In mathematics, specifically geometric group theory, a **geometric group action** is a certain type of action of a discrete group on a metric space.

In geometric group theory, a **geometry** is any proper, geodesic metric space. An action of a finitely-generated group *G* on a geometry *X* is **geometric** if it satisfies the following conditions:

- Each element of
*G*acts as an isometry of*X*. - The action is cocompact, i.e. the quotient space
*X*/*G*is a compact space. - The action is properly discontinuous, with each point having a finite stabilizer.

If a group *G* acts geometrically upon two geometries *X* and *Y*, then *X* and *Y* are quasi-isometric. Since any group acts geometrically on its own Cayley graph, any space on which *G* acts geometrically is quasi-isometric to the Cayley graph of *G*.

Cannon's conjecture states that any hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.

In mathematics and abstract algebra, **group theory** studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

In the mathematical disciplines of topology and geometry, an **orbifold** is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of an Euclidean space.

In mathematics, a **Cayley graph**, also known as a **Cayley colour graph**, **Cayley diagram**, **group diagram**, or **colour group** is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem and uses a specified, usually finite, set of generators for the group. It is a central tool in combinatorial and 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 geometric properties of spaces on which these groups act.

In mathematics, a **hyperbolic metric space** is a metric space satisfying certain metric relations between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called (Gromov-)hyperbolic groups.

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 a distance that measures how efficiently their difference can be expressed as a word whose letters come from a generating set for the group. The word metric on *G* is very closely related to the Cayley graph of *G*: the word metric measures the length of the shortest path in the Cayley graph between two elements of *G*.

In mathematics, the **Teichmüller space** of a (real) topological surface , is a space that parametrizes complex structures on up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from to itself.

In mathematics, **Mostow's rigidity theorem**, or **strong rigidity theorem**, or **Mostow–Prasad rigidity theorem**, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifolds by Mostow (1968) and extended to finite volume manifolds by Marden (1974) in 3 dimensions, and by Prasad (1973) in all dimensions at least 3. Gromov (1981) gave an alternate proof using the Gromov norm. Besson, Courtois, and Gallot (1996) gave the simplest available proof.

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 metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov (1987). The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology, and combinatorial group theory. In a very influential chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others.

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** if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces.

**James W. Cannon** is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.

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 hyperbolic groups are the fundamental groups of complete noncompact hyperbolic manifolds of finite volume.

**Cornelia Druțu** is a Romanian mathematician notable for her contributions in the area of geometric group theory. She is Professor of mathematics at the University of Oxford and Fellow of Exeter College, Oxford.

In mathematics, a **finite subdivision rule** is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals. Subdivision rules have been used in architecture, biology, and computer science, as well as in the study of hyperbolic manifolds. Substitution tilings are a well-studied type of subdivision rule.

In mathematics, the **Gromov boundary** of a δ-hyperbolic space is an abstract concept generalizing the boundary sphere of hyperbolic space. Conceptually, the Gromov boundary is the set of all points at infinity. For instance, the Gromov boundary of the real line is two points, corresponding to positive and negative infinity.

In the mathematical subject of geometric group theory, the **Švarc–Milnor lemma** is a statement which says that a group , equipped with a "nice" discrete isometric action on a metric space , is quasi-isometric to .

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 my Mikhail Gromov in his 1993 monograph *Asymptotic invariants of infinite groups* in the context of geometric group theory, as a quasi-isometry invariant of finitely generated groups. As shown by Guoliang Yu, finitely generated groups of finite homotopy type with finite asymptotic dimension satisfy the Novikov conjecture. Asymptotic dimension has important applications in geometric analysis and index theory.

In the mathematical subject of group theory, a **co-Hopfian group** is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named after Heinz Hopf.

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. This notion generalizes the notions of a hyperbolic group and of a relatively hyperbolic group and includes a significantly wider class of examples, such as mapping class groups and Out(*F _{n}*).

In mathematics, the notion of 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 on those spaces.

- Cannon, James W. (2002). "Geometric Group Theory".
*Handbook of geometric topology*. North-Holland. pp. 261–305. ISBN 0-444-82432-4.

This algebra-related article is a stub. You can help Wikipedia by expanding it. |

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.