Model Theory of Groups and Monoids

We first show that arithmetic is bi-interpretable (with parameters) with the free monoid and with partially commutative monoids with trivial center. This bi-interpretability implies that these monoids have the QFA property and that finitely generated submonoids of these monoids are definable. Moreover, we show that any recursively enumerable language in a finite alphabet X with two or more generators is definable in the free monoid. We also show that for metabelian Baumslag-Solitar groups and for a family of metabelian restricted wreath products, the Diophantine Problem is decidable. That is, we provide an algorithm that decides whether or not a given system of equations in these groups has a solution

On monoids, 2-firs, and semifirs

Several authors have studied the question of when the monoid ring DM of a monoid M over a ring D is a right and/or left fir (free ideal ring), a semifir, or a 2-fir (definitions recalled in section 1). It is known that for M nontrivial, a necessary condition for any of these properties to hold is that D be a division ring. Under that assumption, necessary and sufficient conditions on M are known for DM to be a right or left fir, and various conditions on M have been proved necessary or sufficient for DM to be a 2-fir or semifir. A sufficient condition for DM to be a semifir is that M be a direct limit of monoids which are free products of free monoids and free groups. W.Dicks has conjectured that this is also necessary. However F.Ced\'o has given an example of a monoid M which is not such a direct limit, but satisfies the known necessary conditions for DM to be a semifir. It is an open question whether for this M, the rings DM are semifirs. We note some reformulations of the known necessary conditions for DM to be a 2-fir or a semifir, motivate Ced\'o's construction and a variant, and recover Ced\'o's results for both constructions. Any homomorphism from a monoid M into \Z induces a \Z-grading on DM, and we show that for the two monoids in question, the rings DM are "homogeneous semifirs" with respect to all such nontrivial \Z-gradings; i.e., have (roughly) the property that every finitely generated homogeneous one-sided ideal is free. If M is a monoid such that DM is an n-fir, and N a "well-behaved" submonoid of M, we obtain results on DN. Using these, we show that for M a monoid such that DM is a 2-fir, mutual commutativity is an equivalence relation on nonidentity elements of M, and each equivalence class, together with the identity element, is a directed union of infinite cyclic groups or infinite cyclic monoids. Several open questions are noted.Comment: 28 pages. To appear, Semigroup Forum. Some clarifications and corrections from previous versio

Grothendieck Rings of Theories of Modules

The model-theoretic Grothendieck ring of a first order structure, as defined by Krajic\v{e}k and Scanlon, captures some combinatorial properties of the definable subsets of finite powers of the structure. In this paper we compute the Grothendieck ring, $K_0(M_\mathcal R)$, of a right $R$-module $M$, where $\mathcal R$ is any unital ring. As a corollary we prove a conjecture of Prest that $K_0(M)$ is non-trivial, whenever $M$ is non-zero. The main proof uses various techniques from the homology theory of simplicial complexes.Comment: 42 Page

Introduction to Sofic and Hyperlinear groups and Connes' embedding conjecture

Sofic and hyperlinear groups are the countable discrete groups that can be approximated in a suitable sense by finite symmetric groups and groups of unitary matrices. These notions turned out to be very deep and fruitful, and stimulated in the last 15 years an impressive amount of research touching several seemingly distant areas of mathematics including geometric group theory, operator algebras, dynamical systems, graph theory, and more recently even quantum information theory. Several longstanding conjectures that are still open for arbitrary groups were settled in the case of sofic or hyperlinear groups. These achievements aroused the interest of an increasing number of researchers into some fundamental questions about the nature of these approximation properties. Many of such problems are to this day still open such as, outstandingly: Is there any countable discrete group that is not sofic or hyperlinear? A similar pattern can be found in the study of II_1 factors. In this case the famous conjecture due to Connes (commonly known as the Connes embedding conjecture) that any II_1 factor can be approximated in a suitable sense by matrix algebras inspired several breakthroughs in the understanding of II_1 factors, and stands out today as one of the major open problems in the field. The aim of these notes is to present in a uniform and accessible way some cornerstone results in the study of sofic and hyperlinear groups and the Connes embedding conjecture. The presentation is nonetheless self contained and accessible to any student or researcher with a graduate level mathematical background. An appendix by V. Pestov provides a pedagogically new introduction to the concepts of ultrafilters, ultralimits, and ultraproducts for those mathematicians who are not familiar with them, and aiming to make these concepts appear very natural.Comment: 157 pages, with an appendix by Vladimir Pesto

The Diophantine problem in Chevalley groups

In this paper we study the Diophantine problem in Chevalley groups $G_\pi (\Phi,R)$, where $\Phi$ is an indecomposable root system of rank $> 1$, $R$ is an arbitrary commutative ring with $1$. We establish a variant of double centralizer theorem for elementary unipotents $x_\alpha(1)$. This theorem is valid for arbitrary commutative rings with $1$. The result is principle to show that any one-parametric subgroup $X_\alpha$, $\alpha \in \Phi$, is Diophantine in $G$. Then we prove that the Diophantine problem in $G_\pi (\Phi,R)$ is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in $R$. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.Comment: 44 page