Foliations and Diffeomorphism Groups — Question 73

Definitions and mathematical statement

A group $G$ is called simple if $G\neq\{e\}$ and every normal subgroup $N\trianglelefteq G$ satisfies $N=\{e\}$ or $N=G$.

The question supplied as Question 73 is: is the group $\Gamma_H$ simple? Under the preceding convention, the target is to determine whether \[ \Gamma_H\neq\{e\} \quad\text{and}\quad \forall N\trianglelefteq\Gamma_H,\qquad N=\{e\}\ \text{or}\ N=\Gamma_H. \] The supplied record does not define $H$, $\Gamma_H$, or its group operation, so this target remains conditional on recovering the intended group from the source context.

The preceding source text refers to an unidentified assertion: if that assertion is false, the authors say that the whole group $\Gamma_H$ can be described fairly clearly, while expressing doubt about that possibility; if it is true, they say that $\Gamma_H$ has a $\mathbb Z_{\geq 0}$-valued pseudonorm. The antecedent, proposed description, and pseudonorm are not supplied. These contextual remarks therefore cannot be made into precise additional hypotheses or conclusions.

Short English statement

Question 73 asks whether the intended group $\Gamma_H$ is nontrivial and has no proper nontrivial normal subgroup. Its definition and the context of the preceding conditional remarks are missing. The registry labels the question open; this is supplied source status, not an independently verified assessment.

Sources

• Local UnsolvedMath registry, permanent ID $800132$, source problem code catalogue entry, titled "Foliations and Diffeomorphism Groups --- Question 73." The supplied record attributes the problem to Nariman and Yazdi, with proposed year 2025, and labels it open.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 73. 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 a claim of independent source access or verification.