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 $S$ generate group $H$, and let $G$ be an overgroup for $H$ generated by $S &cup; T$. Then each generating set defines a word metric on the corresponding group; the distortion of $H$ in $G$ is the asymptotic equivalence class of the function $$R\mapsto\frac{\operatorname{diam}_H(B_G(0,R)\cap H)}{\operatorname{diam}_H(B_H(0,R))}\text{,}$$ where $B_{X}(x, r)$ is the ball of radius $r$ about center $x$ in $X$ and $diam(S)$ is the diameter of $S$.

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 $\mathbb{Z} = ⟨b⟩$, embedded as a normal subgroup of the Baumslag–Solitar group $BS(1, 2) = ⟨a, b⟩$. With respect to the chosen generating sets, the element $$b^{2^n}=a^nba^{-n}$$ is distance $2^{n}$ from the origin in $\mathbb{Z}$, but distance $2n + 1$ from the origin in $BS(1, 2)$. In particular, $\mathbb{Z}$ is at least exponentially distorted with base $2$.

On the other hand, any embedded copy of $\mathbb{Z}$ in the free abelian group on two generators $\mathbb{Z}^{2}$ is undistorted, as is any embedding of $\mathbb{Z}$ into itself.

Elementary properties
In a tower of groups $K ≤ H ≤ G$, the distortion of $K$ in $G$ is at least the distortion of $H$ in $G$.

A normal abelian subgroup has distortion determined by the eigenvalues of the conjugation overgroup representation; formally, if $g ∈ G$ acts on $V ≤ G$ with eigenvalue $λ$, then $V$ is at least exponentially distorted with base $λ$. 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 $n ↦ n^{r}$ for some rational $r$.

The denominator in the definition is always $2R$; 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 $n$. The transmitter then encodes $n$ as an element $g ∈ H$ with word length $n$. In a public overgroup $G$ with that distorts $H$, the element $g$ has a word of much smaller length, which is then transmitted to the receiver along with a number of "decoys" from $G \ H$, to obscure the secret subgroup $H$. The receiver then picks out the element of $H$, re-expresses the word in terms of generators of $H$, and recovers $n$.