Length and Area Functions on Groups and Quasi-Isometric Higman Embeddings

Abstract

We survey recent results about asymptotic functions of groups, obtained by the authors in collaboration with J.-C.Birget, V. Guba and E. Rips. We also discuss methods used in the proofs of these results.

Algorithmically complex residually finite groups

“…In fact the set of possible distortion functions of subgroups of F 2 × F 2 coincides, up to a natural equivalence, with the set of Dehn functions of finitely presented groups [47]. Finitely generated subgroups of F 2 × F 2 are equalizers of pairs of homomorphisms φ : F k → G, ψ : F n → G (where F k , F n are some subgroups of F 2 ), i.e.…”

Asymptotic invariants, complexity of groups and related problems

“…It is obvious that this can be done somehow. In [187], we presented a useful and economical way to cut a van Kampen diagram (so that every edge is used at most four times). Since cutting van Kampen diagrams into nice pieces is the main ingredient of many proofs involving van Kampen diagrams (see Sect.…”

“…. , b m } = B into a finitely presented group G = C | R where B ⊂ C. This is done as in [22,187] using an explicitly constructed S-machine. Then we take a copy A = {a 1 , .…”

“…This proves the Higman embedding theorem. Proving that H has isoperimetric function that polynomially depends on the time function of S is of course more complicated (see [187]). Again it involves cutting diagrams into pieces which are treated separately.…”

“…There are very many papers devoted to these problems for various classes of groups. Different aspects of these problems are discussed in many books and surveys (see, for example, [24,35,50,72,78,91,137,147,162,181,187,207,208]). Still there are very many new results and ideas that appeared after the last of these surveys was published.…”