A subset A of a group G is called symmetric with respect to the element g ∈ G if A = gA−1g. It is proved that in any 2-coloring, every infinite group G contains monochrome symmetric subsets of arbitrarily large cardinality < |G|. A topological space is called resolvable if it can be partitioned into two dense subsets [8]. In [4] W. Comfort and J. van Mill proved that each nondiscrete topologica...

A group is small if it has countably many pure n-types for each integer n. It is shown that in a small group, subgroups which are definable with parameters in a finitely generated algebraic closure satisfy local descending chain conditions. An infinite small group has an infinite abelian subgroup, which may not be definable. A nilpotent small group is the central product of a definable divisibl...

This paper investigates the relationship between strata of abelian differentials and various mapping class groups afforded by means topological monodromy representation. Building off prior work authors, we show that fundamental group a stratum surjects onto subgroup which preserves fixed framing underlying Riemann surface, thereby giving complete characterization group. In course our proof also...

Gromov’s Polynomial Growth Theorem [Gro81] states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups. Gromov’s theorem inspired the more general problem (see, e.g. [GdlH91]) of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure—in short, are solvable...

We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigo...

In this short note, we discuss certain examples of Legendrian submanifolds, whose linearized contact (co)homology groups over integers have non-vanishing algebraic torsion. More precisely, for a given arbitrary finitely generated abelian group [Formula: see text] and positive integer text], construct submanifolds the standard vector space text]th computed with respect to augmentation is isomorp...

We study internal Lie algebras in the category of subshifts on a fixed group – or algebraic for short. We show that if acting is virtually polycyclic and underlying vector space has dense homoclinic points, such can be recoded to have cellwise bracket. On other hand there exist (on any finitely-generated non-torsion group) with operations whose bracket cannot cellwi...