Subgroup distortion
In geometric group theory, a discipline of mathematics, subgroup distortion measures the extent to which an overgroup can reduce the complexity of a group's word problem. Like much of geometric group theory, the concept is due to Misha Gromov, who introduced it in 1993.
Formally, let generate group, and let be an overgroup for generated by. Then each generating set defines a word metric on the corresponding group; the distortion of in is the asymptotic equivalence class of the function where is the ball of radius about center in and is the diameter of.
A subgroup with bounded distortion is called undistorted, and is the same thing as a quasi-isometrically embedded subgroup.
Examples
For example, consider the infinite cyclic group, embedded as a normal subgroup of the Baumslag–Solitar group. With respect to the chosen generating sets, the element is distance from the origin in, but distance from the origin in. In particular, is at least exponentially distorted with base.On the other hand, any embedded copy of in the free abelian group on two generators is undistorted, as is any embedding of into itself.
Elementary properties
In a tower of groups, the distortion of in is at least the distortion of in.A normal abelian subgroup has distortion determined by the eigenvalues of the conjugation overgroup representation; formally, if acts on with eigenvalue, then is at least exponentially distorted with base. For many non-normal but still abelian subgroups, the distortion of the normal core gives a strong lower bound.
Known values
Every computable function with at most exponential growth can be a subgroup distortion, but Lie subgroups of a nilpotent Lie group always have distortion for some rational.The denominator in the definition is always ; for this reason, it is often omitted. In that case, a subgroup that is not locally finite has superadditive distortion; conversely every superadditive function can be found this way.