ﻻ يوجد ملخص باللغة العربية
We show that any nonabelian free group $F$ of finite rank is homogeneous; that is for any tuples $bar a$, $bar b in F^n$, having the same complete $n$-type, there exists an automorphism of $F$ which sends $bar a$ to $bar b$. We further study existential types and we show that for any tuples $bar a, bar b in F^n$, if $bar a$ and $bar b$ have the same existential $n$-type, then either $bar a$ has the same existential type as a power of a primitive element, or there exists an existentially closed subgroup $E(bar a)$ (resp. $E(bar b)$) of $F$ containing $bar a$ (resp. $bar b$) and an isomorphism $sigma : E(bar a) to E(bar b)$ with $sigma(bar a)=bar b$. We will deal with non-free two-generated torsion-free hyperbolic groups and we show that they are $exists$-homogeneous and prime. This gives, in particular, concrete examples of finitely generated groups which are prime and not QFA.
Let $Gamma$ be a torsion-free hyperbolic group. We show that the set of solutions of any system of equations with one variable in $Gamma$ is a finite union of points and cosets of centralizers if and only if any two-generator subgroup of $Gamma$ is free.
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 free nilpo
In this paper we determine the torsion free rank of the group of endotrivial modules for any finite group of Lie type, in both defining and non-defining characteristic. On our way to proving this, we classify the maximal rank $2$ elementary abelian $
A theorem by Hall asserts that the multiplication in torsion free nilpotent groups of finite Hirsch length can be facilitated by polynomials. In this note we exhibit explicit Hall polynomials for the torsion free nilpotent groups of Hirsch length at most 5.
We examine residual properties of word-hyperbolic groups, adapting a method introduced by Darren Long to study the residual properties of Kleinian groups.