Geometry of Defining Relations in Groups

ISBN-13: 9780792313946 / Angielski / Twarda / 1991 / 505 str.

A. Iu Ol'shanskii

Opis

'Ht moi ..., si favait su comment en reveniT, One service mathematics hal rendered the je n'y serais point aile.' human race. It has put C"

Spis treści

1 General concepts of group theory.- §1 Definition and examples of groups.- 1 Definition of group.- 2 Examples of groups.- 3 Group isomorphism.- §2 Cyclic groups and subgroups. Generators.- 1 Subgroups.- 2 Cyclic groups.- 3 Subgroups of cyclic groups.- 4 Sets of generators.- §3 Cosets. Factor groups. Homomorphisms.- 1 Decomposition of a group into cosets.- 2 Normal subgroups and factor groups.- 3 Homomorphism theorems.- §4 Relations in groups and free groups.- 1 Free groups.- 2 Defining relations.- 3 Words and subwords.- 2 Main types of groups and subgroups.- §5 p-subgroups in finite and abelian groups.- 1 Conjugacy class. The centre.- 2 p-subgroups of finite groups.- 3 Direct products.- 4 Primary decompositions of abelian groups.- §6 Soluble groups. Laws.- 1 The derived group.- 2 Soluble groups.- 3 Soluble and finite simple groups.- 4 Laws and varieties.- §7 Finiteness conditions in groups.- 1 Local finiteness. The conditions max and min.- 2 Soluble Noetherian and Artinian groups.- 3 The role of involutions.- 3 Elements of two-dimensional topology.- §8 Toplogical spaces.- 1 The definitions of topological and metric spaces.- 2 Continuous mappings.- 3 Quotient spaces.- 4 Compactness.- 5 Connectedness.- §9 Surfaces and their cell decomposition.- 1 The Jordan curve theorem.- 2 The combinatorial definition of a surface.- 3 Comparison of triangulations.- 4 Cell decompositions of surfaces.- 5 Graphs on a surface.- §10 Topological invariants of surfaces.- 1 The Euler characteristic.- 2 Consequences for graphs.- 3 Orientable surfaces.- 4 The fundamental group of a cell decomposition.- 5 Computation of the fundamental groups of surfaces.- 4 Diagrams over groups.- §11 Visual interpretation of the deduction of consequences of defining relations.- 1 Some examples.- 2 The concept of a diagram.- 3 von Kampen’s lemma.- 4 Annular diagrams; subdiagrams.- 5 0-refinements of diagrams.- 6 Cancellable pairs of cells.- §12 Small cancellation theory.- 1 The conditions C’(?) and C(k).- 2 Diagrams over small cancellation groups.- 3 Dehn’s algorithm.- 4 Gol’berg’s example.- 5 Further remarks.- §13 Graded diagrams.- 1 Examples of partitioning sets of relators.- 2 Grading maps and diagrams.- 3 Compatible sections.- 4 Asphericity of presentations.- 5 Atoricity.- 5 A-maps.- §14 Contiguity submaps.- 1 Remarks on graded maps.- 2 Bonds and contiguity submaps.- 3 Distinguished systems of contiguity submaps.- 4 Estimating graphs.- §15 Conditions on the grading.- 1 Auxiliary parameters.- 2 Condition A and smooth sections.- 3 Bonds and contiguity in A-maps.- §16 Exterior arcs and ?-cells.- 1 Definition of the weight function.- 2 Distribution of weights in A-maps.- 3 Existence of a ?-cell.- §17 Paths that are nearly geodesic and cuts on A-maps.- 1 Comparison of the lengths of homotopic paths.- 2 Cutting annular maps.- 3 ?-cells.- 4 Cuts on circular maps.- 6 Relations in periodic groups.- §18 Free Burnside groups of large odd exponent.- 1 Defining relations.- 2 Simple consequences of the inductive hypotheses.- 3 Comparison of periodic words.- 4 Oddness of exponent n.- §19 Diagrams as A-maps. Properties of B(A, n).- 1 Very long periodic words.- 2 Completion of the inductive proof.- 3 Groups of finite exponent.- 7 Maps with partitioned boundaries of cells.- §20 Estimating graphs for B-maps.- 1 Contiguity submaps.- 2 Distinguished contiguity submaps.- 3 Estimating graphs.- 4 B-maps and their smooth sections.- §21 Contiguity and weights in B-maps.- 1 Inequalities for contiguity submaps.- 2 Distribution of weights.- §22 Existence of ?-cells and its consequences.- 1 ?-cells.- 2 “Almost geodesic” paths.- 3 Cuts on the annulus and on the sphere with three holes.- 4 Application of ?-cells.- §23 C-maps.- 1 Condition C.- 2 The weight function for C-maps.- 3 Weights of inner and outer edges.- 4 Structure of C-maps.- §24 Other conditions on the partition of the boundary of a map.- 1 D-maps.- 2 Maps on the sphere with three holes.- 3 Simple paths on the sphere with three holes.- 8 Partitions of relators.- §25 General approach to presenting the groups G(i) and properties of these groups.- 1 Form of the relations.- 2 Analogues of the lemmas in Chapter 6.- 3 Conjugacy and commutativity in rank i.- 4 Diagrams on the sphere with three holes.- §26 Inductive step to G(i+ 1). The group G(?).- 1 Subwords of relators.- 2 Diagrams of rank i+ 1 as B-maps.- 3 Structure of G(?).- 9 Construction of groups with prescribed properties.- §27 Constructing groups with subgroups of bounded order.- 1 Problems on the structure of groups with finiteness conditions.- 2 Relators.- 3 Verification of condition R.- 4 Generating pairs for G(?).- §28 Groups with all subgroups cyclic.- 1 Main theorems.- 2 Algorithmic questions.- 3 Continuously many pairwise non-isomorphic quasi-finite groups.- §29 Group laws other than powers.- 1 A problem in H. Neumann’s book.- 2 Finite groups in a variety M.- 3 Defining relations of M-free groups.- 4 Verification of condition R.- §30 Varieties in which all finite groups are abelian.- 1 Commutators in G(i).- 2 Defining relations of rank i+ 1.- 3 The main theorem.- 10 Extensions of aspherical groups.- §31 Central extensions.- 1 Relations from the commutator subgroup [F,N].- 2 Some torsion-free groups.- 3 A countable non-topologizable group.- 4 The finite basis problem.- 5 Further examples.- §32 Abelian extensions and dependence among relations.- 1 Maximal abelian extensions.- 2 Geometric dependence.- 3 The relation module.- 4 Peiffer transformations.- 5 Algebraic independence of relations of aspherical groups.- 11 Presentations in free products.- §33 Cancellation diagrams over free products.- 1 Free products.- 2 Presentations and diagrams.- 3 Properties of maps.- §34 Presentations with condition R.- 1 Transfer to free products.- 2 Centralizers and finite subgroups.- 3 An example.- 4 A remark on central extensions.- §35 Embedding theorems for groups.- 1 Embedding countable groups without involutions.- 2 Some consequences.- 3 Subgroups of quasi-finite groups.- §36 Operations on groups.- 1 Exact operations.- 2 The operations on.- 3 Construction of the operation ?n on the class of all groups.- 12 Applications to other problems.- §37 Growth functions of groups and their presentations.- 1 Rates of growth.- 2 Amenable groups.- 3 Lemmas on outer cells.- 4 Periodic non-amenable groups.- §38 On group rings of Noetherian groups.- 1 A question of P. Hall.- 2 The Noetherian group G.- 3 Right ideals in K[G].- §39 Further applications of the method.- 1 Subgroups of free n -periodic groups.- 2 Residual properties of free n0-periodic groups.- 3 Characteristic subgroups of free groups.- 4 Residual properties of the groups Fm.- 5 Values of words and verbal subgroups.- 6 Miscellaneous problems.- 13 Conjugacy relations.- §40 Conjugacy cells.- 1 Conditions on maps.- 2 Contiguity in H-maps.- 3 Weight estimates.- 4 Geometry of H-maps.- §41 Finitely generated divisible groups.- 1 Conjugating words.- 2 Cancellation of cells.- 3 Main lemmas.- 4 The inductive step.- 5 Two theorems.- Some notation.- Author Index.