A geometric group action is an isometric group action on a metric space that is proper
and cocompact. For a discrete
group 
, properness means that only finitely many 
satisfy 
for any given compact subset 
. Cocompactness
means that translates of one compact subset
cover the space. For example, 
acts geometrically on 
by integer translations.
A proper group action of a discrete group on a nonempty metric space has finite point stabilizers and hence a finite kernel. For a torsion-free group these stabilizers are trivial. The Cannon conjecture asks for a geometric action on three-dimensional hyperbolic space from a topological condition on the Gromov boundary of a hyperbolic group.
