Algebra VII: Combinatorial Group Theory Applications to Geometry (Hardcover, 1993)

I. Combinatorial Group Theory and Fundamental Groups.- 1. Group Presentations and 2-Complexes.- 1.1. Presentations of Groups.- 1.2. Complexes and Fundamental Groups.- 1.3. Subgroups and Coverings.- 2. Free Groups and Free Products.- 2.1. Free Groups.- 2.2. Amalgamated Free Products and Graphs of Groups.- 2.3. Automorphisms of Free Groups.- 2.4. One-Relator Groups.- 3. Surfaces and Planar Discontinuous Groups.- 3.1. Surfaces.- 3.2. Planar Discontinuous Groups.- 3.3. Subgroups of Planar Groups.- 3.4. Automorphisms of Fuchsian Groups.- 3.5. Relations to Other Theories of Surfaces.- 4. Cancellation Diagrams and Equations Over Groups.- 4.1. Cancellation Diagrams.- 4.2. Locally Indicable Groups and Equations Over Groups.- 5. 3-Manifolds and Knots.- 5.1. Fundamental Groups of 3-Manifolds.- 5.2. Haken Manifolds.- 5.3. On Knots and Their Groups.- 6. Cohomological Methods and Ends.- 6.1. Group Extensions and Cohomology.- 6.2. Ends of Groups.- 7. Decision Problems.- 7.1. Decision Problems and Algorithms.- 7.2. Unsolvable Decision Problems.- 7.3. Automata and Groups.- Index of Notation.- II. Some Questions of Group Theory Related to Geometry.- 1. Equations in Groups and Some Related Questions.- 1. Basic Concepts and the Theorem of Makanin.- 2. Solutions of Systems and Homomorphisms.- 3. Fundamental Sequences and Razborov's Theorem.- 4. On the Structure of the Set of Solutions of Quadratic Equations in Free Groups.- 5. Coefficient-Free Quadratic Equations.- 6. The Classification of Epimorphisms from Surface Groups to Free Groups.- 7. On the Minimal Number of Fixed Points in the Homotopy Class of Mappings and the Width of Elements in Free Groups.- 8. On Quadratic Equations in Hyperbolic Groups.- 2. Splitting Homomorphisms and Some Problems in Topology.- 1. Heegaard Decompositions of 3-Manifolds and their Equivalence.- 2. The Poincare Conjecture and Three Algorithmic Problems Connected with 3-Manifolds.- 3. Information on Aut ?1(T) and Some of its Subgroups and Factor Groups.- 4. On the Problem of the Equivalence of Splitting Homomorphisms.- 5. On an Algebraic Reduction of the Poincare Conjecture and the Algorithmic Poincare Problem.- 6. Some Analogues with the Group of Symplectic Matrices and the Torelli Group.- 7. Algebraic Reduction of the Problem of the Equivalence of Links.- 8. On the Andrews-Curtis Conjecture.- 3. On the Rate of Growth of Groups and Amenable Groups.- 1. On the Growth of Graphs and of Riemannian Manifolds.- 2. On the Notion of Growth of a Finitely Generated Group.- 3. On the Proof of Gromov's Theorem and Some Related Results.- 4. Example of a Group of Intermediate Growth and the Construction Scheme of such a Group.- 5. On the Structure of the Set of Growth Degrees of Groups that are Residually-p Groups.- 6. On an Application of the Theory of Groups of Polynomial Growth to Geometry.- 7. Regularly Filtered Surfaces and Amenable Groups.- Index of Notation.- Author Index.