Simple subgroups of the real space Cremona group

TL;DR

The paper classifies finite simple non‑abelian subgroups of the real space Cremona group Cr_3(\mathbb{R}), showing that only $\mathfrak{A}_5$ and $\mathfrak{A}_6$ can occur. It splits the problem into handling large simple groups (via equivariant MMP, orbifold RR, and representation theory) and the Klein group case (PSL_2(\mathbf{F}_7)) through a detailed analysis of real and complex G_Q‑Fano 3‑folds and their automorphisms. By ruling out embeddings of $\mathrm{SL}_2(\mathbf{F}_8)$, $\mathfrak{A}_7$, and $\mathrm{PSp}_{4}(\mathbf{F}_3)$, and then excluding PSL_2(\mathbf{F}_7) in both Gorenstein and non‑Gorenstein real settings, the authors establish a dichotomy: the only possible finite simple subgroups are $\mathfrak{A}_5$ and $\mathfrak{A}_6$, with a unique real $\,\mathfrak{A}_6$‑subgroup arising from the Segre cubic. The work combines invariant theory, refined MMP techniques, and real forms of Fano 3‑folds to obtain a sharp parallel to the complex classification in the real world.

Abstract

We show that the alternating groups $\mathfrak{A}_5$ and $\mathfrak{A}_6$ are the only finite simple non-abelian subgroups of the group of birational selfmaps of the real three-dimensional projective space.

Theorems & Definitions (212)

• Example
• Conjecture
• Example
• Example : RonanSusanna
• proof : Proof of Theorem A
• proof : Proof of Theorem A$^\prime$
• Corollary I.1.2: CalabiBook
• proof
• Corollary I.1.3
• proof