Is $G$-disjointness of subfields of $\mathbb{C}$ a symmetric relation?

For subfields $E \subseteq F \cap L$ of $\mathbb{C}$, Broudy and Eterović (Ramanujan J. 62:781–795, 2023) define $F$ to be $G(E)$-disjoint from $L$, written $F \perp_E^G L$, where $G = \mathrm{GL}_2$ acts by Möbius transformations: every finite subset of $L$ whose elements lie in distinct $G(E)$-orbits also has its elements in distinct $G(F)$-orbits. Unlike linear disjointness and freeness of fields, which are symmetric relations, it is unknown whether $G$-disjointness is symmetric: does $F \perp_E^G L$ imply $L \perp_E^G F$? The authors prove symmetry only in the special case $E = F \cap L$ with $F$ and $L$ linearly disjoint over $E$ (their Lemma 2.2). The question, posed as open in their Remark 2.1, remains open.

Background

A standard notion in field theory is linear disjointness: extension fields $F$ and $L$ of a common base field $E$ are linearly disjoint over $E$ if every finite $E$-linearly independent tuple from one remains linearly independent over the other; a companion notion, freeness ($E$-algebraically independent tuples staying algebraically independent over the other field) is defined analogously with algebraic independence in place of linear independence. Both relations are symmetric, a classical fact recorded for instance in Lang's Algebra and recalled by Broudy and Eterović in their study of Schanuel-type conjectures.

The motivation comes from transcendence theory. Schanuel's conjecture predicts that for complex numbers $x_1,\dots,x_n$ linearly independent over $\mathbb{Q}$, the field $\mathbb{Q}(x_1,\dots,x_n,e^{x_1},\dots,e^{x_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$. Work quoted in Broudy and Eterović (their reference [CDH+09]) shows that, assuming Schanuel's conjecture, the field generated by all iterated exponentials of algebraic numbers and the field generated by all iterated logarithms are linearly disjoint over $\overline{\mathbb{Q}}$, the field of algebraic numbers. Broudy and Eterović extend this to towers built over more general base fields $F$, and, crucially, to the modular $j$-function (the holomorphic function on the upper half-plane $\mathbb{H}$ whose value at $z$ classifies the complex elliptic curve $\mathbb{C}/(\mathbb{Z} + \mathbb{Z}z)$) under a modular Schanuel conjecture. Because the $j$-function is not a group homomorphism, linear disjointness is no longer the correct notion there; the replacement, introduced earlier by the second author in his 'A Schanuel property for $j$' (Bull. Lond. Math. Soc. 50, 2018, DOI: 10.1112/blms.12135), is $G$-disjointness, defined as follows.

Let $G = \mathrm{GL}_2$ (the group of invertible $2 \times 2$ matrices) act on the Riemann sphere $\mathbb{P}^1(\mathbb{C}) = \mathbb{C} \cup \{\infty\}$ by Möbius transformations: a matrix $\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)$ with coefficients in a subfield $K \subseteq \mathbb{C}$ sends $x$ to $(ax+b)/(cx+d)$. For a subfield $K \subseteq \mathbb{C}$ and a finite set $A \subset \mathbb{C}$, write $\dim_{G(K)}(A)$ for the number of distinct $G(K)$-orbits (under this action) meeting $A$; a set is $G(K)$-independent if each of its elements lies in a different $G(K)$-orbit. Given subfields $E \subseteq F \cap L$ of $\mathbb{C}$, one says that $F$ is $G(E)$-disjoint from $L$, denoted $F \perp_E^G L$, if every finite subset of $L$ that is $G(E)$-independent is also $G(F)$-independent; equivalently, whenever two elements $\ell_1, \ell_2 \in L$ are related by some matrix in $G(F) = \mathrm{GL}_2(F)$, they are already related by a matrix in $G(E) = \mathrm{GL}_2(E)$. This notion is what lets Broudy–Eterović prove (under a modular Schanuel conjecture) that the tower of fields generated by iterated $j$-values and the tower generated by their iterated $j$-preimages are disjoint over the base in the orbit sense; the related closure-operator machinery for $j$ appears in Aslanyan–Eterović–Kirby's 'A closure operator respecting the modular $j$-function'.

The definition of $\perp_E^G$ quantifies only over tuples from one side ($L$), just as linear disjointness is classically defined; but for linear disjointness and freeness the symmetry can be recovered, while for $G$-disjointness the usual exchange/dimension arguments are unavailable, since orbit structure under Möbius transformations is not a matroid-like dimension (a matroid is the abstract independence structure with an exchange law that makes basis-and-dimension counting work for linear and algebraic independence). Broudy and Eterović note two structural facts. Their Lemma 2.3: $F \perp_E^G L$ forces $F \cap L = E$. Their Lemma 2.2: if $E = F \cap L$ and $F$ is linearly disjoint from $L$ over $E$, then both $F \perp_E^G L$ and $L \perp_E^G F$ hold — this is the only general source of symmetry currently known, and it is how the authors convert the freeness obtained from Schanuel-type inequalities into $G$-disjointness of the two $j$-towers in both directions.

Problem Statement

Let $G = \mathrm{GL}_2$ act on the Riemann sphere $\mathbb{P}^1(\mathbb{C}) = \mathbb{C} \cup \{\infty\}$ by Möbius transformations, $\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right) \cdot x = (ax+b)/(cx+d)$. For subfields $K \subseteq \mathbb{C}$, say a finite subset of $\mathbb{C}$ is $G(K)$-independent if no two of its elements lie in a common $G(K)$-orbit. For subfields $E, F, L \subseteq \mathbb{C}$ with $E \subseteq F \cap L$, define $F \perp_E^G L$ to mean: every finite subset of $L$ that is $G(E)$-independent is also $G(F)$-independent (equivalently, any two elements of $L$ related by a matrix in $\mathrm{GL}_2(F)$ are already related by a matrix in $\mathrm{GL}_2(E)$).

Question (Broudy–Eterović, Remark 2.1, Ramanujan J. 62, 2023; posed as open): is $G$-disjointness a symmetric relation? That is, is it true for all subfields $E, F, L \subseteq \mathbb{C}$ with $E \subseteq F \cap L$ that

$$F \perp_E^G L \iff L \perp_E^G F?$$

Symmetry fails exactly when there are subfields $E \subseteq F \cap L \subseteq \mathbb{C}$ with $F \perp_E^G L$ but $L \not\perp_E^G F$; by Lemmas 2.2 and 2.3 of the source, any such triple must satisfy $E = F \cap L$, with $F$ and $L$ not linearly disjoint over $E$.

References

1. Isaac A. Broudy, Sebastian Eterović, Schanuel type conjectures and disjointness, The Ramanujan Journal 62(3):781–795, 2023. DOI: 10.1007/s11139-023-00707-3, https://doi.org/10.1007/s11139-023-00707-3 (preprint arXiv:2211.09556, https://arxiv.org/abs/2211.09556)
2. Sebastian Eterović, A Schanuel property for j, Bulletin of the London Mathematical Society 50(2):293–315, 2018. DOI: 10.1112/blms.12135, https://doi.org/10.1112/blms.12135 (preprint arXiv:1701.05841, https://arxiv.org/abs/1701.05841)
3. Vahagn Aslanyan, Sebastian Eterović, Jonathan Kirby, A closure operator respecting the modular j-function, Israel Journal of Mathematics 253(1):321–357, 2023. DOI: 10.1007/s11856-022-2362-y, https://doi.org/10.1007/s11856-022-2362-y (preprint arXiv:2010.00102, https://arxiv.org/abs/2010.00102)