Free algebras and free groups in Ore extensions and free group algebras in division rings

Abstract

Abstract. Let K be a field of characteristic zero, let σ be an automorphism of K and let δ be a σ-derivation of K. We show that the division ring D = K(x; σ, δ) either has the property that every finitely generated subring satisfies a polynomial identity or D contains a free algebra on two generators over its center. In the case when K is finitely generated over k we then see that for σ a k-algebra automorphism of K and δ a k-linear derivation of K, K(x; σ) having a free subalgebra on two generators is equivalent to σ having infinite order, and K(x; δ) having a free subalgebra is equivalent to δ being nonzero. As an application, we show that if D is a division ring with center k of characteristic zero and D * contains a solvable subgroup that is not locally abelian-by-finite, then D contains a free k-algebra on two generators. Moreover, if we assume that k is uncountable, without any restrictions on the characteristic of k, then D contains the k-group algebra of the free group of rank two.

Free Group Algebras in Division Rings with Valuation II

“…Clearly the involution of Lemma 3.1(iii) is the same as the one in (3). First we prove (2). Let f → f * be any involution in Lemma 3.1(ii).…”

“…Conjecture (A) was formulated independently by L. Makar-Limanov in [28] and T. Stafford. Evidence for conjecture (A) has been provided in many papers, for example [27], [29], [25], [1], [2]. In many division rings in which conjecture (A) holds, D in fact contains a noncommutative free group Z-algebra.…”

“…Concerning conjecture (GA), we are able to prove an extension of a result by Lichtman. More precisely, [25, Theorem 4] is (2) of the following result.…”

Growth alternative for Hecke-Kiselman monoids

“…A more general conjecture, coming from [1], can be stated as follows: if R is a finitely presented algebra then the growth of R is either subexponential or R contains a free noncommutative subalgebra. This problem, as well as related questions concerning free subalgebras of division rings, have attracted a lot of attention, see [4] for example. Most notably, a conjecture formulated independently by Makar-Limanov and Stafford says that a division algebra D either contains a free noncommutative subalgebra over its center or it is a locally PI-algebra.…”

Free algebras in division rings with an involution

“…Makar-Limanov himself provided evidence for this in [13], where it is proved that the division ring of fractions of the first Weyl algebra over the rational numbers contains a free subalgebra of rank 2, and in [14], where the case of the division ring of fractions of a group algebra of a torsion free nonabelian nilpotent groups is tackled. Various authors have dealt with this problem and Makar-Limanov's conjecture has been verified in many families of division rings (see, e.g., [12,16,9,7,18,11,19,2,10,8,5,3,17,4,6,1]).…”