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 existe
ntial 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.
We use basic tools of descriptive set theory to prove that a closed set $mathcal S$ of marked groups has $2^{aleph_0}$ quasi-isometry classes provided every non-empty open subset of $mathcal S$ contains at least two non-quasi-isometric groups. It fol
lows that every perfect set of marked groups having a dense subset of finitely presented groups contains $2^{aleph_0}$ quasi-isometry classes. These results account for most known constructions of continuous families of non-quasi-isometric finitely generated groups. They can also be used to prove the existence of $2^{aleph_0}$ quasi-isometry classes of finitely generated groups having interesting algebraic, geometric, or model-theoretic properties.
We show that the theory of the free group -- and more generally the theory of any torsion-free hyperbolic group -- is $n$-ample for any $ngeq 1$. We give also an explicit description of the imaginary algebraic closure in free groups.
The famous Tits alternative states that a linear group either contains a nonabelian free group or is soluble-by-(locally finite). We study in this paper similar alternatives in pseudofinite groups. We show for instance that an $aleph_{0}$-saturated p
seudofinite group either contains a subsemigroup of rank $2$ or is nilpotent-by-(uniformly locally finite). We call a class of finite groups $G$ weakly of bounded rank if the radical $rad(G)$ has a bounded Prufer rank and the index of the sockel of $G/rad(G)$ is bounded. We show that an $aleph_{0}$-saturated pseudo-(finite weakly of bounded rank) group either contains a nonabelian free group or is nilpotent-by-abelian-by-(uniformly locally finite). We also obtain some relations between this kind of alternatives and amenability.