Geometric Group Theory - Modern Themes and Developments

Modern Themes and Developments

Notable themes and developments in geometric group theory in 1990s and 2000s include:

• Gromov's program to study quasi-isometric properties of groups.

A particularly influential broad theme in the area is Gromov's program of classifying finitely generated groups according to their large scale geometry. Formally, this means classifying finitely generated groups with their word metric up to quasi-isometry. This program involves:

1. The study of properties that are invariant under quasi-isometry. Examples of such properties of finitely generated groups include: the growth rate of a finitely generated group; the isoperimetric function or Dehn function of a finitely presented group; the number of ends of a group; hyperbolicity of a group; the homeomorphism type of the boundary of a hyperbolic group; asymptotic cones of finitely generated groups (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.
2. 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.
3. 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 Farb and 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 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 Bowditch, in the 1990s. The study of relatively hyperbolic groups gained prominence in 2000s.
• Interactions with mathematical logic and the study of 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 Kharlampovich and 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 generate group.
• The study of isoperimetric inequalities, Dehn functions and their generalizations for finitely presented group. This includes, in particular, the work of Birget, Ol'shanskii, Rips and Sapir essentially characterizing the possible Dehn functions of finitely presented groups, as well as results providing explicit constructions of groups with fractional Dehn functions.
• Development of the theory of JSJ-decompositions for finitely generated and finitely presented 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 and the Baum-Connes conjecture and the development and study of related group-theoretic notions such as topological amenability, asymptotic dimension, uniform embeddability into Hilbert spaces, rapid decay property, and so on (see, for example,).
• Interactions with the theory of quasiconformal analysis on metric spaces, particularly in relation to Cannon's Conjecture about characterization of hyperbolic groups with boundary homeomorphic to the 2-sphere.
• Finite subdivision rules, also in relation to Cannon's Conjecture.

• Interactions with topological dynamics in the contexts of studying actions of discrete groups on various compact spaces and group compactifications, particularly convergence group methods
• Development of the theory of group actions on -trees (particularly the Rips machine), and its applications.
• The study of group actions on CAT(0) spaces and CAT(0) cubical complexes, motivated by ideas from Alexandrov geometry.
• Interactions with low-dimensional topology and hyperbolic geometry, particularly the study of 3-manifold groups (see, e.g.,), mapping class groups of surfaces, braid groups and Kleinian groups.
• Introduction of probabilistic methods to study algebraic properties of "random" group theoretic objects (groups, group elements, subgroups, etc.). A particularly important development here is the work of Gromov who used probabilistic methods to prove the existence of a finitely generated group that is not uniformly embeddable into a Hilbert space. Other notable developments include introduction and study of the notion of generic-case complexity for group-theoretic and other mathematical algorithms and algebraic rigidity results for generic groups.
• The study of automata groups and iterated monodromy groups as groups of automorphisms of infinite rooted trees. In particular, Grigorchuk's groups of intermediate growth, and their generalizations, appear in this context.
• The study of measure-theoretic properties of group actions on measure spaces, particularly introduction and development of the notions of measure equivalence and orbit equivalence, as well as measure-theoretic generalizations of Mostow rigidity.
• The study of unitary representations of discrete groups and Kazhdan's property (T)
• The study of Out(F_n) (the outer automorphism group of a free group of rank n) and of individual automorphisms of free groups. Introduction and the study of Culler-Vogtmann's outer space and of the theory of train tracks for free group automorphisms played a particularly prominent role here.
• Development of Bass–Serre theory, particularly various accessibility results and the theory of tree lattices. Generalizations of Bass–Serre theory such as the theory of complexes of groups.
• The study of random walks on groups and related boundary theory, particularly the notion of Poisson boundary (see, e.g.,). The study of amenability and of groups whose amenability status is still unknown.
• Interactions with finite group theory, particularly progress in the study of subgroup growth.
• Studying subgroups and lattices in linear groups, such as, and of other Lie Groups, via geometric methods (e.g. buildings), algebro-geometric tools (e.g. algebraic groups and representation varieties), analytic methods (e.g. unitary representations on Hilbert spaces) and arithmetic methods.
• Group cohomology, using algebraic and topological methods, particularly involving interaction with algebraic topology and the use of morse-theoretic ideas in the combinatorial context; large-scale, or coarse (e.g. see ) homological and cohomological methods.
• Progress on traditional combinatorial group theory topics, such as the Burnside problem, the study of Coxeter groups and Artin groups, and so on (the methods used to study these questions currently are often geometric and topological).