Strongly translation-discrete group

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.

Weaker properties

 * Translation-discrete group
 * Translation-separable group