Items related to Geometry of Defining Relations in Groups
Synopsis
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
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