(see e.g.); amenability of a finitely generated group; being virtually abelian (that is, having an abelian subgroup of finite index); being virtually nilpotent; being virtually free; being finitely presentable; being a finitely presentable group with solvable Word Problem; and others. Theorems which use quasi-isometry invariants to prove algebraic results about groups, for example: Gromov's polynomial growth theorem; Stallings' ends theorem; Mostow rigidity theorem. Quasi-isometric rigidity theorems, in which one classifies algebraically all groups that are quasi-isometric to some given group or metric space. This direction was initiated by the work of Schwartz on quasi-isometric rigidity of rank-one lattices and the work of Benson Farb and Lee Mosher on quasi-isometric rigidity of Baumslag–Solitar groups. The theory of word-hyperbolic and relatively hyperbolic groups. A particularly important development here is the work of Zlil Sela in 1990s resulting in the solution of the isomorphism problem for word-hyperbolic groups. The notion of a relatively hyperbolic groups was originally introduced by Gromov in 1987 and refined by Farb and Brian Bowditch, in the 1990s. The study of relatively hyperbolic groups gained prominence in the 2000s. Interactions with mathematical logic and the study of the first-order theory of free groups. Particularly important progress occurred on the famous Tarski conjectures, due to the work of Sela as well as of Olga Kharlampovich and Alexei Myasnikov. The study of limit groups and introduction of the language and machinery of non-commutative algebraic geometry gained prominence. Interactions with computer science, complexity theory and the theory of formal languages. This theme is exemplified by the development of the theory of automatic groups, a notion that imposes certain geometric and language theoretic conditions on the multiplication operation in a finitely generated group. The study of isoperimetric inequalities, Dehn functions and their generalizations for finitely presented group. This includes, in particular, the work of Jean-Camille Birget, Aleksandr Olʹshanskiĭ, Eliyahu Rips and Mark Sapir essentially characterizing the possible Dehn functions of finitely presented groups, as well as results providing explicit constructions of groups with fractional Dehn functions. The theory of toral or JSJ-decompositions for 3-manifolds was originally brought into a group theoretic setting by Peter Kropholler. This notion has been developed by many authors for both finitely presented and finitely generated groups. Connections with geometric analysis, the study of C*-algebras associated with discrete groups and of the theory of free probability. This theme is represented, in particular, by considerable progress on the Novikov conjecture