Ampleness in the Farey graph
We show that the first-order theory of the Farey graph is $1$-ample but not $2$-ample. Along the way, we prove that it weakly eliminates imaginaries and observe that forking is trivial.
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arXiv subjects
Rizos Sklinos
Publications and source records attributed to Rizos Sklinos.
Model theory of term algebras revisited
Building on work of Maltsev on locally free algebras in finite purely functional languages, we revisit the model theory of (absolutely free) term algebras and their completions. Maltsev's analysis yields a natural axiomatization together with quantifier elimination to positive Boolean combinations of special formulas, and shows that the complete extensions are parametrized exactly by the number $k\in\{0,1,\dots,ω\}$ of indecomposable elements; for $1\le k\leω$ the standard model is the free term algebra on $k$ generators. We give a new, quantifier-elimination--free proof of completeness using Ehrenfeucht--Fraïssé games, and we establish several further structural properties of the standard models and theories. In particular, for $1\le k\leω$ we prove first-order rigidity and atomicity of the standard model. For every $0\le k\leω$ we show that the corresponding theory does not have the finite cover property and weakly eliminates imaginaries. We also provide new proofs of stability-theoretic features previously obtained by Belegradek: the theories are stable but not superstable, normal (hence $1$-based), and have trivial forking; consequently, no infinite group is interpretable in any model. Finally, we analyze model completeness and show that $T_0$ is the model companion of the theory of locally free algebras, while the theories with $k\ge 1$ are not model complete.
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First-order sentences in random groups III
We show that a first-order sentence is almost surely true in a random group of density d<1/2 if and only if it is true in a non-abelian free group.
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Fields interpretable in the free group
We prove that no infinite field is interpretable in the first-order theory of nonabelian free groups. We also obtain a characterization of Abelian groups interpretable in this theory.
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Profinite rigidity of affine Coxeter groups
We prove that the irreducible affine Coxeter groups are first-order rigid and deduce from this that they are profinitely rigid in the absolute sense. We then show that the first-order theory of any irreducible affine Coxeter group does not have a prime model. Finally, we prove that universal Coxeter groups of finite rank are homogeneous, and that the same applies to every hyperbolic (in the sense of Gromov) one-ended right-angled Coxeter group.
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Nonequational Stable Groups
We introduce a combinatorial criterion for verifying whether a formula is not the conjunction of an equation and a co-equation. Using this, we give a proof for the nonequationality of the free group. Furthermore, we generalize the latter result to the first-order theory of any free product of groups of the form $G*\mathbb{F}_ω$.
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First-order sentences in random groups II: $\forall\exists$-sentences
We prove that a random group, in Gromov's density model with $d < 1/16$ satisfies with overwhelming probability a universal-existential first-order sentence $σ$ (in the language of groups) if and only if $σ$ is true in a nonabelian free group.
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Towers and the first-order theory of hyperbolic groups
This paper is devoted to the first-order theory of torsion-free hyperbolic groups. One of its purposes is to review some results and to provide precise and correct statements and definitions, as well as some proofs and new results. A key concept is that of a tower (Sela) or NTQ system (Kharlampovich-Myasnikov). We discuss them thoroughly. We state and prove a new general theorem which unifies several results in the literature: elementarily equivalent torsion-free hyperbolic groups have isomorphic cores (Sela); if $H$ is elementarily embedded in a torsion-free hyperbolic group $G$, then $G$ is a tower over $H$ relative to $H$ (Perin); free groups (Perin-Sklinos, Ould-Houcine), and more generally free products of prototypes and free groups, are homogeneous. The converse to Sela and Perin's results just mentioned is true. This follows from the solution to Tarski's problem on elementary equivalence of free groups, due independently to Sela and Kharlampovich-Myasnikov, which we treat as a black box throughout the paper. We present many examples and counterexamples, and we prove some new model-theoretic results. We characterize prime models among torsion-free hyperbolic groups, and minimal models among elementarily free groups. Using Fraïssé's method, we associate to every torsion-free hyperbolic group $H$ a unique homogeneous countable group $\mathcal{M}$ in which any hyperbolic group $H'$ elementarily equivalent to $H$ has an elementary embedding. In an appendix we give a complete proof of the fact, due to Sela, that towers over a torsion-free hyperbolic group $H$ are $H$-limit groups.
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Fraïssé limits of limit groups
We modify the notion of a Fraïssé class and show that various interesting classes of groups, notably the class of nonabelian limit groups and the class of finitely generated elementary free groups, admit Fraïssé limits. Furthermore, we rediscover Lyndon's $\Z[t]$-exponential completions of countable torsion-free CSA groups, as Fraïssé limits with respect to extensions of centralizers. Dedicated to the memory of Charles Sims.
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Some lower bounds on Shelah rank in the free group
We give some lower bounds on the Shelah rank of varieties in the free group whose coordinate groups are hyperbolic towers.
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Forking and JSJ decompositions in the free group II
We give a complete characterization of the forking independence relation over any set of parameters in the free groups of finite rank, in terms of the $JSJ$ decompositions relative to those parameters.
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The free group does not have the finite cover property
We prove that the first order theory of nonabelian free groups eliminates the "there exists infinitely many" quantifier (in eq). Equivalently, since the theory of nonabelian free groups is stable, it does not have the finite cover property. We also extend our results to torsion-free hyperbolic groups under some conditions.
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Elementary equivalence vs commensurability for hyperbolic groups
We study to what extent torsion-free (Gromov)-hyperbolic groups are elementarily equivalent to their finite index subgroups. In particular, we prove that a hyperbolic limit group either is a free product of cyclic groups and surface groups, or admits infinitely many subgroups of finite index which are pairwise non elementarily equivalent.
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On Superstable Expansions of Free Abelian Groups
We prove that $(\Z,+,0)$ has no proper superstable expansions of finite Lascar rank. Nevertheless, this structure equipped with a predicate defining powers of a given natural number is superstable of Lascar rank $ω$. Additionally, our methods yield other superstable expansions such as $(\Z,+,0)$ equipped with the set of factorial elements.
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The free group has the dimensional order property
We prove that the common theory of nonabelian free groups has the dimensional order property, or DOP, implying, for example, that there is no reasonable structure theorem for $\aleph_1$-saturated models of this theory.
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Fields definable in the free group
We prove that no infinite field is definable in the theory of the free group
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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 $X(\mathbb F_n)=X(\mathbb F_ω)$.
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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.
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