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.
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 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.
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, 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
- {{cite conference | last = Cannon | first = James W. | title = Geometric Group Theory | book-title = Handbook of geometric topology | pages = 261–305 | publisher = North-Holland | date = 2002 | isbn = 0-444-82432-4}}
Category:Geometric group theory
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