A characterization of Clifford parallelism by automorphisms

TL;DR

This paper refines the understanding of Clifford parallelism in real projective space by establishing that it is uniquely characterized by invariance under groups of dimension at least 4, improving previous bounds.

Contribution

The paper improves the known bound for the invariance of Clifford parallelism from 5 to 4, identifying 3 as the critical dimension for such invariance.

Findings

Clifford parallelism is uniquely characterized by invariance under groups of dimension at least 4.

The bound for the group dimension invariance is improved from 5 to 4.

Examples exist of parallelisms with group invariance of dimension 3, marking it as the critical threshold.

Abstract

Betten and Riesinger have shown that Clifford parallelism on real projective space is the only topological parallelism that is left invariant by a group of dimension at least 5. We improve the bound to 4. Examples of different parallelisms admitting a group of dimension 3 are known, so 3 is the "critical dimension".

Equations

$$T_0=\{⟨q⟩\mapsto ⟨aqb⟩|a,b\in \mathbb{C},|a|=|b|=1\}.$$

$$⟨(z,w)⟩\mapsto ⟨(azb,aw\bar{b})⟩.$$

Full text

MSC 2000: 51H10, 51A15, 51M30

Consider $\mathbb{R}^{4}$ as the quaternion skew field $\mathbb{H}$. Then the orthogonal group $\mathop{\rm SO}(4,\mathbb{R})$ may be described as the product of two commuting copies $\tilde{\Lambda},\tilde{\Phi}$ of the unitary group $\mathop{\rm U}(2,\mathbb{C})$, consisting of the maps $q\mapsto aq$ and $q\mapsto qb$, respectively, where $a$, $b$ are quaternions of norm one and multiplication is quaternion multiplication. The intersection of the two factors is of order two, containing the map $-\rm id$. Thus, passing to projective space, we get $\mathop{\rm PSO}(4,\mathbb{R})=\Lambda\times\Phi$, a direct product of two copies of $\rm SO(3,\mathbb{R})$. The left and right Clifford parallelisms are defined as the equivalence relations on the line space of $\rm PG(3,\mathbb{R})$ formed by the orbits of $\Lambda$ and $\Phi$, respectively.

The two Clifford parallelisms are equivalent under quaternion conjugation $q\to\bar{q}$; this is immediate from their definition in view of the fact that conjugation does not change the norm and is an anti-automorphism, i.e., that $\overline{pq}=\bar{q}\bar{p}$. Note that both $\Lambda$ and $\Phi$ are transitive on the point set of projective space. Since they centralize one another, each acts transitively on the parallelism defined by the other, and the group $\mathop{\rm PSO}(4,\mathbb{R})$ leaves both parallelisms invariant (we say that it consists of *automorphisms *of these parallelisms). For more information on Clifford parallels, see [1], [8], [4]. For generalizations to other dimensions, compare also [10].

The notion of a *topological parallelism *on real projective 3-space $\mathop{\rm PG}(3,\mathbb{R})$ generalizes this example. A *spread *is a set $\cal C$ of lines such that every point is incident with exactly one of them, and a topological parallelism may be defined as a compact set $\Pi$ of compact spreads such that every line belongs to exactly one of them; see, e.g., [5] for details. Many examples of different topological parallelisms have been constructed in a series of papers by Betten and Riesinger, see, e.g., [2] and references given therein.

The group $\Sigma=\mathop{\rm Aut}\Pi$ of automorphisms of a topological parallelism is a closed subgroup of $\mathop{\rm PGL}(4,\mathbb{R})$. The identity component $\Delta=\Sigma^{1}$ is compact [7], and hence (conjugate to) a subgroup of $\mathop{\rm PSO}(4,\mathbb{R})\cong\mathop{\rm SO}(3,\mathbb{R})\times\mathop{\rm SO}(3,\mathbb{R})$. Hence, in the case of the Clifford parallelism, $\Delta$ is the 6-dimensional group $\mathop{\rm PSO}(4,\mathbb{R})$ that we used to define the parallelism. Betten and Riesinger [5] proved that no other topological parallelism has a group of dimension $\dim\Sigma\geq 5$. Examples of parallelisms with 1-, 2- or 3-dimensional automorphism groups are known, see [6], [2], [3]. Here we consider parallelisms with a 4-dimensional group.

*Proof. *Recall that a topological parallelism $\Pi$ is homeomorphic to the real projective plane in the Hausdorff topology on the space of compact sets of lines, and that every equivalence class is a compact spread and homeomorphic to the 2-sphere, compare [5].

It suffices to consider the identity component $\Delta=\Sigma^{1}$, and we may assume that $\dim\Delta=4$. Further, up to equivalence, we may assume that $\Delta=\Lambda\cdot\Gamma$, where $\Gamma\leq\Phi$ is the subgroup defined by restricting the factor $b$ to be a complex number (here we use the notation of the introduction.) Since $\Lambda$ does not have any one-dimensional coset spaces, we know that $\Lambda$ acts on $\Pi$ either transitively or trivially. If it acts trivially, then the classes of $\Pi$ are the $\Lambda$-orbits of lines, and we have the Clifford parallelism. Observe here that every $\Lambda$-orbit is contained in a single class, and both the orbit and the class are 2-spheres.

In what follows, assume therefore that $\Lambda$ acts transitively on $\Pi$. There is only one possibility for this action namely, the standard transitive action of $\mathop{\rm SO}(3,\mathbb{R})$ on the real projective plane. Every 2-dimensional subgroup of $\Delta$ contains $\Gamma$. Hence, there is no effective action of $\Delta$ on the projective plane $\Pi$, and the kernel can only be $\Gamma$ since the only other proper normal subgroup is $\Lambda$, which is transitive. If ${\cal C}\in\Pi$ is any equivalence class, then the stabilizer $\Lambda_{\cal C}$ is a product of a 1-torus and a group of order two. Hence $\Delta_{\cal C}$ contains a 2-torus $T$. There is only one conjugacy class of 2-tori in $\Delta$, represented by the group

[TABLE]

Here, $\langle q\rangle$ denotes the one-dimensional real vector space spanned by $q$. We may assume that $T=T_{0}$. Write quaternions as pairs of complex numbers with multiplication $(x,y)(u,v)=(xu-\bar{v}y,vx+y\bar{u})$, see [9], 11.1. Then complex numbers become pairs $(a,0)$, and the elements of $T$ are now given by

[TABLE]

The kernel of ineffectivity of $T$ on the 2-sphere $\cal C$ must be a 1-torus $\Xi$, and the elements of the kernel other than the identity cannot have eigenvalue 1 — otherwise they would be axial collineations of the translation plane defined by the spread $\cal C$ and would act non-trivially on $\cal C$. There are only two subgroups of the 2-torus satisfying these conditions, given by $b=1$ and by $a=1$, respectively. In other words, the kernel $\Xi$ is a subgroup either of $\Lambda$ or of $\Phi$. In both cases, $\cal C$ consists of the fixed lines of $\Xi$. If $\Xi\leq\Phi$, then $\Lambda$ permutes these lines, contrary to the transitivity of $\Lambda$ on $\Pi$. If $\Xi\leq\Lambda$, then $\Phi$ permutes the fixed lines, which means that $\cal C$ is a $\Phi$-orbit. Now $\Lambda$ is transitive both on $\Pi$ and on the set of $\Phi$-orbits, hence $\Pi$ equals the Clifford parallelism formed by the $\Phi$-orbits.