Strongly translation-discrete group
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This term is related to: geometric group theory
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Definition
A finitely generated group is said to be strongly translation-discrete if it satisfies the following equivalent conditions:
- There exists a finite generating set such that the group is translation-separable with respect to that generating set, and further, for any , the number of conjugacy classes for which the translation number is , is finite.
- For every finite generating set of the group, the group is translation-separable with respect to that generating set, and further, for any , the number of conjugacy classes for which the translation number is , is finite.
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Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finitely generated free group | finitely generated free implies strongly translation-discrete | |FULL LIST, MORE INFO | ||
| finitely generated abelian group | finitely generated abelian implies strongly translation-discrete | |FULL LIST, MORE INFO |