The Atiyah conjecture predicts that the L-Betti numbers of a finite CW -complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for positive 1-relator g...

We give new examples of FA presentable torsion-free abelian groups. Namely, for every n > 2, we construct a rank n indecomposable torsion-free abelian group which has an FA presentation. We also construct an FA presentation of the group (Z,+) in which every nontrivial cyclic subgroup is not FA recognizable.

Let $A$ be a nite dimensional $k-$algebra and $R$ be a locally bounded category such that $R rightarrow R/G = A$ is a Galois covering dened by the action of a torsion-free group of automorphisms of $R$. Following [30], we provide criteria on the convex subcategories of a strongly simply connected category R in order to be a cycle- nite category and describe the module category of $A$. We p...

In this article, we formalize in Mizar [7] the definition of “torsion part” of Z-module and its properties. We show Z-module generated by the field of rational numbers as an example of torsion-free non free Z-modules. We also formalize the rank-nullity theorem over finite-rank free Z-modules (previously formalized in [1]). Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lo...

We show that if G is a non-elementary torsion-free word hyperbolic group then there exists another word hyperbolic group G∗, such that G is a subgroup of G∗ but G is not quasiconvex in G∗ . We also prove that any non-elementary subgroup of a torsion-free word hyperbolic group G contains a free group of rank two which is malnormal and quasiconvex in G.

The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsion-free groups is proved. In a special case, this generalized version become a multivariable variant of this theorem. For unimodular equations over torsion free groups, we prove an analogue of Magnus...

Adding two generators and one arbitrary relator to a nontrivial torsion-free group, we always obtain an SQ-universal group. In the course of the proof of this theorem, we obtain some other results of independent interest. For instance, adding one generator and one relator in which the exponent sum of the additional generator is one to a free product of two nontrivial torsion-free groups, we als...

We consider the Dirac equation in flat Minkowski 3–space and rewrite it as the Maxwell equation in Minkowski 4–space with torsion. The torsion tensor is defined as the dual of the electromagnetic vector potential. Our model clearly distinguishes the electron and the positron without resorting to “negative frequencies”: we produce a real scalar invariant (charge) which indicates whether we are l...