Aspects of Free Actions Based on Dependent Elements in Group Rings

On dependent elements in rings

out, R will represent an associative ring with center Z(R). As usual the commutator xy−yx will be denoted by [x,y]. We will use basic commutator identities [xy,z] = [x,z]y+x[y,z] and [x,yz]= [x,y]z+y[x,z]. Recall that a ring R is prime if aRb = (0) implies that a = 0 or b = 0, and is semiprime if aRa = (0) implies a = 0. An additive mapping x x∗ on a ring R is called involution in case (xy)∗ = ...

Algebraic Elements in Group Rings

In this brief note, we study algebraic elements in the complex group algebra C[G]. Specifically, suppose £ e C[G] satisfies /(£) = 0 for some nonzero polynomial f(x) e C[x]. Then we show that a certain fairly natural function of the coefficients of Z is bounded in terms of the complex roots of f(x). For G finite, this is a recent observation of [HLP], Thus the main thrust here concerns infinite...

Distortion Elements in Group actions on surfaces

If G is a finitely generated group with generators {g1, . . . , gj} then an infinite order element f ∈ G is a distortion element of G provided lim inf n→∞ |f|/n = 0, where |f| is the word length of f in the generators. Let S be a closed orientable surface and let Diff(S)0 denote the identity component of the group of C diffeomorphisms of S. Our main result shows that if S has genus at least two...

Cyclic Group Actions on Polynomial Rings

We consider a cyclic group of order p acting on a module incharacteristic p and show how to reduce the calculation of the symmetric algebra to that of the exterior algebra. Consider a cyclic group of order p acting on a polynomial ring S = k[x1, . . . , xr], where k is a field of characteristic p; this is equivalent to the symmetric algebra S∗(V ) on the module V generated by x1, . . . , xr. We...

Dependent Elements of Derivations on Semiprime Rings