Alternatives for pseudofinite 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 pseudofinite 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.

Ampleness in the free group

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.

Finitely presented groups with infinitely many non-homeomorphic asymptotic cones

We construct a finitely presented group with infinitely many non-homeomorphic asymptotic cones. We also show that the existence of cut points in asymptotic cones of finitely presented groups does, in general, depend on the choice of scaling constants and ultrafilters.

Homogeneity and prime models in torsion-free hyperbolic groups

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.

The monomorphism problem in free groups

Let $F$ be a free group of finite rank. We say that the monomorphism problem in $F$ is decidable if for any two elements $u$ and $v$ in $F$, there is an algorithm that determines whether there exists a monomorphism of $F$ that sends $u$ to $v$. In this paper we show that the monomorphism problem is decidable and we provide an effective algorithm that solves the problem.

One Variable Equations in Torsion-Free Hyperbolic Groups

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.

Note on the Cantor-Bendixson rank of limit groups

We show that the Cantor-Bendixson rank of a limit group is finite as well as that of a limit group of a linear group.

Superstable groups acting on trees

We study superstable groups acting on trees. We prove that an action of an $omega$-stable group on a simplicial tree is trivial. This shows that an HNN-extension or a nontrivial free product with amalgamation is not $omega$-stable. It is also shown that if $G$ is a superstable group acting nontrivially on a $Lambda$-tree, where $Lambda=mathbb Z$ or $Lambda=mathbb R$, and if $G$ is either $alpha$-connected and $Lambda=mathbb Z$, or if the action is irreducible, then $G$ interprets a simple group having a nontrivial action on a $Lambda$-tree. In particular if $G$ is superstable and splits as $G=G_1*_AG_2$, with the index of $A$ in $G_1$ different from 2, then $G$ interprets a simple superstable non $omega$-stable group. We will deal with minimal superstable groups of finite Lascar rank acting nontrivially on $Lambda$-trees, where $Lambda=mathbb Z$ or $Lambda=mathbb R$. We show that such groups $G$ have definable subgroups $H_1 lhd H_2 lhd G$, $H_2$ is of finite index in $G$, such that if $H_1$ is not nilpotent-by-finite then any action of $H_1$ on a $Lambda$-tree is trivial, and $H_2/H_1$ is either soluble or simple and acts nontrivially on a $Lambda$-tree. We are interested particularly in the case where $H_2/H_1$ is simple and we show that $H_2/H_1$ has some properties similar to those of bad groups.

On Finitely Generated Models of Theories with at Most Countably Many Nonisomorphic Finitely Generated Models

We study finitely generated models of countable theories, having at most countably many nonisomorphic finitely generated models. We intro- duce a notion of rank of finitely generated models and we prove, when T has at most countably many nonisomorphic finitely generated models, that every finitely generated model has an ordinal rank. This rank is used to give a prop- erty of finitely generated models analogue to the Hopf property of groups and also to give a necessary and sufficient condition for a finitely generated model to be prime of its complete theory. We investigate some properties of limit groups of equationally noetherian groups, in respect to their ranks.

Subgroup theorem for valuated groups and the CSA property

A valuated group with normal forms is a group with an integer-valued length function satisfying some Lyndons axioms and an additional axiom considered by Hurley. We prove a subgroup theorem for valuated groups with normal forms analogous to Grushko-Neumanns theorem. We study also the CSA property in such groups.