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.
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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