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 . Subgroups with constant distortion are called quasiconvex. Examples
For example, consider the infinite cyclic group , embedded as a normal subgroup of the Baumslag–Solitar group . Then  is distance  from the origin in , but distance  from the origin in . In particular,  is at least exponentially distorted with base . Similarly, the same infinite cyclic group, embedded in the free abelian group on two generators , has linear distortion; the embedding in itself as  only produces constant distortion. 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 (up to asymptotic equivalence) can be found this way. In cryptography
The simplification in a word problem induced by subgroup distortion suffices to construct a cryptosystem, algorithms for encoding and decoding secret messages. Formally, the plaintext message is any object (such as text, images, or numbers) that can be encoded as a number . The transmitter then encodes  as an element  with word length . In a public overgroup  with that distorts , the element  has a word of much smaller length, which is then transmitted to the receiver along with a number of "decoys" from , to obscure the secret subgroup . The receiver then picks out the element of , re-expresses the word in terms of generators of , and recovers . References 

Geometric group theory
Low-dimensional topology