In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically
geometric group theory, a geometric group action is a certain type of
action 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 a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
.
Definition
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
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'' me ...
of ''X''.
# The action is
cocompact, 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. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
.
# The action is
properly discontinuous, with each point having a finite
stabilizer.
Uniqueness
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
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathematics), group. Its definition is sug ...
, 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 with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.
References
*
Geometric group theory
Discrete groups
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