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 where BX(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]

  1. ^ Broaddus, Nathan; Farb, Benson; Putman, Andrew (2011). "Irreducible Sp-representations and subgroup distortion in the mapping class group". Commentarii Mathematici Helvetici. 86: 537–556. arXiv:0707.2262. doi:10.4171/CMH/233. S2CID 7665268.
  2. ^ a b Gromov, M. (1993). Asymptotic Invariants of Infinite Groups. London Mathematical Society lecture notes 182. Cambridge University Press. OCLC 842851469.
  3. ^ Druţu, Cornelia; Kapovich, Michael (2018). Geometric Group Theory. American Mathematical Society, Providence, RI. p. 285. ISBN 978-1-4704-1104-6.