# 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:

1. Each element of G acts as an isometry of X.
2. The action is cocompact, i.e. the quotient space X/G is a compact space.
3. 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

• Cannon, James W. (2002). "Geometric Group Theory". Handbook of geometric topology. North-Holland. pp. 261–305. ISBN 0-444-82432-4.