# Translation-separable 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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## History

The notion of translation-separability was introduced, and used, by Gregory R. Conner in his paper Discreteness properties of translation numbers in solvable groups.

## Definition

### Symbol-free definition

A finitely generated group is said to be translation-separable or translation-proper if it satisfies the following equivalenyt conditions:

• There exists a generating set with respect to which the set of translation numbers has zero as an isolated point. In other words, there is a generating set, and a positive real number $\epsilon$, such that there is no translation number betwee 0 and $\epsilon$.
• For any generating set, the set of translation numbers has zero as an isolated point. In other words, for any generating set, and a positive real number $\epsilon$, such that there is no translation number betwee 0 and $\epsilon$.

## References

• Discreteness properties of translation numbers in solvable groups by Gregory R. Conner, J. Group Theory 3 (2000), no. 1, 77–94