In this paper, we verify the solvability of free product of finite cyclic groups with topological methods. We use Cayley graphs and Everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. In particular, using these methods, we prove that the commutator subgroup of $mathbb{Z}_{m}*mathbb{Z}_{n}$ is free of rank $(m-1)(n-1)$ fo...

Amalgamated free products of von Neumann algebras were first used by S. Popa ([26]) to construct an irreducible inclusion of (non-AFD) type II1 factors with an arbitrary (admissible) Jones index. Further investigation in this direction was made by K. Dykema ([10]) and F. Rădulescu ([27, 29]) based on Voiculescu’s powerful machine ([40, 41, 44]), and F. Boca ([4]) discussed the Haagerup approxim...

We show that there are sets of integers with asymptotic density arbitrarily close to 1 in which there is no solution to the equation ab = c, with a, b, c in the set. We also consider some natural generalizations, as well as a specific numerical example of a product-free set of integers with asymptotic density greater than 1/2.

Let (A; ') be the reduced free product of innnitely many C {algebras (A ; ') with respect to faithful states. Assume that the A are not too small, in a speciic sense. If ' is a trace then the positive cone of K 0 (A) is determined entirely by K 0 ('). If, furthermore, the image of K 0 (') is dense in R, then A has real rank zero. On the other hand, if ' is not a trace then A is simple and purel...

If A is a finite subset of a free group with at least two noncommuting elements then |A · A · A| ≥ |A| 2 (log |A|)O(1) . More generally, the same conclusion holds in an arbitrary virtually free group, unless A generates a virtually cyclic subgroup. The central part of the proof of this result is carried on by estimating the number of collisions in multiple products A1 · . . . · Ak. We include a...

Consider the reduced free product of C {algebras, (A; ') = (A 1 ; ' 1) (A 2 ; ' 2), with respect to states ' 1 and ' 2 that are faithful. If ' 1 and ' 2 are traces, if the so{called Avitzour conditions are satissed, (i.e. A 1 and A 2 are not \too small" in a speciic sense) and if A 1 and A 2 are nuclear, then it is shown that the positive cone, K 0 (A) + , of the K 0 {group of A consists of tho...