Translation-separable group

From Groupprops
Jump to: navigation, search
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This term is related to: geometric group theory
View other terms related to geometric group theory | View facts related to geometric group theory

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