Foliations and Diffeomorphism Groups — Question 61

Definitions and mathematical statement

Identify the circle with $S^1={\mathbb{R}}/{\mathbb{Z}}$. A rotation is a map $r_t(x)=x+t$ for $t\in{\mathbb{R}}/{\mathbb{Z}}$. The source specifies a rotation subgroup $R_1\leq G_1$ and, for each positive integer $n$, a subgroup \[ R_n=p_n^{-1}(R_1)\leq G_n. \] However, the supplied record does not define the groups $G_n$ or the maps $p_n$. It asserts that every $R_n$ is isomorphic as a topological group to ${\mathbb{R}}/{\mathbb{Z}}$ and instructs that these isomorphisms be fixed; the relevant group topologies and the particular isomorphisms are not supplied.

To express the stated construction conditionally, let $\alpha_i:{\mathbb{R}}/{\mathbb{Z}}\longrightarrow R_i$ denote the intended fixed isomorphisms. With the embeddings into $G_i$ obtained by composing $\alpha_i$ with subgroup inclusion, form the abstract amalgamated free products \[ G^{(n)}=G_1*_{{\mathbb{R}}/{\mathbb{Z}}}G_2*_{{\mathbb{R}}/{\mathbb{Z}}}\cdots*_{{\mathbb{R}}/{\mathbb{Z}}}G_n, \qquad n\geq 1, \] where the images of each $t\in{\mathbb{R}}/{\mathbb{Z}}$ in all factors are identified. Using the resulting natural inclusions, set \[ G=\varinjlim_{n\geq 1}G^{(n)}. \] The source also displays an initial term $G^{(0)}$ in this direct system, but does not define it.

The source asks whether $G$ is dense in $\mathrm{Diff}^{\omega}(S^1)_0$, the intended identity component of the group of real analytic circle diffeomorphisms. A precise density assertion additionally requires an ambient group topology $\tau$ and the intended injective homomorphism \[ \iota:G\hookrightarrow\mathrm{Diff}^{\omega}(S^1)_0. \] With these data specified, the requested assertion is \[ \overline{\iota(G)}^{\,\tau}=\mathrm{Diff}^{\omega}(S^1)_0. \] Neither $\tau$ nor $\iota$, nor the definitions of $G_n$ and $p_n$ needed to construct them, is present in the supplied record. Thus this formula records the intended question conditionally; the supplied evidence does not determine a complete unconditional mathematical statement.

Short English statement

Question 61 asks whether the direct limit of certain groups amalgamated along their circle rotation subgroups is dense in the identity component of the real analytic circle diffeomorphism group. The factor groups, covering or other maps, ambient realization, and topology must be recovered before this density question is fully specified. The local registry labels the problem open; that label is supplied metadata, not an independently verified assessment.

Sources

• UnsolvedMath local registry, permanent ID $800120$, source problem code catalogue entry, title "Foliations and Diffeomorphism Groups --- Question 61." The supplied record attributes the problem to Nariman and Yazdi, with proposed year $2025$, and labels its status as open.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups ($2025$), Question 61. Supplied source URL: https://arxiv.org/abs/2503.18274. This source attribution and link are provided by the record and have not been independently checked here.
• Supplied provenance: accessed 2026-10-16; extraction method arxiv_latex. This access date describes supplied provenance and does not claim independent source consultation or literature verification.