Related papers: A note on pseudofinite groups of finite centralise…
In this paper we study ends of finitely generated semigroups. The ends we are working with are the ends of the undirected graphs of Cayley graphs of finitely generated semigroups. We prove that the number of ends is preserved for…
We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity…
A pseudomodular group is a finite coarea nonarithmetic Fuchsian group whose cusp set is exactly $\mathbb{P}^1(\mathbb{Q})$. Long and Reid constructed finitely many of these by considering Fricke groups, i.e., those that uniformize…
It is proven that if a finitely presented group is one ended it has asymptotic dimension bigger than one. It follows that finitely presented groups with asdim 1 are virtually free. A counterexample is given for the finitely generated case.
We construct a family of finite special 2-groups which have commuting graph of increasing diameter
It is shown that there exists a finitely generated infinite simple group of infinite commutator width, and that the commutator width of a finitely generated infinite boundedly simple group can be arbitrarily large. Besides, such groups can…
Necessary and sufficient conditions for finite commutative semihypergroups to be built from abelian groups of the same order are established.
We observe that the nonstandard finite cardinality of a definable set in a strongly minimal pseudofinite structure D is a polynomial over the integers in the nonstandard finite cardinality of D. We conclude that D is unimodular, hence also…
Universality has been an important concept in computable structure theory. A class $\mathcal{C}$ of structures is universal if, informally, for any structure, of any kind, there is a structure in $\mathcal{C}$ with the same…
We show that countable direct limits of finite-dimensional Lie groups do not have small subgroups. The same conclusion is obtained for suitable direct limits of infinite-dimensional Lie groups.
We consider formal maps in any finite dimension $d$ with coefficients in an integral domain $K$ with identity. Those invertible under formal composition form a group $\mathcal{G}$. We consider the centraliser $C_g$ of an element…
It is known that in any free group the isolator of finitely generated subgroup is finitely generated subgroup. A very simple proof of this statement is proposed.
We develop local NIP group theory in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure $G$ expanding a group, and left invariant NIP formula $\delta(x;\bar{y})$, we prove various…
We provide explicit identity bases for finite cyclic semigroups.
We investigate structural properties of non-sofic groups, assuming that such groups exist. We introduce and study two classes: minimal non-sofic groups and $\omega$-non-sofic groups. For minimal non-sofic groups, we establish strong…
In this paper we prove that free solvable groups have finite Krull dimension. In fact, this is true for much wider class of solvable groups, termed rigid groups. Along the way we study the algebraic structure of the limit solvable groups…
The main result of this paper is a probabilistic construction of finite rigid structures. It yields a finitely axiomatizable class of finite rigid structures where no L^omega_{infty, omega} formula with counting quantifiers defines a linear…
We classify quasi-simple finite groups of essential dimension 3.
In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.
We study pseudorandomness and pseudorandom generators from the perspective of logical definability. Building on results from ordinary derandomization and finite model theory, we show that it is possible to deterministically construct, in…