mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically

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 group (mathematics), groups and topology, topological and geometry, geometric pro ...

, a geometric group action is a certain type of

action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...

of a

discrete group In mathematics, a topological group ''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). Equivalently, the group ''G'' is discrete if and on ...

on a

metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

Definition

In geometric group theory, a geometry is any

proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...

geodesic metric space In the mathematical study of metric spaces, one can consider the arclength of paths in the space. If two points are at a given distance from each other, it is natural to expect that one should be able to get from the first point to the second alo ...

. An action of a

finitely-generated group In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of s ...

''G'' on a geometry ''X'' is geometric if it satisfies the following conditions: # Each element of ''G'' acts as an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

of ''X''. # The action is

cocompact Cocompact may refer to: * Cocompact group action * Cocompact Coxeter group * Cocompact embedding * Cocompact lattice {{dab ...

, i.e. the quotient space ''X''/''G'' is a

compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...

. # The action is

properly discontinuous In mathematics, a group action on a space (mathematics), space is a group homomorphism of a given group (mathematics), group into the group of transformation (geometry), transformations of the space. Similarly, a group action on a mathematical ...

, with each point having a finite stabilizer.

Uniqueness

If a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

''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 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 by Cayley's theorem (named after Arthur Cayle ...

, any space on which ''G'' acts geometrically is quasi-isometric to the Cayley graph of ''G''.

Examples

Cannon's conjecture states that any

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 metric satisfying certain properties abstra ...

with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.

References

* Geometric group theory Discrete groups {{algebra-stub