# Compact symmetry

A hyperbolic symmetry group is **compact** if its fundamental domain is finite.

A polytope may be called compact if its symmetry group is compact. A compact polytope never has ideal or ultra-ideal vertices.

## Paracompact symmetry[edit | edit source]

A symmetry group is **paracompact** if its fundamental domain has finite area. Some authors may additionally require it to not be compact.

In 2 dimensions, the paracompact Coxeter groups are precisely those Coxeter groups with an ∞ in their diagram.

## Hypercompact symmetry[edit | edit source]

A symmetry group is **noncompact** if it is not compact or paracompact. Thus the fundamental domain of a noncompact symmetry group has infinite area and noncompact polytopes have ultra-ideal vertices.

Some authors may call this condition **hypercompactness** in which case the term "noncompact" simply refers to any symmetry group which is not compact.

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