# Category: Geometric group theory

Length function
In the mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.
Finiteness properties of groups
In mathematics, finiteness properties of a group are a collection of properties that allow the use of various algebraic and topological tools, for example group cohomology, to study the group. It is m
Iterated monodromy group
In geometric group theory and dynamical systems the iterated monodromy group of a covering map is a group describing the monodromy action of the fundamental group on all iterations of the covering. A
Fully irreducible automorphism
In the mathematical subject geometric group theory, a fully irreducible automorphism of the free group Fn is an element of Out(Fn) which has no periodic conjugacy classes of proper free factors in Fn
Coxeter complex
In mathematics, the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are the basic objects that allow t
Flexagon
In geometry, flexagons are flat models, usually constructed by folding strips of paper, that can be flexed or folded in certain ways to reveal faces besides the two that were originally on the back an
Quasi-isometry
In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric
Gromov boundary
In mathematics, the Gromov boundary of a δ-hyperbolic space (especially a hyperbolic group) is an abstract concept generalizing the boundary sphere of hyperbolic space. Conceptually, the Gromov bounda
Bass–Serre theory
Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates grou
Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case
Free group
In mathematics, the free group FS over a given set S consists of all words that can be built from members of S, considering two words to be different unless their equality follows from the group axiom
Gromov's theorem on groups of polynomial growth
In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nil
Free factor complex
In mathematics, the free factor complex (sometimes also called the complex of free factors) is a free group counterpart of the notion of the curve complex of a finite type surface.The free factor comp
Muller–Schupp theorem
In mathematics, the Muller–Schupp theorem states that a finitely generated group G has context-free word problem if and only if G is virtually free. The theorem was proved by David Muller and Paul Sch
Kazhdan's property (T)
In mathematics, a locally compact topological group G has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology. Informally, this means th
Teichmüller space
In mathematics, the Teichmüller space of a (real) topological (or differential) surface , is a space that parametrizes complex structures on up to the action of homeomorphisms that are isotopic to the
Discrete group
In mathematics, a topological group like G is called a discrete group if there is no limit point in it (i.e., for each element in G, there is a neighborhood which only contains that element). Equivale
Buekenhout geometry
In mathematics, a Buekenhout geometry or diagram geometry is a generalization of projective spaces, Tits buildings, and several other geometric structures, introduced by .
Dehn function
In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the area of a relation in th
Out(Fn)
In mathematics, Out(Fn) is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory.
Thurston boundary
In mathematics, the Thurston boundary of Teichmüller space of a surface is obtained as the boundary of its closure in the projective space of functionals on simple closed curves on the surface. The Th
Hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metr
Relatively hyperbolic group
In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group. The motivating examples of relatively hyperboli
Asymptotic dimension
In metric geometry, asymptotic dimension of a metric space is a large-scale analog of Lebesgue covering dimension. The notion of asymptotic dimension was introduced by Mikhail Gromov in his 1993 monog
Convergence group
In mathematics, a convergence group or a discrete convergence group is a group acting by homeomorphisms on a compact metrizable space in a way that generalizes the properties of the action of Kleinian
Graph of groups
In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups
Tits alternative
In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.
Baumslag–Gersten group
In the mathematical subject of geometric group theory, the Baumslag–Gersten group, also known as the Baumslag group, is a particular one-relator group exhibiting some remarkable properties regarding i
Stallings theorem about ends of groups
In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G has more than one end if and only if the group G admits a nontrivial de
Švarc–Milnor lemma
In the mathematical subject of geometric group theory, the Švarc–Milnor lemma (sometimes also called Milnor–Švarc lemma, with both variants also sometimes spelling Švarc as Schwarz) is a statement whi
Small cancellation theory
In the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have "smal
Building (mathematics)
In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projecti
Curve complex
In mathematics, the curve complex is a simplicial complex C(S) associated to a finite-type surface S, which encodes the combinatorics of simple closed curves on S. The curve complex turned out to be a
Mapping class group of a surface
In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface vie
Presentation complex
In geometric group theory, a presentation complex is a 2-dimensional cell complex associated to any presentation of a group G. The complex has a single vertex, and one loop at the vertex for each gene
Grigorchuk group
In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a fini
Train track map
In the mathematical subject of geometric group theory, a train track map is a continuous map f from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice c
Ultralimit
In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces Xn a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite c
Random group
In mathematics, random groups are certain groups obtained by a probabilistic construction. They were introduced by Misha Gromov to answer questions such as "What does a typical group look like?" It so
In the mathematical subject of group theory, the Adian–Rabin theorem is a result that states that most "reasonable" properties of finitely presentable groups are algorithmically undecidable. The theor
Amenable group
In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The origin
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested
Kazhdan–Margulis theorem
In Lie theory, an area of mathematics, the Kazhdan–Margulis theorem is a statement asserting that a discrete subgroup in semisimple Lie groups cannot be too dense in the group. More precisely, in any
Rips machine
In geometric group theory, the Rips machine is a method of studying the action of groups on R-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991. An R-tree is a uniquely arcwis
Van Kampen diagram
In the mathematical area of geometric group theory, a Van Kampen diagram (sometimes also called a Lyndon–Van Kampen diagram ) is a planar diagram used to represent the fact that a particular word in t
Outer space (mathematics)
In the mathematical subject of geometric group theory, the Culler–Vogtmann Outer space or just Outer space of a free group Fn is a topological space consisting of the so-called "marked metric graph st
Von Neumann conjecture
In mathematics, the von Neumann conjecture stated that a group G is non-amenable if and only if G contains a subgroup that is a free group on two generators. The conjecture was disproved in 1980. In 1
Geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometr
Kurosh subgroup theorem
In the mathematical field of group theory, the Kurosh subgroup theorem describes the algebraic structure of subgroups of free products of groups. The theorem was obtained by Alexander Kurosh, a Russia
(2,3,7) triangle group
In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces
Subgroup distortion
In geometric group theory, a discipline of mathematics, subgroup distortion measures the extent to which an overgroup can reduce the complexity of a group's word problem. Like much of geometric group
Grushko theorem
In the mathematical subject of group theory, the Grushko theorem or the Grushko–Neumann theorem is a theorem stating that the rank (that is, the smallest cardinality of a generating set) of a free pro
Triangle group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangl
Acylindrically hyperbolic group
In the mathematical subject of geometric group theory, an acylindrically hyperbolic group is a group admitting a non-elementary 'acylindrical' isometric action on some geodesic hyperbolic metric space
Hanna Neumann conjecture
In the mathematical subject of group theory, the Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was pose
Approximate group
In mathematics, an approximate group is a subset of a group which behaves like a subgroup "up to a constant error", in a precise quantitative sense (so the term approximate subgroup may be more correc
Weyl distance function
In combinatorial geometry, the Weyl distance function is a function that behaves in some ways like the distance function of a metric space, but instead of taking values in the positive real numbers, i
Word metric
In group theory, a word metric on a discrete group is a way to measure distance between any two elements of . As the name suggests, the word metric is a metric on , assigning to any two elements , of
Følner sequence
In mathematics, a Følner sequence for a group is a sequence of sets satisfying a particular condition. If a group has a Følner sequence with respect to its action on itself, the group is amenable. A m
Cannon–Thurston map
In mathematics, a Cannon–Thurston map is any of a number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group
Geometric group action
In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space.
Haagerup property
In mathematics, the Haagerup property, named after Uffe Haagerup and also known as Gromov's a-T-menability, is a property of groups that is a strong negation of Kazhdan's property (T). Property (T) is
Boundedly generated group
In mathematics, a group is called boundedly generated if it can be expressed as a finite product of cyclic subgroups. The property of bounded generation is also closely related with the congruence sub