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Existentially closed groups are, informally, groups that contain solutions to every consistent finite system of equations and inequations. They were introduced in 1951 in an algebraic context and subsequent research elucidated deep…
We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through…
In [arXiv:1405.6274, Question 5.2 & Question 5.3] Aschenbrenner, Friedl and Wilton ask: (1) Is the equation problem solvable for the fundamental group of any $3$-manifold? and (2) Is the first-order theory of the fundamental group of any…
Let $\C(\Gamma)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $\Gamma$. We investigate the extent to which $\C(\Gamma)$ determines $\Gamma$ when $\Gamma$ is a group of geometric interest. If…
The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent…
The general decomposition theory of exponential operators is briefly reviewed. A general scheme to construct independent determining equations for the relevant decomposition parameters is proposed using Lyndon words. Explicit formulas of…
We prove that all finitely generated fully residually free groups (limit groups) have a sequence of finite dimensional unitary representations that `strongly converge' to the regular representation of the group. The corresponding statement…
In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker condition:…
Let G be a torsion free hyperbolic group. We prove that the elementary theory of G is decidable and admits an effective quantifier elimination to boolean combination of AE-formulas. The existence of such quantifier elimination was…
A class of determinants is introduced. Different kind of mathematical objects, such as Fibonacci, Lucas, Tchebychev, Hermite, Laguerre, Legendre polynomials, sums and covergents are represented as determinants from this class. A closed…
We construct examples of finitely presented simple groups whose Dehn functions are at least exponential. To the best of our knowledge, these are the first such examples known. Our examples arise from R\"over-Nekrashevych groups, using…
We develop techniques at the interface between differential algebra and model theory to study the following problems of exponential algebraicity: Does a given algebraic differential equation admits an exponentially algebraic solution, that…
We study the exponential stability of evolutionary equations. The focus is laid on second order problems and we provide a way to rewrite them as a suitable first order evolutionary equation, for which the stability can be proved by using…
A description of the endomorphisms of semidirect products of two groups as a group of $2\times 2$ matrices of maps is already known. Using this description, we have studied the concept of determinant for the endomorphisms of semidirect…
We solve the twisted conjugacy problem on Thompson's group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut_+(F) are orbit decidable provided a certain conjecture on Thompson's group T is true.…
Free exponential families have been previously introduced as a special case of the q-exponential family. We show that free exponential families arise also from a procedure analogous to the definition of exponential families by using the…
A finitely generated group $G$ is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety $\mathcal{UV}$, where $\mathcal{U}$ (respectively,…
A famous result of Hall asserts that the multiplication and exponentiation in finitely generated torsion free nilpotent groups can be described by rational polynomials. We describe an algorithm to determine such polynomials for all torsion…
We give a positive answer, in the measurable-group-theory context, to von Neumann's problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group…
It is shown that there exist finitely generated infinite simple groups of infinite commutator width and infinite square width on which there exists no stably unbounded conjugation-invariant norm, and in particular stable commutator length…