Some first order sentences for free groups.

Question

I'm looking for some examples of sentences in the elementary theory of free groups. For example, let $F$ be the free group on 3 generators, how can I capture the idea of freeness in a first order sentence ? Can the universal mapping property be expressed someway ? I saw it was asked something similar but the answer involved ultrafilters (too specific) ecc... Thanks!!

Answer 1

No. Let $G$ be the free group on 3 elements $a, b, c \in G$. Consider the language of groups with three distinguished elements. By the upward Lowenheim-Skolem theorem, we can construct a group $H$, together with three elements $e, f, g \in H$, that satisfies the same first-order theory as $G, a, b, c$, but where $|H| > |G|$.

Answer 2

The paper The Tarski Theorems and Elementary Free Groups by Benjamin Fine includes this statement (p.7, section 6):

The proof of the Tarski theorems provided a complete characterizations of those finitely generated groups that have exactly the same first order theory as the non‐abelian free groups. Such groups are called elementary free groups and extend beyond the class of purely non‐abelian free groups.

Note that the free group of rank 1 is abelian, but all free groups of higher rank are non-abelian. As being abelian is a first-order property, it makes sense to consider the theory of just the non-abelian free groups.

So the answer appears to be no: if $T$ is the set of first-order statements satisfied by all finitely-generated nonabelian free groups, then there are models of $T$ that are finitely-generated but not free. Thus freeness cannot be captured by any first-order sentence, or even any collection of first-order sentences.

Fine's paper mentions some examples of these non-free models of $T$: the fundamental groups of orientable surfaces of genus greater than 1, and of non-orientable surfaces of genus greater than 3.

Fine's paper is a survey paper. The papers proving these results are much longer. The topic appears (to my eyes) to be technical and difficult.

Remarkably, all finitely-generated non-abelian free groups have the exact same first-order theory, namely $T$, and this theory is decidable. In other words, $T=\text{Th}(F)$ for any finitely-generated non-abelian free group $F$. That's the content of the so-called first Tarski Theorem (conjectured by Tarski in 1945, proved by Kharlampovich and A. Myasnikov and independently by Z. Sela after 60 years).

Since the elementary theory of any single structure is complete, it follows that $T$ is complete. It was also proved that $T$ is decidable.

You asked for some examples of first-order sentences in $\text{Th}(F)$, i.e., in $T$. For starters, all free groups are torsion-free, so we have $$\forall x(x\neq 1\to x^2\neq 1\wedge x^3\neq 1\wedge\ldots\wedge x^n\neq 1)$$ for all $n>1$. Here I'm taking the usual liberties with syntax: $x^3$, for example, would strictly be written as $x\cdot(x\cdot x)$.

For $F_2$, the free group of rank 2, we can say things like this (a somewhat random example): $$\exists x\exists y(x\neq y\wedge x^2\neq y\wedge\ldots\wedge xy^2x\neq yx)$$ The idea is, if $\phi(x,y)$ and $\psi(x,y)$ are two distinct reduced terms, then $\exists x\exists y(\phi(x,y)\neq\psi(x,y))$. If we were working in a language that allowed infinite conjunctions (like $L_{\infty\omega}$)) we could write a single sentence capturing the "freeness" of the pair $x,y$, thus: first make an enumeration of all pairs of distinct reduced words, $\{(\phi_i,\psi_i)|i\in\mathbb{N}\}$. Then write $$\exists x\exists y\Big(\bigwedge_i \phi_i(x,y)\neq\psi_i(x,y)\Big)$$ Since we're not allowed infinite conjunctions, the next best thing is an infinite sequence of axioms, putting in longer and longer initial segments of the infinite conjunction.

With an infinite disjunction one can express the fact that a pair of elements $x,y$ generate the whole group, but I don't know of any way to turn that into a collection of first-order sentences. My guess is that it can't be done.

What about $F_3$, the free group on three generators? Obviously we can play the same game, listing all pairs of distince reduced words $\phi(x,y,z),\psi(x,y,z)$. But as Fine points out (p.3), $F_2$ contains a subgroup isomorphic to $F_3$ and vice versa. So all these sentences hold for $F_2$, and all the $\phi(x,y)\neq\psi(x,y)$ sentences hold for $F_3$.

My reading of the Fine paper is that the proof of the Tarski conjectures involved a detailed analysis of such equations and inequalities. So far I've mentioned only existential and universal sentences. Are there sentences of $T$ with an $\forall\exists$ prefix that don't follow from the sentences above? I don't know.