Consider the free algebra An generated over Q by n generators x1, . . . , xn. Interesting objects attached to A = An are members of its lower central series, Li = Li(A), defined inductively by L1 = A, Li+1 = [A,Li], and their associated graded components Bi = Bi(A) defined as Bi = Li/Li+1. These quotients Bi for i ≥ 2, as well as the reduced quotient B̄1 = A/(L2 + AL3), exhibit a rich geometric ...

Let A be a noncommutative associative algebra, viewed as a Lie algebra via definition of the Lie bracket [x, y] = xy − yx. Define the lower central series filtration inductively by L1(A) = A, and Li+1(A) = [A,Li(A)]. Denote the components of its associated graded space by Bi(A) = Li(A)/Li+1(A). The components of the associated graded space are well understood in the cases when A = An = the free...

We continue the study of the lower central series and its associated graded components for a free associative algebra with n generators, as initiated in [FS]. We establish a linear bound on the degree of tensor field modules appearing in the Jordan-Hölder series of each graded component, which is conjecturally tight. We also bound the leading coefficient of the Hilbert polynomial of each graded...

Denote by Mod(S) the mapping class group of a compact, oriented surface S = Sg,1 of genus g ≥ 2 with one boundary component; i.e. Mod(S) is the group of homeomorphisms of S fixing ∂S pointwise up to isotopies fixing ∂S pointwise. A basic question to contemplate is: what topological or dynamical data of a mapping class can be extracted from various kinds of algebraic data? Since pseudo-Anosovs a...

We give various estimates of the minimal number of self-intersections of a nontrivial element of the kth term of the lower central series and derived series of the fundamental group of a surface. As an application, we obtain a new topological proof of the fact that free groups and fundamental groups of closed surfaces are residually nilpotent. Along the way, we prove that a nontrivial element o...

This paper examines the behavior of the successive quotients Ni(A) of the lower central series ideals Mi(A) of a nitely generated associative algebra A over Z. We de ne the lower central series Li(A) by L1(A) = A, Li+1(A) = [A,Li(A)], Mi(A) = A · Li(A) · A, and Ni(A) = Mi(A)/Mi+1(A). We decompose the Ni into its free and torsion components using the structure theorem of nitely generated abelian...

for entire functions, the notions of their growth indicators such as ritt order are classical in complex analysis. but the concepts of relative ritt order of entire functions and as well as their technical advantages of not comparing with the growths of $exp exp z$ are not at all known to the researchers of this area. therefore the studies of the growths of entire functions in the light of thei...

Echinoids fauna are common and distributed in the Lower Miocene deposits of the Bagher Abad area, northeast Isfahan, Central Iran.There are six Echinoid taxa belonging to the Echinoidea Class that can be described as: Clypeaster intermedius, Prinocidaris sp. andspines related to: Eucidaris zaemays, Stylocidaris Polyacantha, Spatangoid sp. and Prinocidaris sp. Bivalves, Foraminifers,Bryozoans, B...