On distorted and isometric embeddings of groups
by
A.Yu. Olshanskii
Moscow State

Assume a group G is a subgroup of a group H with a finite set of generators X. Then obviously the function l(g) = |g|_X (where g in G, and |g|_X is the length of g in H) satisfies the following D-condition. (D1) ( for allg in G)  l(g^-1) = l(g), and l(g) = 0 iff g = 1. (D2) ( for allg, h in G)  l(gh) <= l(g) + l(h). (D3) ( existsa > 1) ( for allr in N)  card{g in G | l(g) <= r} <= a^r. We proved that any function l: G --> N with D-condition can be realized (up to an equivalence) as the length of elements by an embedding of G into suitable finitely generated group H. In case of infinite cyclic group G = Z, this solves Gromov's "realization problem" for embeddings into finitely generated groups. If there is an algorithm to compute a function l with D-condition (i.e. l is a computable function, for example, l(n) = [|n|^{\surd2 -1}] for G=Z), the group H can be chosen finitely presented. To obtain the latter statement, we proved an isometric version of the Higman Embedding Theorem (in 1997). Very recently J.-C. Birget, E.Rips, M.Sapir and the author jointly completed a work, the main theorem of which says that the word problem of a finitely generated group G is NP if and only if G is a subgroup of a finitely presented group H with a polynomial isoperimetric function. (The embedding G <= H can be chosen with a bounded distortion.)

Date received: March 16, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabg-13.