51 research outputs found
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 with non-quadratic Dehn function
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 with growth at least (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
with 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 (and much more) for
all real computable in reasonable time, for example,
or , or 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 .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 with two
non--equivalent asymptotic cones.Comment: 6 page
On flat submaps of maps of non-positive curvature
We prove that for every if a non-positively curved -map
contains no flat submaps of radius , then the area of does not exceed
for some constant . This strengthens a theorem of Ivanov and Schupp.
We show that an infinite -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
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