Geometric Group Theory
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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/. -
18.703 Modern Algebra, Presentations and Groups of Small
12. Presentations and Groups of small order Definition-Lemma 12.1. Let A be a set. A word in A is a string of 0 elements of A and their inverses. We say that the word w is obtained 0 from w by a reduction, if we can get from w to w by repeatedly applying the following rule, −1 −1 • replace aa (or a a) by the empty string. 0 Given any word w, the reduced word w associated to w is any 0 word obtained from w by reduction, such that w cannot be reduced any further. Given two words w1 and w2 of A, the concatenation of w1 and w2 is the word w = w1w2. The empty word is denoted e. The set of all reduced words is denoted FA. With product defined as the reduced concatenation, this set becomes a group, called the free group with generators A. It is interesting to look at examples. Suppose that A contains one element a. An element of FA = Fa is a reduced word, using only a and a−1 . The word w = aaaa−1 a−1 aaa is a string using a and a−1. Given any such word, we pass to the reduction w0 of w. This means cancelling as much as we can, and replacing strings of a’s by the corresponding power. Thus w = aaa −1 aaa = aaaa = a 4 = w0 ; where equality means up to reduction. Thus the free group on one generator is isomorphic to Z. The free group on two generators is much more complicated and it is not abelian. -
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". -
The Subgroups of a Free Product of Two Groups with an Amalgamated Subgroup!1)
transactions of the american mathematical society Volume 150, July, 1970 THE SUBGROUPS OF A FREE PRODUCT OF TWO GROUPS WITH AN AMALGAMATED SUBGROUP!1) BY A. KARRASS AND D. SOLITAR Abstract. We prove that all subgroups H of a free product G of two groups A, B with an amalgamated subgroup V are obtained by two constructions from the inter- section of H and certain conjugates of A, B, and U. The constructions are those of a tree product, a special kind of generalized free product, and of a Higman-Neumann- Neumann group. The particular conjugates of A, B, and U involved are given by double coset representatives in a compatible regular extended Schreier system for G modulo H. The structure of subgroups indecomposable with respect to amalgamated product, and of subgroups satisfying a nontrivial law is specified. Let A and B have the property P and U have the property Q. Then it is proved that G has the property P in the following cases: P means every f.g. (finitely generated) subgroup is finitely presented, and Q means every subgroup is f.g.; P means the intersection of two f.g. subgroups is f.g., and Q means finite; P means locally indicable, and Q means cyclic. It is also proved that if A' is a f.g. normal subgroup of G not contained in U, then NU has finite index in G. 1. Introduction. In the case of a free product G = A * B, the structure of a subgroup H may be described as follows (see [9, 4.3]): there exist double coset representative systems {Da}, {De} for G mod (H, A) and G mod (H, B) respectively, and there exists a set of elements t\, t2,.. -
Some Remarks on Semigroup Presentations
SOME REMARKS ON SEMIGROUP PRESENTATIONS B. H. NEUMANN Dedicated to H. S. M. Coxeter 1. Let a semigroup A be given by generators ai, a2, . , ad and defining relations U\ — V\yu2 = v2, ... , ue = ve between these generators, the uu vt being words in the generators. We then have a presentation of A, and write A = sgp(ai, . , ad, «i = vu . , ue = ve). The same generators with the same relations can also be interpreted as the presentation of a group, for which we write A* = gpOi, . , ad\ «i = vu . , ue = ve). There is a unique homomorphism <£: A—* A* which maps each generator at G A on the same generator at G 4*. This is a monomorphism if, and only if, A is embeddable in a group; and, equally obviously, it is an epimorphism if, and only if, the image A<j> in A* is a group. This is the case in particular if A<j> is finite, as a finite subsemigroup of a group is itself a group. Thus we see that A(j> and A* are both finite (in which case they coincide) or both infinite (in which case they may or may not coincide). If A*, and thus also Acfr, is finite, A may be finite or infinite; but if A*, and thus also A<p, is infinite, then A must be infinite too. It follows in particular that A is infinite if e, the number of defining relations, is strictly less than d, the number of generators. 2. We now assume that d is chosen minimal, so that A cannot be generated by fewer than d elements. -
Knapsack Problems in Groups
MATHEMATICS OF COMPUTATION Volume 84, Number 292, March 2015, Pages 987–1016 S 0025-5718(2014)02880-9 Article electronically published on July 30, 2014 KNAPSACK PROBLEMS IN GROUPS ALEXEI MYASNIKOV, ANDREY NIKOLAEV, AND ALEXANDER USHAKOV Abstract. We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time decidable in hyperbolic groups and give var- ious examples of finitely presented groups where the subset sum problem is NP-complete. 1. Introduction 1.1. Motivation. This is the first in a series of papers on non-commutative discrete (combinatorial) optimization. In this series we propose to study complexity of the classical discrete optimization (DO) problems in their most general form — in non- commutative groups. For example, DO problems concerning integers (subset sum, knapsack problem, etc.) make perfect sense when the group of additive integers is replaced by an arbitrary (non-commutative) group G. The classical lattice problems are about subgroups (integer lattices) of the additive groups Zn or Qn, their non- commutative versions deal with arbitrary finitely generated subgroups of a group G. The travelling salesman problem or the Steiner tree problem make sense for arbitrary finite subsets of vertices in a given Cayley graph of a non-commutative infinite group (with the natural graph metric). The Post correspondence problem carries over in a straightforward fashion from a free monoid to an arbitrary group. This list of examples can be easily extended, but the point here is that many classical DO problems have natural and interesting non-commutative versions. -
The Free Product of Groups with Amalgamated Subgroup Malnorrnal in a Single Factor
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF PURE AND APPLIED ALGEBRA Journal of Pure and Applied Algebra 127 (1998) 119-136 The free product of groups with amalgamated subgroup malnorrnal in a single factor Steven A. Bleiler*, Amelia C. Jones Portland State University, Portland, OR 97207, USA University of California, Davis, Davis, CA 95616, USA Communicated by C.A. Weibel; received 10 May 1994; received in revised form 22 August 1995 Abstract We discuss groups that are free products with amalgamation where the amalgamating subgroup is of rank at least two and malnormal in at least one of the factor groups. In 1971, Karrass and Solitar showed that when the amalgamating subgroup is malnormal in both factors, the global group cannot be two-generator. When the amalgamating subgroup is malnormal in a single factor, the global group may indeed be two-generator. If so, we show that either the non-malnormal factor contains a torsion element or, if not, then there is a generating pair of one of four specific types. For each type, we establish a set of relations which must hold in the factor B and give restrictions on the rank and generators of each factor. @ 1998 Published by Elsevier Science B.V. All rights reserved. 0. Introduction Baumslag introduced the term malnormal in [l] to describe a subgroup that intersects each of its conjugates trivially. Here we discuss groups that are free products with amalgamation where the amalgamating subgroup is of rank at least two and malnormal in at least one of the factor groups. -
Arxiv:Hep-Th/9404101V3 21 Nov 1995
hep-th/9404101 April 1994 Chern-Simons Gauge Theory on Orbifolds: Open Strings from Three Dimensions ⋆ Petr Horavaˇ ∗ Enrico Fermi Institute University of Chicago 5640 South Ellis Avenue Chicago IL 60637, USA ABSTRACT Chern-Simons gauge theory is formulated on three dimensional Z2 orbifolds. The locus of singular points on a given orbifold is equivalent to a link of Wilson lines. This allows one to reduce any correlation function on orbifolds to a sum of more complicated correlation functions in the simpler theory on manifolds. Chern-Simons theory on manifolds is known arXiv:hep-th/9404101v3 21 Nov 1995 to be related to 2D CFT on closed string surfaces; here I show that the theory on orbifolds is related to 2D CFT of unoriented closed and open string models, i.e. to worldsheet orb- ifold models. In particular, the boundary components of the worldsheet correspond to the components of the singular locus in the 3D orbifold. This correspondence leads to a simple identification of the open string spectra, including their Chan-Paton degeneration, in terms of fusing Wilson lines in the corresponding Chern-Simons theory. The correspondence is studied in detail, and some exactly solvable examples are presented. Some of these examples indicate that it is natural to think of the orbifold group Z2 as a part of the gauge group of the Chern-Simons theory, thus generalizing the standard definition of gauge theories. E-mail address: [email protected] ⋆∗ Address after September 1, 1994: Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544. 1. Introduction Since the first appearance of the notion of “orbifolds” in Thurston’s 1977 lectures on three dimensional topology [1], orbifolds have become very appealing objects for physicists. -
Amenable Actions, Free Products and a Fixed Point Property
Submitted exclusively to the London Mathematical Society doi:10.1112/S0000000000000000 AMENABLE ACTIONS, FREE PRODUCTS AND A FIXED POINT PROPERTY Y. GLASNER and N. MONOD Abstract We investigate the class of groups admitting an action on a set with an invariant mean. It turns out that many free products admit interesting actions of that kind. A complete characterization of such free products is given in terms of a fixed point property. 1. Introduction 1.A. In the early 20th century, the construction of Lebesgue’s measure was followed by the discovery of the Banach-Hausdorff-Tarski paradoxes ([20], [16], [4], [30]; see also [18]). This prompted von Neumann [32] to study the following general question: Given a group G acting on a set X, when is there an invariant mean on X ? Definition 1.1. An invariant mean is a G-invariant map µ from the collection of subsets of X to [0, 1] such that (i) µ(A ∪ B) = µ(A) + µ(B) when A ∩ B = ∅ and (ii) µ(X) = 1. If such a mean exists, the action is called amenable. Remarks 1.2. (1) For the study of the classical paradoxes, one also considers normalisations other than (ii). (2) The notion of amenability later introduced by Zimmer [33] and its variants [3] are different, being in a sense dual to the above. 1.B. The thrust of von Neumann’s article was to show that the paradoxes, or lack thereof, originate in the structure of the group rather than the set X. He therefore proposed the study of amenable groups (then “meßbare Gruppen”), i.e. -
Combinatorial Group Theory
Combinatorial Group Theory Charles F. Miller III March 5, 2002 Abstract These notes were 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 for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2002 by C. F. Miller. Contents 1 Free groups and presentations 3 1.1 Free groups . 3 1.2 Presentations by generators and relations . 7 1.3 Dehn’s fundamental problems . 9 1.4 Homomorphisms . 10 1.5 Presentations and fundamental groups . 12 1.6 Tietze transformations . 14 1.7 Extraction principles . 15 2 Construction of new groups 17 2.1 Direct products . 17 2.2 Free products . 19 2.3 Free products with amalgamation . 21 2.4 HNN extensions . 24 3 Properties, embeddings and examples 27 3.1 Countable groups embed in 2-generator groups . 27 3.2 Non-finite presentability of subgroups . 29 3.3 Hopfian and residually finite groups . 31 4 Subgroup Theory 35 4.1 Subgroups of Free Groups . 35 4.1.1 The general case . 35 4.1.2 Finitely generated subgroups of free groups . 35 4.2 Subgroups of presented groups . 41 4.3 Subgroups of free products . 43 4.4 Groups acting on trees . 44 5 Decision Problems 45 5.1 The word and conjugacy problems . 45 5.2 Higman’s embedding theorem . 51 1 5.3 The isomorphism problem and recognizing properties . 52 2 Chapter 1 Free groups and presentations In introductory courses on abstract algebra one is likely to encounter the dihedral group D3 consisting of the rigid motions of an equilateral triangle onto itself. -
On Just-Infiniteness of Locally Finite Groups and Their -Algebras
Bull. Math. Sci. DOI 10.1007/s13373-016-0091-4 On just-infiniteness of locally finite groups and their C∗-algebras V. Belyaev1 · R. Grigorchuk2 · P. Shumyatsky3 Received: 14 August 2016 / Accepted: 23 September 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We give a construction of a family of locally finite residually finite groups with just-infinite C∗-algebra. This answers a question from Grigorchuk et al. (Just- infinite C∗-algebras. https://arxiv.org/abs/1604.08774, 2016). Additionally, we show that residually finite groups of finite exponent are never just-infinite. Keywords Just infinite groups · C*-algebras Mathematics Subject Classification 46L05 · 20F50 · 16S34 For Alex Lubotzky on his 60th birthday. Communicated by Efim Zelmanov. R. Grigorchuk was supported by NSA Grant H98230-15-1-0328. P. Shumyatsky was supported by FAPDF and CNPq. B P. Shumyatsky [email protected] V. Belyaev [email protected] R. Grigorchuk [email protected] 1 Institute of Mathematics and Mechanics, S. Kovalevskaja 16, Ekaterinburg, Russia 2 Department of Mathematics, Texas A&M University, College Station, TX, USA 3 Department of Mathematics, University of Brasília, 70910 Brasília, DF, Brazil 123 V. Belyaev et al. 1 Introduction A group is called just-infinite if it is infinite but every proper quotient is finite. Any infi- nite finitely generated group has just-infinite quotient. Therefore any question about existence of an infinite finitely generated group with certain property which is pre- served under homomorphic images can be reduced to a similar question in the class of just-infinite groups. -
Topics for the NTCIR-10 Math Task Full-Text Search Queries
Topics for the NTCIR-10 Math Task Full-Text Search Queries Michael Kohlhase (Editor) Jacobs University http://kwarc.info/kohlhase December 6, 2012 Abstract This document presents the challenge queries for the Math Information Retrieval Subtask in the NTCIR Math Task. Chapter 1 Introduction This document presents the challenge queries for the Math Information Retrieval (MIR) Subtask in the NTCIR Math Task. Participants have received the NTCIR-MIR dataset which contains 100 000 XHTML full texts of articles from the arXiv. Formulae are marked up as MathML (presentation markup with annotated content markup and LATEX source). 1.1 Subtasks The MIR Subtask in NTCIR-10 has three challenges; queries are given in chapters 2-4 below, the format is Formula Search (Automated) Participating IR systems obtain a list of queries consisting of formulae (possibly) with wildcards (query variables) and return for every query an ordered list of XPointer identifiers of formulae claimed to match the query, plus possible supporting evidence (e.g. a substitution for query variables). Full-Text Search (Automated) This is like formula search above, only that IR results are ordered lists of \hits" (i.e. XPointer references into the documents with a highlighted result fragments plus supporting evidence1). EdN:1 Open Information Retrieval (Semi-Automated) In contrast to the first two challenges, where the systems are run in batch-mode (i.e. without human intervention), in this one mathe- maticians will challenge the (human) participants to find specific information in a document corpus via human-readable descriptions (natural language text), which are translated by the participants to their IR systems.