On identities in the products of group varieties

Let ${\cal B}_n$ be the variety of groups satisfying the law $x^n=1$. It is proved that for every sufficiently large prime $p$, say $p>10^{10}$, the product ${\cal B}_p{\cal B}_p$ 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 ${\cal B}_p{\cal B}_p$. 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

On Volumes of Permutation Polytopes

This paper focuses on determining the volumes of permutation polytopes associated to cyclic groups, dihedral groups, groups of automorphisms of tree graphs, and Frobenius groups. We do this through the use of triangulations and the calculation of Ehrhart polynomials. We also present results on the theta body hierarchy of various permutation polytopes.Comment: 19 pages, 1 figur

Knapsack Problems in Groups

We generalize the classical knapsack and subset sum problems to arbitrary groups and study the computational complexity of these new problems. We show that these problems, as well as the bounded submonoid membership problem, are P-time decidable in hyperbolic groups and give various examples of finitely presented groups where the subset sum problem is NP-complete.Comment: 28 pages, 12 figure

Constants of Weitzenb\"ock derivations and invariants of unipotent transformations acting on relatively free algebras

In commutative algebra, a Weitzenb\"ock derivation is a nonzero triangular linear derivation of the polynomial algebra $K[x_1,...,x_m]$ in several variables over a field $K$ of characteristic 0. The classical theorem of Weitzenb\"ock states that the algebra of constants is finitely generated. (This algebra coincides with the algebra of invariants of a single unipotent transformation.) In this paper we study the problem of finite generation of the algebras of constants of triangular linear derivations of finitely generated (not necessarily commutative or associative) algebras over $K$ assuming that the algebras are free in some sense (in most of the cases relatively free algebras in varieties of associative or Lie algebras). In this case the algebra of constants also coincides with the algebra of invariants of some unipotent transformation. \par The main results are the following: 1. We show that the subalgebra of constants of a factor algebra can be lifted to the subalgebra of constants. 2. For all varieties of associative algebras which are not nilpotent in Lie sense the subalgebras of constants of the relatively free algebras of rank $\geq 2$ are not finitely generated. 3. We describe the generators of the subalgebra of constants for all factor algebras $K/I$ modulo a $GL_2(K)$-invariant ideal $I$. 4. Applying known results from commutative algebra, we construct classes of automorphisms of the algebra generated by two generic $2\times 2$ matrices. We obtain also some partial results on relatively free Lie algebras.Comment: 31 page