Fundamental group, presentations, free differential calculus, Generators, relations, and presentations of groups, Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations, finitely-generated groups, Topological methods in group theory, relatively hyperbolic groups, negative curvature, Word problems, other decision problems, connections with logic and automata (group-theoretic aspects), coset graphs, Cayley graphs, Hyperbolic groups and nonpositively curved groups, malnormal subgroups, hyperbolic manifolds with cusps, negatively curved metric spaces, fundamental groups