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.^[1] Like much of geometric group theory, the concept is due to Misha Gromov, who introduced it in 1993.^[2]

Formally, let S generate group H, and let G be an overgroup for H generated by S ∪ 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 $${\displaystyle R\mapsto {\frac {\operatorname {diam} _{H}(B_{G}(0,R)\cap H)}{\operatorname {diam} _{H}(B_{H}(0,R))}}{\text{,}}}$$ where B_X(x, r) is the ball of radius r about center x in X and diam(S) is the diameter of S.^[2]: 49

A subgroup with bounded distortion is called undistorted, and is the same thing as a quasi-isometrically embedded subgroup.^[3]