Geometric group action
From Wikipedia, the free encyclopedia
In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space.
[edit] 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.
[edit] 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.
[edit] References
- Cannon, James W. (2002). "Geometric Group Theory". Handbook of geometric topology: 261-305, North-Holland. ISBN 0444824324.
