A theorem of Myasnikov and Roman'kov says that any verbally closed subgroup of a finitely generated free group is a retract. We prove that all free (and many virtually free) verbally closed subgroups are retracts in any finitely generated group.
In this work we construct a harmonic analysis on free Abelian groups of rank 2, namely: we construct and investigate spaces of functions and distributions, Fourier transforms and actions of discrete and extended discrete Heisenberg groups. In the case of the rank-2 value group of a two-dimensional local field with finite last residue field we connect this harmonic analysis with harmonic analysis on the two-dimensional local field, where the latter harmonic analysis was constructed in earlier works by the authors
Let k be an algebraically closed field of characteristic zero and Ga = (k, +) the additive group of k. An algebraic variety X is said to be flexible if the tangent space at every regular point of X is generated by the tangent vectors to orbits of various regular actions of Ga. In 1972, Vinberg and Popov introduced the class of affine S-varieties which are also known as affine horospherical varieties. These are varieties on which a connected algebraic group acts with an open orbit in such a way that the stationary subgroup of each point in the orbit contains a maximal unipotent subgroup of G. In this paper the flexibility of affine horospherical varieties of semisimple groups is proved.
The model of statistical physics on a countable amenable group G is considered. For the natural action of G on the space of configurations S^G, and for any closed invariant set X we prove that there exists pressure which corresponds to a potential with finite norm on X (in the sense of the limit with respect to any Følner sequence of sets in G). The existence of an equilibrium measure is established.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.
Let X be an affine algebraic variety endowed with an action of complexity one of an algebraic torus T. It is well known that homogeneous locally nilpotent derivations on the algebra of regular functions K[X] can be described in terms of proper polyhedral divisors corresponding to T-variety X. We prove that homogeneous locally nilpotent derivations commute if an only if some combinatorial criterion holds. These results are used to describe actions of unipotent groups of dimension two on affine T-varieties.
We study the number f of connected components in the complement to a finite set (arrangement) of closed submanifolds of codimension 1 in a closed manifold M. In the case of arrangements of closed geodesics on an isohedral tetrahedron, we find all possible values for the number fof connected components. We prove that the set of numbers that cannot be realized by the number f of an arrangement of n ≥ 71 projective planes in the three-dimensional real projective space is contained in the similar known set of numbers that are not realizable by arrangements of n lines on the projective plane. For Riemannian surfaces M we express the number f via a regular neighbourhood of a union of immersed circles and the multiplicities of their intersection points. For m-dimensional Lobachevskii space we find the set of all possible numbers f for hyperplane arrangements.