# 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

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

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