Foliations and Diffeomorphism Groups — Question 62

Definitions and mathematical statement

Write $S^1={\mathbb{R}}/{\mathbb{Z}}$, and let $D=\mathrm{Diff}^{\omega}(S^1)$ denote the source's ambient group of real-analytic circle diffeomorphisms; its orientation convention is unspecified. The source names subgroups \[ A_k=\mathrm{PSL}(2)^{(k)},\qquad A_l=\mathrm{PSL}(2)^{(l)} \] of $D$, but the supplied record does not define this notation, the permitted parameters $k,l$, or the subgroup embeddings. Let $\langle A_k,A_l\rangle$ denote the smallest subgroup of $D$ containing both subgroups.

To interpret the amalgamated product, one must specify injective homomorphisms $\iota_k:H\to A_k$ and $\iota_l:H\to A_l$, where \[ H=\mathrm{SO}(2)=\{R\in M_2({\mathbb{R}}):R^{\mathsf T}R=I_2,\ \det R=1\}. \] For specified embeddings, define \[ A_k*_{H}A_l =(A_k*A_l)\big/\left\langle\!\left\langle \iota_k(h)\iota_l(h)^{-1}:h\in H \right\rangle\!\right\rangle, \] where $A_k*A_l$ is the free product and the double brackets denote normal closure.

The supplied question, conditional on recovering these missing definitions and parameter restrictions, is whether \[ \langle A_k,A_l\rangle\cong A_k*_{H}A_l \] for the intended values of $k,l$. No quantifier over these parameters can be recovered from the supplied record. If the two embeddings of $H$ agree after inclusion into $D$, there is a canonical surjective homomorphism from this amalgamated product onto $\langle A_k,A_l\rangle$. Whether the source specifically asks that this canonical map be an isomorphism, rather than merely asking for an abstract group isomorphism, is also unresolved.

Short English statement

The question asks whether two specified subgroups of the real-analytic circle diffeomorphism group generate a group isomorphic to their amalgamated free product over $\mathrm{SO}(2)$. The supplied record omits the definitions of these subgroups, their common subgroup embeddings, and the restrictions on $k,l$, so this remains a conditional statement requiring clarification.

Sources

• Local UnsolvedMath registry: permanent ID $800121$, source problem code catalogue entry, titled Foliations and Diffeomorphism Groups --- Question 62. The supplied record attributes the problem to Nariman and Yazdi, with proposed year 2025.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 62. Supplied source URL: https://arxiv.org/abs/2503.18274. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not an independent source visit or verification.
• The local record labels the problem open. This is a supplied registry classification, not an independently verified assessment of the literature.