Geometric group theory and combinatorial group theory are two closely related branches of mathematics, which study infinite discrete groups. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ...

Geometric group theory uses topological and geometric methods to study groups; the main philosophy is to deduce information about a group by analyzing how it acts on topological spaces. Combinatorial group theory studies discrete groups as quotients of free groups, typically described using presentations. In the early 20th century, pioneering work of Dehn, Nielsen, Reidemeister and Schreier amongst others established a close correspondence between the two subjects. While some problems and methods are still discernibly "more geometric" or "more combinatorial" than others, the fields are inextricably intertwined; they are now generally considered the same area of mathematics. Other closely related fields include algebraic topology, geometric topology and computational group theory. A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... In mathematics, a symmetry group describes all symmetries of objects. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... In mathematics, one method of defining a group is by a presentation. ... Max Dehn (November 13, 1878 – June 27, 1952) was a German mathematician. ... Jakob Nielsen (October 15, 1890 – August 3, 1959) was a Danish mathematician known for his work on automorphisms of surfaces. ... Kurt Werner Friedrich Reidemeister (October 13, 1893 - July 8, 1971) was a mathematician born in Brunswick, Germany. ... Otto Schreier (born March 3, 1901 in Vienna, Austria; died June 2, 1929 in Hamburg, Germany) was an Austrian mathematician who made major contributions in combinatorial group theory. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. ... In mathematics, computational group theory is the study of groups by means of computers. ...

Since the early 1980's there have developed important broad themes which motivate the study of arbitrary finitely generated groups. Particularly influential 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: Mikhail Leonidovich Gromov Russian: Михаил Леонидович Громов (born December 23, 1943, also known as Mikhael Gromov, Michael Gromov, or Misha Gromov) is a mathematician known for important contributions in many different areas of geometry, especially metric geometry, symplectic geometry, and geometric group theory. ... In mathematics, the word metric is a metric defined on a group, depending on a set of generators for the group. ... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ...

1) A description of properties that are invariant under quasi-isometry, for example

• the growth rate of a finitely generated group

• the isoperimetric function or Dehn function of a finitely generated group

• ends of a group

• hyperbolicity of a group, and the boundary of a hyperbolic group

2) Theorems which use quasi-isometry invariants to prove algebraic results about groups, for example In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. ... In group theory, a hyperbolic group, also called negatively curved group, word-hyperbolic group, Gromov-hyperbolic group, -hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry. ...

• 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. In mathematics, Gromovs theorem on groups of polynomial growth, named for Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index. ... In mathematics, Mostows rigidity theorem, sometimes called the strong rigidity theorem, essentially states that the geometry of a finite volume hyperbolic manifold (for dimension greater than two) is determined by the fundamental group and hence unique. ...

Examples

The following examples are often studied in geometric group theory:

• The infinite cyclic group Z
• Free groups
• Free products
• Outer automorphism groups Out(F_n) (via Outer space)
• Hyperbolic groups
• Mapping class groups (automorphisms of surfaces)
• Symmetric groups
• Braid groups
• Coxeter groups
• General Artin groups
• Thompson's group F
• CAT(0) groups
• Arithmetic groups
• Automatic groups
• Kleinian groups, and other lattices acting on symmetric spaces.
• Wallpaper groups
• Baumslag-Solitar groups

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... The integers are commonly denoted by the above symbol. ... The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... In abstract algebra, the free product of groups constructs a group from two or more given ones. ... In mathematics, the outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G). ... In mathematics, Out(Fn) is the outer automorphism group of a free group on n generators. ... In group theory, a hyperbolic group, also called negatively curved group, word-hyperbolic group, Gromov-hyperbolic group, -hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry. ... In mathematics, in the sub-field of geometric topology, the mapping class group is an important algebraic invariant of a topological space. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics, the braid group on n strands, denoted by Bn, is a certain group which has a nice geometrical representation and in a sense generalizes the symmetric group Sn. ... In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. ... In mathematics, an Artin group (or generalized braid group) is a group with a presentation of the form where . For , represents an alternating product of and of length , beginning with . ... Th see Thompson group (finite). ... In mathematics, a CAT(k) group is a group that acts discretely, cocompactly and isometrically on a CAT(k) space. ... In mathematics, an arithmetic group (arithmetic subgroup) in a linear algebraic group G defined over a number field K is a subgroup Γ of G(K) that is commensurable with G(O), where O is the ring of integers of K. Here two subgroups A and B of a group... In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. ... In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i. ... Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. ... In the mathematical field of group theory, the Baumslag–Solitar groups are examples of two-generator one-relator groups that play an important role in combinatorial group theory and geometric group theory as (counter)examples and test-cases. ...

See also

• The Cayley graph, the "canonical" choice of space for a group action
• The Ping-Pong lemma, a useful way to exhibit a group as a free product
• Amenability

The Cayley graph of the free group on two generators a and b In mathematics, a Cayley graph, named after Arthur Cayley, is a graph that encodes the structure of a group. ... In mathematics, an amenable group is a topological group G carrying a kind of averaging operation, that is invariant under translations by group elements. ...

External links

• The Geometric Group Theory Page
• What is Geometric Group Theory?
• Open Problems in combinatorial and geometric group theory