Jairo Z. Gonçalves

Are there free groups in division rings

The following conjecture is investigated: a noncentral subnormal subgroup of the multiplicative group of a division ring contains a noncyclic free subgroup. Special cases are proved, entailing several known commutativity theorems. Also a new framework is presented for some kinds of commutativity theorems, based on the existence of (group) words for which one can always find an appropriate substitution by elements of such a subnormal subgroup that yields a noncentral element. Several families of such words are given; one gets commutativity theorems imposing some restrictions (like periodicity) to the image of these words.

UNITARY UNITS IN GROUP ALGEBRAS

LetK[G] denote the group algebra of the finite groupG over the non-absolute fieldK of characteristic ≠ 2, and let *:K[G] →K[G] be theK-involution determined byg*=g−1 for allg ∈G. In this paper, we study the group A = A(K[G]) of unitary units ofK[G] and we classify those groupsG for which A contains no nonabelian free group. IfK is algebraically closed, then this problem can be effectively studied via the representation theory ofK[G]. However, for general fields, it is preferable to take an approach which avoids having to consider the division rings involved. Thus, we use a result of Tits to construct fairly concrete free generators in numerous crucial special cases.

On free group algebras in division rings with uncountable center

Let D be a division algebra with uncountable center k. If D contains a noncommutative free k-algebra, then D also contains the k-group algebra of the free group of rank 2.

A Survey on Free Objects in Division Rings and in Division Rings with an Involution

Let D be a division ring with center k, and let D † be its multiplicative group. We investigate the existence of free groups in D †, and free algebras and free group algebras in D. We also go through the case when D has an involution * and consider the existence of free symmetric and unitary pairs in D †.

FREE SYMMETRIC AND UNITARY PAIRS IN DIVISION RINGS WITH INVOLUTION

Let D be a division ring with an involution and characteristic different from 2. Then, up to a few exceptions, D contains a pair of symmetric elements freely generating a free subgroup of its multiplicative group provided that (a) it is finite-dimensional and the center has a finite sufficiently large transcendence degree over the prime field, or (b) the center is uncountable, but not algebraically closed in D. Under conditions (a), if the involution is of the first kind, it is also shown that the unitary subgroup of the multiplicative group of D contains a free subgroup, with one exception. The methods developed are also used to exhibit free subgroups in the multiplicative group of a finite-dimensional division ring provided the center has a sufficiently large transcendence degree over its prime field.

Construction of free subgroups in the group of units of modular group algebras

Let KGbe the group algebra of a p1 -group Gover a field Kof characteristic p > 0, and let U(KG)be its group of units. If KGcontains a nontrivial bicyclic unit and if Kis not algebraic over its prime field, then we prove that the free product Zp∗ Zp∗ Zpcan be embedded in U(KG).

Involutions and free pairs of bicyclic units in integral group rings

Abstract If * : G → G is an involution on the finite group G, then * extends to an involution on the integral group ring ℤ[G]. In this paper, we consider whether bicyclic units u ∈ ℤ[G] exist with the property that the group 〈u, u*〉 generated by u and u* is free on the two generators. If this occurs, we say that (u, u*) is a free bicyclic pair. It turns out that the existence of u depends strongly upon the structure of G and on the nature of the involution. One positive result here is that if G is a nonabelian group with all Sylow subgroups abelian, then for any involution *, ℤ[G] contains a free bicyclic pair.

Bicyclic units, Bass cyclic units and free groups

Abstract Let G be a finite group and ℤG its integral group ring. We show that if α is a nontrivial bicyclic unit of ℤG, then there are bicyclic units β and γ of different types, such that 〈α, β〉 and 〈α, γ〉 are non-abelian free groups. In the case when G is non-abelian of order coprime to 6 we prove the existence of a bicyclic unit u and a Bass cyclic unit v in ℤG, such that 〈um , v〉 is a free non-abelian group for all sufficiently large positive integers m.

EXPLICIT FREE GROUPS IN DIVISION RINGS

Let D be a division ring of characteristic 6= 2 and suppose that the multiplicative group D• = D \ {0} has a subgroup G isomorphic to the Heisenberg group. Then we use the generators of G to construct an explicit noncyclic free subgroup of D•. The main difficulty occurs here when D has characteristic 0 and the commutators in G are algebraic over Q.

INVOLUTIONS AND FREE PAIRS OF BASS CYCLIC UNITS IN INTEGRAL GROUP RINGS

Let ℤG be the integral group ring of the finite nonabelian group G over the ring of integers ℤ, and let * be an involution of ℤG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (uk, m(x), uk, m(x*)) or (uk, m(x), uk, m(y)), formed respectively by Bass cyclic and *-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ℤG.