Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants

This survey is devoted to investigations concerning topological, algebraic, and combinatorial characteristics as well as metric invariants for arbitrary groups of homeomorphisms of the line and the circle. Relationships between these characteristics are established, the most important metric invariants are studied (in the form of invariant, projectively invariant, and $\omega$-projectively invariant measures), and the main ‘obstructions’ to the existence of metric invariants of this kind are described.