In
mathematics, specifically
geometric group theory, 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 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 set ...
.
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 fo ...
,
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
Cocompact may refer to:
* Cocompact group action
* Cocompact Coxeter group
* Cocompact embedding
* Cocompact lattice
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space wi ...
, 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", ...
.
# The action is
properly discontinuous, 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 Cayl ...
, 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
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