, b] = aba b , obviously, is | not a non-trivial free product with amalgamation. Other examples are given by groups −1 n £ Gn = ¢ a, b aba = b . For any n, the group Gn is solvable and using results from [3], | ¡ it is easy to show that for n = 1 the group G is not decomposable into a non-trivial − n free product with amalgamation. The following conjecture was stated in [4]. m £ ¤ Conjecture 1 Let G = ¢ a, b R (a, b) = 1 , m 2, be a group with two generators | and one relation with torsion. Then G is a non-trivial free product with amalgamation. -
Lecture 4, Geometric and Asymptotic Group Theory
Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Quasi-isometry Limit groups Free actions Lecture 4, Geometric and asymptotic group theory Olga Kharlampovich NYC, Sep 16 1 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The universal property of free groups allows one to describe arbitrary groups in terms of generators and relators. Let G be a group with a generating set S. By the universal property of free groups there exists a homomorphism ': F (S) ! G such that '(s) = s for s 2 S. It follows that ' is onto, so by the first isomorphism theorem G ' F (S)=ker('): In this event ker(') is viewed as the set of relators of G, and a group word w 2 ker(') is called a relator of G in generators S. If a subset R ⊂ ker(') generates ker(') as a normal subgroup of F (S) then it is termed a set of defining relations of G relative to S. 2 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The pair hS j Ri is called a presentation of G, it determines G uniquely up to isomorphism. The presentation hS j Ri is finite if both sets S and R are finite. A group is finitely presented if it has at least one finite presentation. Presentations provide a universal method to describe groups. Example of finite presentations 1 G = hs1;:::; sn j [si ; sj ]; 81 ≤ i < j ≤ ni is the free abelian group of rank n. -
Combinatorial Group Theory
Combinatorial Group Theory Charles F. Miller III 7 March, 2004 Abstract An early version of these notes was prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated and expanded many times for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2004 by C. F. Miller III. Contents 1 Preliminaries 3 1.1 About groups . 3 1.2 About fundamental groups and covering spaces . 5 2 Free groups and presentations 11 2.1 Free groups . 12 2.2 Presentations by generators and relations . 16 2.3 Dehn’s fundamental problems . 19 2.4 Homomorphisms . 20 2.5 Presentations and fundamental groups . 22 2.6 Tietze transformations . 24 2.7 Extraction principles . 27 3 Construction of new groups 30 3.1 Direct products . 30 3.2 Free products . 32 3.3 Free products with amalgamation . 36 3.4 HNN extensions . 43 3.5 HNN related to amalgams . 48 3.6 Semi-direct products and wreath products . 50 4 Properties, embeddings and examples 53 4.1 Countable groups embed in 2-generator groups . 53 4.2 Non-finite presentability of subgroups . 56 4.3 Hopfian and residually finite groups . 58 4.4 Local and poly properties . 61 4.5 Finitely presented coherent by cyclic groups . 63 1 5 Subgroup Theory 68 5.1 Subgroups of Free Groups . 68 5.1.1 The general case . 68 5.1.2 Finitely generated subgroups of free groups . -
Arxiv:1711.08410V2
LATTICE ENVELOPES URI BADER, ALEX FURMAN, AND ROMAN SAUER Abstract. We introduce a class of countable groups by some abstract group- theoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing ℓ2- Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group Γ in this class we determine the general structure of the possible lattice embeddings of Γ, i.e. of all compactly generated, locally compact groups that contain Γ as a lattice. This leads to a precise description of possible non-uniform lattice embeddings of groups in this class. Further applications include the determination of possi- ble lattice embeddings of fundamental groups of closed manifolds with pinched negative curvature. 1. Introduction 1.1. Motivation and background. Let G be a locally compact second countable group, hereafter denoted lcsc1. Such a group carries a left-invariant Radon measure unique up to scalar multiple, known as the Haar measure. A subgroup Γ < G is a lattice if it is discrete, and G/Γ carries a finite G-invariant measure; equivalently, if the Γ-action on G admits a Borel fundamental domain of finite Haar measure. If G/Γ is compact, one says that Γ is a uniform lattice, otherwise Γ is a non- uniform lattice. The inclusion Γ ֒→ G is called a lattice embedding. We shall also say that G is a lattice envelope of Γ. The classical examples of lattices come from geometry and arithmetic. Starting from a locally symmetric manifold M of finite volume, we obtain a lattice embedding π1(M) ֒→ Isom(M˜ ) of its fundamental group into the isometry group of its universal covering via the action by deck transformations. -
Finitely Generated Subgroups of Lattices in Psl2 C
FINITELY GENERATED SUBGROUPS OF LATTICES IN PSL2 C YAIR GLASNER, JUAN SOUTO, AND PETER STORM Abstract. Let Γ be a lattice in PSL2(C). The pro-normal topology on Γ is defined by taking all cosets of non-trivial normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every finitely generated subgroup ∆ < Γ is closed in the pro-normal topology. As a corollary we deduce that if H is a maximal subgroup of a lattice in PSL2(C) then either H is finite index or H is not finitely generated. 1. Introduction In most infinite groups, it is virtually impossible to understand the lattice of all subgroups. Group theorists therefore focus their attention on special families of subgroups such as finite index subgroups, normal subgroups, or finitely generated subgroups. This paper will study the family of finitely generated subgroups of lattices in PSL2(C). Recall that lattices in PSL2(C) are the fundamental groups of finite volume hyperbolic 3-orbifolds. For these groups we establish a connection between finitely generated subgroups and normal subgroups. This connection is best expressed in the following topological terms. If a family of subgroups N is invariant under conjugation and satisfies the con- dition, (1) for all N1,N2 ∈ N ∃ N3 ∈ N such that N3 ≤ N1 ∩ N2 then one can define an invariant topology in which the given family of groups constitutes a basis of open neighborhoods for the identity element. The most famous example is the pro-finite topology, which is obtained by taking N to be the family of (normal) finite index subgroups. -
Relative Subgroup Growth and Subgroup Distortion
Relative Subgroup Growth and Subgroup Distortion Tara C. Davis, Alexander Yu. Olshanskii October 29, 2018 Abstract We study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embed- dings. We explore which functions are equivalent to the relative growth functions and distortion functions of finitely generted subgroups. We also study the connections between these two asymptotic invariants of group embeddings. We give conditions under which a length function on a finitely generated group can be extended to a length function on a larger group. 1 Introduction and Background We will use the notation that for a group G with finite generating set X, and for an element g G, then g X represents the word length of ∈ | | the element g with respect to the generating set X. For this paper, N = 1, 2,..., . For r N we use the notation that BG(r) represents the ball { } ∈ in G centered at the identity with radius r. That is, BG(r) = g G : { ∈ g X r . | | ≤ } Definition 1.1. Let G be a group finitely generated by a set X with any subset H. The growth function of H in G is G g : N N : r # h H : h X r = #(BG(r) H). H → → { ∈ | | ≤ } ∩ arXiv:1212.5208v1 [math.GR] 20 Dec 2012 The growth functions of subsets and other functions of this type be- come asymptotic invariants independent of the choice of a finite set X if one takes them up to a certain equivalence (see the definitions of , , ∼ ∼ and in Section 3.) ≈ The growth of a subgroup H with respect to a finite set of generators of a bigger group G is called the relative growth of H in G and denoted by G G gH . -
Myasnikov.1.Free.Groups.Pdf
Free groups Contents 1 Free groups 1 1.1 De¯nitions and notations . 1 1.2 Construction of a free group with basis X ........... 2 1.3 The universal property of free groups. 6 1.4 Presentations of groups . 8 1.5 Rank of free groups . 9 1 Free groups 1.1 De¯nitions and notations Let G be a group. If H is a subgroup of G then we write H · G; if H is a normal subgroup of G we write H E G. For a subset A ⊆ G by hAi we denote the subgroup of G generated by A (the intersection of all subgroups of G containing A). It is easy to see that "1 "n hAi = fai1 ; : : : ; ain j aij 2 A; "j 2 f1; ¡1g; n 2 Ng "1 "n To treat expressions like ai1 ; : : : ; ain a little bit more formally we need some terminology. Let X be an arbitrary set. A word in X is a ¯nite sequence of elements (perhaps, empty) w which we write as w = y1 : : : yn ( yi 2 X). The number n is called the length of the word w, we denote it by jwj. We denote the empty word by ² and put j²j = 0. Consider X¡1 = fx¡1jx 2 Xg; where x¡1 is just a formal expression obtained from x and ¡1. If x 2 X then the symbols x and x¡1 are called literals in X. Denote by X§1 = X [ X¡1 1 2 the set of all literals in X. For a literal y 2 X§1 we de¯ne y¡1 as ½ x¡1; if y = x 2 X; y¡1 = x; if y = x¡1 2 X. -
On Finitely Generated Profinite Groups, I 173
Annals of Mathematics, 165 (2007), 171–238 On finitely generated profinite groups, I: strong completeness and uniform bounds By Nikolay Nikolov* and Dan Segal Abstract We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is deter- mined by the algebraic structure. This is deduced from the main result about finite groups: let w be a ‘locally finite’ group word and d ∈ N. Then there exists f = f(w, d) such that in every d-generator finite group G, every element of the verbal subgroup w(G) is equal to a product of fw-values. An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the lower central series is closed. The proofs rely on some properties of the finite simple groups, to be established in Part II. Contents 1. Introduction 2. The Key Theorem 3. Variations on a theme 4. Proof of the Key Theorem 5. The first inequality: lifting generators 6. Exterior squares and quadratic maps 7. The second inequality, soluble case 8. Word combinatorics 9. Equations in semisimple groups, 1: the second inequality 10. Equations in semisimple groups, 2: powers 11. Equations in semisimple groups, 3: twisted commutators *Work done while the first author held a Golda-Meir Fellowship at the Hebrew University of Jerusalem. 172 NIKOLAY NIKOLOV AND DAN SEGAL 1. Introduction A profinite group G is the inverse limit of some inverse system of finite groups. Thus it is a compact, totally disconnected topological group; prop- erties of the original system of finite groups are reflected in properties of the topological group G. -
2.4 Direct Products and Finitely Generated Abelian Groups
2.4. DIRECT PRODUCTS AND FINITELY GENERATED ABELIAN GROUPS31 2.4 Direct Products and Finitely Generated Abelian Groups 2.4.1 Direct Products 4 Def 2.43. The Cartesian product of sets S1,S2, ··· ,Sn is the set of all ordered n-tuples (a1, ··· , an) where ai ∈ Si for i = 1, 2 ··· , n. n Y Si = S1 × S2 × · · · × Sn i=1 := {(a1, a2, ··· , an) | ai ∈ Si for i = 1, 2, ··· , n}. Caution: The notation (a1, a2, ··· , an) here has different meaning from the cycle notation in permutation groups. Thm 2.44. If G1,G2, ··· ,Gn are groups with the corresponding multiplica- Qn tions, then i=1 Gi is a group under the following multiplication: (a1, a2, ··· , an)(b1, b2, ··· , bn) := (a1b1, a2b2, ··· , anbn) Qn for (a1, a2, ··· , an) and (b1, b2, ··· , bn) in i=1 Gi. Proof. Qn 1. Closed: ai ∈ Gi, bi ∈ Gi ⇒ aibi ∈ Gi ⇒ (a1b1, ··· , anbn) ∈ i=1 Gi. 2. Associativity: By the associativity in each Gi. 3. Identity: Let ei be the identity in Gi. Then e := (e1, e2, ··· , en) is the identity of G. −1 −1 4. Inverse: The inverse of (a1, ··· , an) is (a1 , ··· , an ). Ex 2.45. R2, R3 are abelian groups. (Every finite dimensional real vector space is an abelian group that is iso- morphic to certain Rn.) Ex 2.46. Z2 = {(a, b) | a, b ∈ Z} is a subgroup of R2. Qn Qn (If Hi ≤ Gi for i = 1, ··· , n, then i=1 Hi ≤ i=1 Gi). 41st HW: 1, 2, 14, 34, 46 32 CHAPTER 2. PERMUTATIONS, COSETS, DIRECT PRODUCTS Ex 2.47.