On the (non) superstable part of the free group
In this short note we prove that a definable set $X$ over $\mathbb F_n$ is superstable only if $X(\mathbb F_n)=X(\mathbb F_ω)$.
math.LO↗
arXiv subjects
Rizos Sklinos
Publications and source records attributed to Rizos Sklinos.
On ampleness and pseudo-Anosov homeomorphisms in the free group
We use pseudo-Anosov homeomorphisms of surfaces in order to prove that the first order theory of non abelian free groups, $T_{fg}$, is $n$-ample for any $n\inω$. This result adds to the work of Pillay, that proved that $T_{fg}$ is non CM -trivial. The sequence witnessing ampleness is a sequence of primitive elements in $F_ω$. Our result provides an alternative proof to the main result of a preprint by Ould Houcine-Tent. We also add an appendix in which we make a few remarks on Sela's paper on imaginaries in torsion free hyperbolic groups. In particular we give alternative transparent proofs concerning the non-elimination of certain imaginaries.
math.LO↗
Saturated free algebras revisited
We give an exposition of results of Baldwin-Shelah on saturated free algebras, at the level of generality of complete first order theories $T$ with a saturated model $M$ which is in the algebraic closure of an indiscernible set. We then make some new observations when $M$ is a saturated free algebra, analogous to (more difficult) results for the free group, such as a description of forking.
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Forking and JSJ decompositions in the free group
We give a description of the model theoretic relation of forking independence in terms of the notion of JSJ decompositions in non abelian free groups.
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On groups and fields interpretable in torsion-free hyperbolic groups
We prove that the generic type of a non-cyclic torsion-free hyperbolic group G is foreign to any interpretable abelian group, hence also to any interpretable field. This result depends, among other things, on the definable simplicity of a non-cyclic torsion-free hyperbolic group, and we take the opportunity to give a proof of the latter using Sela's description of imaginaries in torsion-free hyperbolic groups. We also use the description of imaginaries to prove that if F is a free group of rank > 2 then no orbit of a finite tuple from F under Aut(F) is definable.
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Hyperbolic towers and independent generic sets in the theory of free groups
We use hyperbolic towers to answer some model theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type $p_0$, but that there is a finitely generated model which omits $p_0^{(2)}$. We exhibit a finitely generated model in which there are two maximal independent sets of realizations of the generic type which have different cardinalities. We also show that a free product of homogeneous groups is not necessarily homogeneous.
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A note on ampleness in the theory of non abelian free groups
Recently Ould Houcine-Tent (see arXiv:1205.0929v2 [math.GR]) proved that the theory of non abelian free groups is $n$-ample for any $n<ω$. We give a sequence of single elements in $F_ω$ witnessing the above mentioned result. Our proof is not independent from the one given in arXiv:1205.0929v2 [math.GR], as we essentially use some theorems from there. On the other hand our witnessing sequence is much simpler.
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On the Generic Type of the Free Group
We answer a question raised by Pillay, that is whether the infinite weight of the generic type of the free group is witnessed in $F_ω$. We also prove that the set of primitive elements in finite rank free groups is not uniformly definable. As a corollary, we observe that the generic type over the empty set is not isolated. Finally, we show that uncountable free groups are not $\aleph_1$-homogeneous.
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Homogeneity in the free group
We show that any non abelian free group $\F$ is strongly $\aleph_0$-homogeneous, i.e. that finite tuples of elements which satisfy the same first-order properties are in the same orbit under $\Aut(\F)$. We give a characterization of elements in finitely generated groups which have the same first-order properties as a primitive element of the free group. We deduce as a consequence that most hyperbolic surface groups are not $\aleph_0$-homogeneous.
math.GR↗