Strongly translation-discrete group

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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 r \in \R^+, the number of conjugacy classes [g] for which the translation number is \le r, 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 r \in \R^+, the number of conjugacy classes [g] for which the translation number is \le r, is finite.

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

Weaker properties