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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.
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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 follows 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 apply results proved in [Li19] to the linear order expansions of non-trivial free homogeneous structures and the universal n-linear order for $ngeq 2$, and prove the simplicity of their automorphism 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.
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.
We prove that an inverse-free equation is valid in the variety LG of lattice-ordered groups (l-groups) if and only if it is valid in the variety DLM of distributive lattice-ordered monoids (distributive l-monoids). This contrasts with the fact that, as proved by Repnitskii, there exist inverse-free equations that are valid in all Abelian l-groups but not in all commutative distributive l-monoids, and, as we prove here, there exist inverse-free equations that hold in all totally ordered groups but not in all totally ordered monoids. We also prove that DLM has the finite model property and a decidable equational theory, establish a correspondence between the validity of equations in DLM and the existence of certain right orders on free monoids, and provide an effective method for reducing the validity of equations in LG to the validity of equations in DLM.
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