Geometry of Defining Relations in Groups

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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.