POWER SERIES OVER THE GROUP RING OF A FREE GROUP AND APPLICATIONS TO NOVIKOV-SHUBIN INVARIANTS

2003 ◽  
Author(s):  
ROMAN SAUER
Keyword(s):  
Power Series ◽  
Free Group ◽  
Group Ring

Homology of the Fermat Tower and Universal Measures for Jacobi Sums

2016 ◽  
Vol 59 (3) ◽  
pp. 624-640
Author(s):  
Noriyuki Otsubo
Keyword(s):  
Power Series ◽  
Group Ring ◽  
Simple Proof ◽  
Homology Group ◽  
Interpolation Property ◽  
Cyclic Module ◽  
Precise Description ◽  
Fermat Curve ◽  
Jacobi Sums ◽  
Definition Of

AbstractWe give a precise description of the homology group of the Fermat curve as a cyclic module over a group ring. As an application, we prove the freeness of the profinite homology of the Fermat tower. This allows us to define measures, an equivalent of Anderson’s adelic beta functions, in a manner similar to Ihara’s definition of ℓ-adic universal power series for Jacobi sums. We give a simple proof of the interpolation property using a motivic decomposition of the Fermat curve.

Polynomial Ideals in Group Rings

1973 ◽  
Vol 25 (6) ◽  
pp. 1174-1182 ◽  
Author(s):  
M. M. Parmenter ◽  
I. B. S. Passi ◽  
S. K. Sehgal
Keyword(s):  
Commutative Ring ◽  
Free Group ◽  
Group Ring ◽  
Multiplicative Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Polynomial Ideals

Letf(x1, x2, … , xn) be a polynomial in n non-commuting variables x1, x2, … , xn and their inverses with coefficients in the ring Z of integers, i.e. an element of the integral group ring of the free group on X1, x2, … , xn. Let R be a commutative ring with unity, G a multiplicative group and R(G) the group ring of G with coefficients in R.

BASS CYCLIC UNITS AS FACTORS IN A FREE GROUP IN INTEGRAL GROUP RING UNITS

2011 ◽  
Vol 21 (04) ◽  
pp. 531-545 ◽  
Author(s):  
JAIRO Z. GONÇALVES ◽  
ÁNGEL DEL RÍO
Keyword(s):  
Finite Group ◽  
Free Group ◽  
Group Ring ◽  
Prime Order ◽  
Infinite Order ◽  
Integral Group Ring ◽  
Integral Group ◽  
Cyclic Unit ◽  
Central Elements ◽  
Group A

Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ℤG of a solvable and finite group G, such that u has infinite order modulo the center of U(ℤG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ℤG which is either a Bass cyclic unit or a bicyclic unit.

The Cohn localization of the free group ring

1992 ◽  
Vol 111 (3) ◽  
pp. 433-443 ◽  
Author(s):  
M. Farber ◽  
P. Vogel
Keyword(s):  
Free Group ◽  
Group Ring

In [1] P. Cohn suggested the construction of a localization of a ring with respect to a class of square matrices. Let us briefly recall the definitions.

scholarly journals A sufficient condition for the second derived factor group to be finite

1987 ◽  
Vol 30 (1) ◽  
pp. 83-89
Author(s):  
J. R. Howse
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Factor Group ◽  
Sufficient Condition

This paper concerns an application of an algorithm for the second derived factor group as described by Howse and Johnson in [3]. This algorithm has as its basis theFox derivative (see [1]), a mapping from the free group F to the group-ring ℤF, definedas follows: let X be a set of generators of a group G, and let w = y1…yk with each yi∈X±1.

scholarly journals The subgroup determined by a certain ideal in a free group ring

2016 ◽  
Vol 449 ◽  
pp. 400-407 ◽  
Author(s):  
Roman Mikhailov ◽  
Inder Bir S. Passi
Keyword(s):  
Free Group ◽  
Group Ring

Idempotents in Noetherian Group Rings

1973 ◽  
Vol 25 (2) ◽  
pp. 366-369 ◽  
Author(s):  
Edward Formanek
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Related Result ◽  
Group Rings ◽  
Ordered Group ◽  
Torsion Free Group ◽  
Zero Divisors ◽  
Torsion Free

If G is a torsion–free group and F is a field, is the group ring F[G] a ring without zero divisors? This is true if G is an ordered group or various generalizations thereof - beyond this the question remains untouched. This paper proves a related result.

Free Groups in Subnormal Subgroups and the Residual Nilpotence of the Group of Units of Groups Rings

1984 ◽  
Vol 27 (3) ◽  
pp. 365-370 ◽  
Author(s):  
Jairo Z. Gonçalves
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Free Groups ◽  
Unit Group ◽  
Subnormal Subgroup ◽  
Group Of Units ◽  
Residual Nilpotence ◽  
Subnormal Subgroups

AbstractLet KG be the group ring of the group G over the field K and U(KG) its unit group. When G is finite we derive conditions which imply that every noncentral subnormal subgroup of U(KG) contains a free group of rank two. We also show that residual nilpotence of U(KG) coincides with nilpotence, this being no longer true if G is infinite.We can answer partially the following question: when is G sub-normal in U(KG)?

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