On pairs of finitely generated subgroups in free groups

We prove that for arbitrary two finitely generated subgroups A and B having infinite index in a free group F, there is a subgroup H of finite index in B such that the subgroup generated by A and H has infinite index in F. The main corollary of this theorem says that a noncyclic free group of finite rank admits a faithful highly transitive action on an infinite set, whereas the restriction of this action to any finitely generated subgroup of infinite index in F has no infinite orbits.Comment: 12 pages, 2 figure

Groups with quadratic-non-quadratic Dehn functions

We construct a finitely presented group $G$ with non-quadratic Dehn function $f$ majorizable by a quadratic function on arbitrary long intervals.Comment: 16 page

Polynomially-bounded Dehn functions of groups

On the one hand, it is well known that the only subquadratic Dehn function of finitely presented groups is the linear one. On the other hand there is a huge class of Dehn functions $d(n)$ with growth at least $n^4$ (essentially all possible such Dehn functions) constructed in \cite{SBR} and based on the time functions of Turing machines and S-machines. The class of Dehn functions $n^{\alpha}$ with $\alpha\in (2; 4)$ remained more mysterious even though it has attracted quite a bit of attention (see, for example, \cite{BB}). We fill the gap obtaining Dehn functions of the form $n^{\alpha}$ (and much more) for all real $\alpha\ge 2$ computable in reasonable time, for example, $\alpha=\pi$ or $\alpha= e$, or $\alpha$ is any algebraic number. As in \cite{SBR}, we use S-machines but new tools and new way of proof are needed for the best possible lower bound $d(n)\ge n^2$.Comment: 98 pages, 18 figures, replaced figures, correction

Linear automorphism groups of relatively free groups

Let G be a free group in a variety of groups, but G is not absolutely free. We prove that the group of automorphisms Aut(G) is linear iff G is a virtually nilpotent group.Comment: 4 page

On products of T-ideals in free algebras and free group algebras

Let F be a field and A a free associative F-algebra or a group algebra of a free group with an infinite set X of generators. We find a necessary and sufficient condition for the inclusion I' into I, where I=I_1...I_k and I'=I'_1...I'_l are any products of T-ideals in A. A canonical reformulation in terms of products of group representation varieties answers a question posed in 1986Comment: 7 page

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

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.Comment: 33 page

A finitely presented group with two non-homeomorphic asymptotic cones

We give an example of a finitely presented group $G$ with two non-$\pi_1$-equivalent asymptotic cones.Comment: 6 page

On flat submaps of maps of non-positive curvature

We prove that for every $r>0$ if a non-positively curved $(p,q)$-map $M$ contains no flat submaps of radius $r$, then the area of $M$ does not exceed $Crn$ for some constant $C$. This strengthens a theorem of Ivanov and Schupp. We show that an infinite $(p,q)$-map which tessellates the plane is quasi-isometric to the Euclidean plane if and only if the map contains only finitely many non-flat vertices and faces. We also generalize Ivanov and Schupp's result to a much larger class of maps, namely to maps with angle functions.Comment: v1: 25 page

On finite presentability of group F/[M,N]

The characterization of normal subgroups M, N of free group F for which the quotient group F/[M,N] is finitely presented is given.Comment: 6 page

The Conjugacy Problem and Higman Embeddings

For every finitely generated recursively presented group G we construct a finitely presented group H containing G such that G is (Frattini) embedded into H and the group H has solvable conjugacy problem if and only if G has solvable conjugacy problem. Moreover G and H have the same r.e. Turing degrees of the conjugacy problem. This solves a problem by D. Collins