35 research outputs found
Subgroup Distortion in Wreath Products of Cyclic Groups
We study the effects of subgroup distortion in the wreath products A wr Z,
where A is finitely generated abelian. We show that every finitely generated
subgroup of A wr Z has distortion function equivalent to some polynomial.
Moreover, for A infinite, and for any polynomial l^k, there is a 2-generated
subgroup of A wr Z having distortion function equivalent to the given
polynomial. Also a formula for the length of elements in arbitrary wreath
product H wr G easily shows that the group Z_2 wr Z^2 has distorted subgroups,
while the lamplighter group Z_2 wr Z has no distorted (finitely generated)
subgroups
Filtrations and Distortion in Infinite-Dimensional Algebras
A tame filtration of an algebra is defined by the growth of its terms, which
has to be majorated by an exponential function. A particular case is the degree
filtration used in the definition of the growth of finitely generated algebras.
The notion of tame filtration is useful in the study of possible distortion of
degrees of elements when one algebra is embedded as a subalgebra in another. A
geometric analogue is the distortion of the (Riemannian) metric of a (Lie)
subgroup when compared to the metric induced from the ambient (Lie) group. The
distortion of a subalgebra in an algebra also reflects the degree of complexity
of the membership problem for the elements of this algebra in this subalgebra.
One of our goals here is to investigate, mostly in the case of associative or
Lie algebras, if a tame filtration of an algebra can be induced from the degree
filtration of a larger algebra
On identities in the products of group varieties
Let be the variety of groups satisfying the law . It is
proved that for every sufficiently large prime , say , the
product cannot be defined by a finite set of identities.
This solves the problem formulated by C.K. Gupta and A.N. Krasilnikov in 2003.
We also find the axiomatic and the basis ranks of the variety . For this goal, we improve the estimate for the basis rank of the product
of group varieties obtained by G. Baumslag, B.H. Neumann, H. Neumann and P.M.
Neumann long ago.Comment: 9 page
Large Restricted Lie Algebras
We establish some results about large restricted Lie algebras similar to
those known in the Group Theory. As an application we use this group-theoretic
approach to produce some examples of restricted as well as ordinary Lie
algebras which can serve as counterexamples for various Burnside-type
questions