Editing Group action (section)

=== Topological properties ===
Assume that {{math|''X''}} is a [[topological space]] and the action of {{math|''G''}} is by [[homeomorphism]]s.

The action is ''wandering'' if every {{math|''x'' ∈ ''X''}} has a [[Neighbourhood (mathematics)|neighbourhood]] {{math|''U''}} such that there are only finitely many {{math|''g'' ∈ ''G''}} with {{math|''g''&sdot;''U'' ∩ ''U'' ≠ ∅}}.{{sfn|Thurston|1997|loc=Definition 3.5.1(iv)}}

More generally, a point {{math|''x'' ∈ ''X''}} is called a point of discontinuity for the action of {{math|''G''}} if there is an open subset {{math|''U'' ∋ ''x''}} such that there are only finitely many {{math|''g'' ∈ ''G''}} with {{math|''g''&sdot;''U'' ∩ ''U'' ≠ ∅}}. The ''domain of discontinuity'' of the action is the set of all points of discontinuity. Equivalently it is the largest {{math|''G''}}-stable open subset {{math|Ω ⊂ ''X''}} such that the action of {{math|''G''}} on {{math|Ω}} is wandering.{{sfn|Kapovich|2009|loc=p. 73}} In a dynamical context this is also called a ''[[wandering set]]''.

The action is ''properly discontinuous'' if for every [[Compact space|compact]] subset {{math|''K'' ⊂ ''X''}} there are only finitely many {{math|''g'' ∈ ''G''}} such that {{math|''g''&sdot;''K'' ∩ ''K'' ≠ ∅}}. This is strictly stronger than wandering; for instance the action of {{math|'''Z'''}} on {{math|'''R'''<sup>2</sup> &setminus; {{mset|(0, 0)}}}} given by {{math|1=''n''&sdot;(''x'', ''y'') = (2<sup>''n''</sup>''x'', 2<sup>−''n''</sup>''y'')}} is wandering and free but not properly discontinuous.{{sfn|Thurston|1980|p=176}}

The action by [[deck transformation]]s of the [[fundamental group]] of a locally [[simply connected space]] on a [[Covering space#Universal covering|universal cover]] is wandering and free. Such actions can be characterized by the following property: every {{math|''x'' ∈ ''X''}} has a neighbourhood {{math|''U''}} such that {{math|1=''g''&sdot;''U'' ∩ ''U'' = ∅}} for every {{math|''g'' ∈ ''G'' &setminus; {{mset|''e''<sub>''G''</sub>}}}}.{{sfn|Hatcher|2002|loc=p. 72}} Actions with this property are sometimes called ''freely discontinuous'', and the largest subset on which the action is freely discontinuous is then called the ''free regular set''.{{sfn|Maskit|1988|loc=II.A.1, II.A.2}}

An action of a group {{math|''G''}} on a [[locally compact space]] {{math|''X''}} is called ''[[Cocompact group action|cocompact]]'' if there exists a compact subset {{math|''A'' ⊂ ''X''}} such that {{math|1=''X'' = ''G'' &sdot; ''A''}}. For a properly discontinuous action, cocompactness is equivalent to compactness of the [[Quotient space (topology)|quotient space]] {{math|''X'' / ''G''}}.