Semi-simple groups of compact 16-dimensional planes

TL;DR

This paper investigates the structure of automorphism groups of compact 16-dimensional projective planes, establishing conditions under which these groups are Lie groups, especially focusing on semi-simple groups and their fixed elements.

Contribution

It provides new bounds on the dimensions of semi-simple automorphism groups that guarantee they are Lie groups, refining previous results based on fixed elements and group structure.

Findings

Semi-simple groups fixing exactly one line are Lie groups if dimension ≥ 11.

Semi-simple groups with dimension ≥ 25 are Lie groups.

Sharper bounds depend on the fixed elements and group structure.

Abstract

The automorphism group $\Sigma$ of a compact topological projective plane with a $16$-dimensional point space is a locally compact group. If the dimension of $\Sigma$ is at least $29$, then $\Sigma$ is known to be a Lie group. For the connected component $\Delta$ of $\Sigma$ the condition $\dim\Delta{\,\ge\,}27$ suffices. Depending on the structure of $\Delta$ and the configuration of the fixed elements of $\Delta$ sharper bounds are obtained here. Example: If $\Delta$ is semi-simple and fixes exactly one line and possibly several points on this line, then $\Delta$ is a Lie group if $\dim\Delta{\,\ge\,}11$. Any semi-simple group $\Delta$ which satisfies $\dim\Delta{\,\ge\,}25$ is a Lie group.

Full text

1. Introduction

A compact connected topological projective plane has a point space $P$ of (covering) dimension $\dim P{\,=\,}d{\,|\,}16$, provided $d{\,<\,}\infty$. For properties and resušts concerning $16$-dimensional planes see the treatise [19] and a more recent update [17]. With the compact-open topology (the topology of uniform convergence), the automorphism group ${\Sigma}{\,=\,}\mathop{{\rm Aut}}{\cal P}$ of a compact $16$-dimensional plane ${\cal P}{\,=\,}(P,{{L}})$ is a locally compact transformation group of $P$ as well as of the line space ${{L}}$, see [19] 44.3. In the case of the classical plane ${\cal O}$ over the octonion algebra ${O}$ the group ${\Sigma}$ is a simple Lie group of dimension $78$  ([19] 18.19). If ${\cal P}$ is not classical, then $\dim\mathop{{\rm Aut}}{\cal P}{\,\leq\,}40$  ([19] 87.7). All planes with $\dim{\Sigma}{\,\geq\,}35$ are known explicitly, provided ${\Sigma}$ does not fix exactly one incident point-line pair ([17]). If $\dim{\Sigma}{\,\geq\,}27$, then the connected component ${\Delta}$ of ${\Sigma}$ is a Lie group  ([10]). More detailed results can be found in [17]. For semi-simple groups sharper bounds will be obtained here.

2. Preliminaries

In the following, ${\cal P}{\,=\,}(P,{{L}})$ will always mean a compact $16$-dimensionl projective plane if not stated otherwise; ${\Delta}$ denotes a connected closed subgroup of $\mathop{{\rm Aut}}{\cal P}$.

Notation is more or less standard and agrees with that in the book [19]. A flag is an incident point-line pair, a double flag consists of two points, say $u,v$, their join $uv$, and a second line in the pencil ${{L}}_{v}$. A $2$-dimensional plane will also be called flat. Homeomorphism is indicated by $\approx$. The topological dimension dim of a set $M{\subseteq}P$ or a group, the covering dimension, coincides with the inductive dimension, see [19] § 92, 93.5. As customary, ${\rm Cs}_{\hskip 1.0pt{\Delta}}{\hskip 0.5pt{\Gamma}}$ or just ${\rm Cs}_{\hskip 1.0pt}{\hskip 0.5pt{\Gamma}}$ is the centralizer of ${\Gamma}$ in ${\Delta}$. Distinguish between the commutator subgroup ${\Gamma}^{\prime}$ and the connected component ${\Gamma}^{1}$ of the topological group ${\Gamma}$. Local isomorphy of groups is symbolized by $\circeq$. The coset space ${\Delta}/{\Gamma}{\,=\,}\{{\Gamma}\delta\mid\delta{\,\in\,}{\Delta}\}$ has the dimension ${\Delta}{:}{\Gamma}{\hskip 1.0pt=\hskip 1.0pt}\dim{\Delta}{-}\dim{\Gamma}$. If ${\Gamma}$ is any subset of ${\Delta}$, then ${\cal F}_{\Gamma}$ denotes the configuration of all fixed elements (points and lines) of ${\Delta}$. The group ${\Delta}_{[c,A]}$ consists of the axial collineations in ${\Delta}$ with axis $A$ and center $c$. A collineation group ${\Gamma}$ is said to be straight if each orbit $x^{\Gamma}$ is contained in some line. In this case a theorem of Baer [1] asserts that either ${\Gamma}{=\,}{\Gamma}_{\hskip-2.0pt[c,A]}$ is a group of axial collineations or ${\cal F}_{\Gamma}$ is a Baer subplane  (cf. 2.2 below).

2.1 Topology. Each line $L$ of ${\cal P}$ is homotopy equivalent $(\simeq)$ to an $8$-sphere ${S}_{8}$, see [19] 54.11. So far, no example with $L{\,\not\approx\,}{S}_{8}$ has been found. The Lefschetz fixed point theorem implies that each homeomorphism $\varphi{\,:\,}P{\,\to\,}P$ has a fixed point. By duality, each automorphism of ${\cal P}$ fixes a point and a line, see [19] 55. 19, 45.

2.2 Baer subplanes. Each $8$-dimensional $closed$ subplane ${\cal B}$ of ${\cal P}$ is a Baer subplane, i.e., each point of ${\cal P}$ is incident with a line of ${\cal B}$ (and dually, each line of ${\cal P}$ contains a point of ${\cal B}$), see [14] § 3 or [19] 55.5 for details. By a theorem of Löwen [8] Th. B, any two Baer subplanes of a compact connected projective plane of finite dimension have a point and a line in common. $\langle{\cal M}\rangle$ will denote the smallest closed subplane of ${\cal P}$ containing the set ${\cal M}$ of points and lines. We write {\cal B}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P} if ${\cal B}$ is a Baer subplane and {\cal B}{\,\leq\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 2.0pt\hbox{\scriptscriptstyle\bullet}\mskip-3.0mu\,}{\cal P} if $\,8|\dim{\cal B}$.

2.3 Groups. Any connected subgroup ${\Delta}$ of ${\Sigma}{\,=\,}\mathop{{\rm Aut}}{\cal P}$ with $\dim{\Delta}{\,\geq\,}27$ is a Lie group, see [10]. For $\dim{\Delta}{\,\geq\,}35$, the result has been proved in [19] 87.1. In particular, ${\Delta}$ is then either semi-simple, or ${\Delta}$ has a central torus subgroup or a minimal normal vector subgroup ${\Xi}{\,\cong\,}{R}^{k}$, see [19] 94.26.

2.4 Involutions. Each involution $\iota$ is either a reflection or it is planar ({\cal F}_{\hskip-1.5pt\iota}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}), see [19] 55.29. Commuting involutions with the same fixed point set are identical ([19] 55.32). Let ${Z}_{2}^{\hskip 3.0ptr}{\,\cong\,}{\Phi}{\,\leq}\mathop{{\rm Aut}}{\cal P}$ and $\dim P{\,=\,}2^{m}$. Then $r{\,\leq\,}m{+}1$; if ${\Phi}$ is generated by reflections, then $r{\,\leq\,}2$, see [19] 55.34(c,b). If $r{\,\geq\,}m\ ({\,>\,}2)$, then ${\Phi}$ contains a reflection (a planar involution), cf. [19] 55.34(d,b). Any torus group in $\mathop{{\rm Aut}}{\cal P}$ has dimension at most $m$,  see [19] 55.37, for $m{\,=\,}4$ in particular, $\rm rk\,{\Delta}{\,\leq\,}4$, see also [4]. The orthogonal group ${\rm SO}_{5}{{R}}$ cannot act non-trivially on ${\cal P}\,$ ([19] 55.40).

2.5 Stiffness refers to the fact that the dimension of the stabilizer ${\Lambda}$ of a (non-degenerate) quadrangle ${{e}}$ cannot be very large. If ${\cal P}$ is the classical plane, then ${\Lambda}$ is the $14$-dimensional exceptional compact simple Lie group ${\rm G}_{2}{\,\cong\,}\mathop{{\rm Aut}}{O}$ for each choice of ${{e}}$, cf. [19] 11. 30–35. In the general case, let ${\Lambda}$ denote the connected component of ${\Delta}_{{e}}$. Then the following holds:

(a) ${\Lambda}{\,\cong\,}{\rm G}_{2}$* and ${\cal F}_{\Lambda}$ is flat, or $\dim{\Lambda}{\,<\,}14$* ([19] 83. 23, 24),

(â) ${\Lambda}{\,\cong\,}{\rm G}_{2}$* or $\dim{\Lambda}{\,\leq\,}11$* ([2] 4.1),

(b) if ${\cal F}_{\Lambda}$ is a Baer subplane, then ${\Delta}_{{e}}$ is compact and $\dim{\Lambda}{\,\leq\,}7$   ([19] 83.6),

(b̂) if {\cal F}_{\Lambda}{\;\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P} and ${\Lambda}$ is a Lie group, then ${\Lambda}{\,\cong\,}{\rm Spin}_{3}{{R}}$ or $\dim{\Lambda}{\,\leq\,}1$  ([19] 83.22),

(c) if there exists a subplane ${\cal B}$ such that {\cal F}_{\Lambda}{\;\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal B}{\;\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}, then ${\Lambda}$ is compact

([12] 2.2, [19] 83.9),

(ĉ) if ${\Lambda}$ contains a pair of commuting invo lutions, then ${\Delta}_{{e}}$ is compact  ([19] 83.10),

(d) if ${\Lambda}$ is compact or semi-simple or if ${\cal F}_{\Lambda}$ is connected, then ${\Lambda}{\,\cong\,}{\rm G}_{2}$ or $\dim{\Lambda}{\,\leq\,}10$

([5] XI.9.8, [12] 4.1, [2] 3.5),

(e) if ${\Lambda}$ is compact, or if $\dim{\cal F}_{\Lambda}{\,=\,}4$, or if ${\Lambda}$ is a Lie group and ${\cal F}_{\Lambda}$ is connected, then

${\Lambda}{\,\cong\,}{\rm G}_{2}$* or ${\Lambda}{\,\cong\,}{\rm SU}_{3}{{C}}$ or $\dim{\Lambda}{\,<\,}8$*  ([12] 2.1; [13], [2] 3.5; [3]),

(ê) if ${\Lambda}$ is a compact Lie group, then ${\Lambda}{\hskip 1.0pt\cong\,}{\rm G}_{2}$, ${\rm SU}_{3}{{C}},\;{\rm SO}_{4}{{R}}$, or $\dim{\Lambda}{\,\leq\,}4$  ([12] 2.1).

2.6 Groups with open orbits. Let $X$ denote the point space $P$ of ${\cal P}$ or a line $L$ of ${\cal P}$. If $U{\,\subseteq\,}X$ is a ${\Delta}$-orbit which is open in $X$ or, equivalently, satisfies $\dim U{\,=\,}\dim X$, then $X$ is a manifold and ${\Delta}$ induces a Lie group on $U$. If $X{\,=\,}L$, then it follows that all lines are manifolds homeomorphic to ${S}_{8}$. If $X{\,=\,}P$, then ${\Sigma}$ is a Lie group.  ([19] 53.2).

Remark. The result [19] 53.2 depends on a theorem of Szenthe for which a correct proof has been given in the meantime by Hofmann and Kramer [6] 5.5 Corollary.

2.7 Lemma. If ${\cal F}_{\hskip-2.0pt{\Delta}}{\,=\,}\emptyset$, then ${\Delta}$ is semi-simple with trivial center, or ${\Delta}$ induces a centerfree semi-simple group on some connected proper closed subplane. In particular, ${\cal F}_{\hskip-2.0pt{\Delta}}{\,\neq\,}\emptyset$ whenever ${\Delta}$ is commutative.

Proof. As a flat plane does not have a proper closed subplane, the Lemma is true for flat planes by [19] 33.1, and each commutative connected subgroup fixes a point or a line. Suppose the Lemma has been shown for all planes of smaller dimension thane the plane on which ${\Delta}$ acts. If ${\Delta}$ contains a central element $\zeta{\,\neq\,}1\kern-3.0pt{\rm l}$ or if ${\Delta}$ is not semi-simple and hence has a commutative normal subgroup ${\Xi}$, then there is a point $p$ such that $p^{\zeta}{\,=\,}p$ or $p^{\Xi}{\,=\,}p$ (up to duality). By assuption, $p^{\Delta}{\,\neq\,}p$ and $p^{\Delta}$ is not contained in a line. Obviously $\zeta|_{p^{\Delta}}{\,=\,}1\kern-3.0pt{\rm l}$, and normality of ${\Xi}$ implies ${\Xi}|_{p^{\Delta}}{\,=\,}1\kern-3.0pt{\rm l}$. Therefore ${\cal E}{\,=\,}\langle p^{\Delta}\rangle$ is a proper connected subplane and $\overline{\Delta}{\,=\,}{\Delta}|_{\cal E}$ satisfies ${\cal F}_{\hskip-1.0pt\overline{\Delta}}{\,=\,}\emptyset$. Now the Lemma can be applied to $\overline{\Delta}$. $\square$

Note that the Lemma holds for all compact connected planes of finite dimension.

2.8 Addendum. If ${\cal F}_{\Delta}{\,=\,}\emptyset$ and if there is no ${\Delta}$-invariant proper subplane, then each involution in a proper simple factor of ${\Delta}$ is planar. This is true for each compact plane of dimension $d\,|\,16$.

Proof. By 2.7 the group ${\Delta}$ is semi-simple with trivial center. Suppose that ${\Delta}{\,=\,}{\Gamma}{\times}{\Psi}$ is a product of two proper factors. If ${\Gamma}$ contains a reflection $\gamma{\,\in\,}{\Gamma}_{\hskip-2.0pt[c,A]}$, then $c^{\Psi}{=\,}c{\,\neq\,}c^{\Gamma}$ and ${\Psi}|_{c^{\Gamma}}{\,=\,}1\kern-3.0pt{\rm l}$. A line $L$ containing $c^{\Gamma}$ would be fixed by ${\Gamma}{\Psi}{\,=\,}{\Delta}$. Therefore $\langle c^{\Gamma}\rangle{\,=\,}{\cal E}{\,\leq\,}{\cal F}_{\Psi}$, and ${\cal E}^{\Gamma}{=\,}c^{\Psi}{~=\,}c^{\Delta}$ contrary to the assumtion. Hence each involution in a proper factor is planar.

Remark. The statement is not true if there is an invariant subplane: in the classical quaternion plane ${\cal H}$, the group ${\Delta}$ consisting of all collineations which map the real subplane onto itself is the direct product of the automorphism group ${\Gamma}{\,\cong\,}{\rm SO}_{3}{{R}}$ of the quaternions and the group ${P}{\,\cong\,}{\rm SL}_{3}{{R}}$ of real collineations. The involutions in ${P}$ are reflections of ${\cal H}$.

2.9 Fact. If ${\cal E}^{\Delta}{\,=\,}{\cal E}{\,<\,}{\cal P}$, if ${\Delta}|_{\cal E}{\,=\,}{\Delta}/{K}$ is a semi-simple Lie group, and if ${\Lambda}{\,=\,}{K}^{1}$, then ${\Delta}$ contains a covering group ${\Gamma}$ of ${\Delta}/{K}$ such that ${\Gamma}|_{\cal E}{\,\cong\,}{\Delta}/{K}$ and ${\Delta}{\,=\,}{\Gamma}{\Lambda}$. In particular, ${\Delta}$ is a Lie group whenever ${\Lambda}$ is.

Proof. The first claim is a special case of [19] 94.27; the second follows from $\dim{\Gamma}{\,=\,}{\Delta}{:}{K}=\dim{\Delta}{\,-\,}\dim{\Lambda}$ because ${\Delta}$ is connected. $\square$

3. No fixed elements

The following slightly improves results in [17]:

3.1. If ${\Delta}$ is semi-simple, ${\cal F}_{\mskip-3.0mu{\Delta}}{\,=\,}\emptyset$ and if $\dim{\Delta}{\,\geq\,}20$, then ${\Delta}$ is a Lie group. For $\dim{\Delta}{\,\geq\,}21$ the assertion has been proved by different methods in [17] 3.0.

Proof. Suppose that ${\Delta}$ is not a Lie group.

(a) By the Lemma above, ${\Delta}$ is a direct product of simple groups with trivial center, or it induces such a group on some ${\Delta}$-invariant connected subplane ${\cal E}$. Choose ${\cal E}$ in such a way that the dimension of ${\cal E}$ is minimal. In the case ${\cal E}{\,=\,}{\cal P}$ the center of ${\Delta}$ is trivial and ${\Delta}$ is a Lie group. If ${\cal E}$ is flat, then ${\Delta}|_{\cal E}{\,=\,}{\Delta}/{K}$ is strictly simple of dimension $3$ or ${\Delta}|_{\cal E}{\,\cong\,}{\rm SL}_{3}{{R}}$  (cf. [19] 33.6,7); by Stiffness, ${\Lambda}{\,=\,}{K}^{1}{\,\cong\,}{\rm G}_{2}$ is a Lie group and 2.9 applies, or $\dim{K}{\,\leq\,}10$ and $\dim{\Delta}{\,\leq\,}18$.

(b) Next, assume that {\cal E}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. All almost simple groups of dimension at least $10$ acting on ${\cal E}$ have been determined by Stroppel [20]; for those satisfying ${\cal F}_{\Delta}{\,=\,}\emptyset$ see also [16] 2.1. Suppose first that ${\Delta}^{\mskip-3.0mu*}{=\,}{\Delta}|_{\cal E}{\,=\,}{\Delta}/{K}{\,=\,}{\Gamma}{\times}{\Psi}$ is not simple, and that ${\Gamma}$ is a simple factor. By the addendum above, each involution in ${\Gamma}$ is planar. As the center of ${\Gamma}$ is trivial, there are distinct involutions $\alpha,\beta{\,\in\,}{\Gamma}$. Because of Löwen’s theorem (see 2.2 above) there is some point $c{\,\in}{\cal F}_{\alpha,\beta}$, and the point space $F_{\alpha,\beta}$ of ${\cal F}_{\alpha,\beta}$ has dimension at most $2$. If $c^{\Psi}{=\,}c$, then $\langle c^{\Delta}\rangle{\,=\,}\langle c^{\Gamma}\rangle{\,\leq\,}{\cal F}_{\Psi}{\,<\,}{\cal E}$ contrary to the assumption. Similarly, if $c^{\Psi}{\neq\,}c$ is contained in a line $L$ of ${\cal E}$, then $L{\,=\,}L^{\Psi}$ induces a line of ${\cal F}_{\alpha}$ and of ${\cal F}_{\beta}$, a contradiction dual to the previous one. Hence $\langle c^{\Psi}\rangle$ is a flat subplane, and ${\Psi}$ is simple of dimension $3$ or $8$. Moreover, ${\Lambda}{\,=\,}{K}^{1}$ is a compact normal, hence semi-simple subgroup, in fact a Lie group (see [19] 93.11). Stiffness implies $\dim{\Lambda}{\,\leq\,}6$. Interchanging the rôles of ${\Gamma}$ and ${\Psi}$, it follows that both ${\Gamma}$ and ${\Psi}$ have dimension $8$, in fact, ${\Gamma}{\,\cong\,}{\Psi}{\,\cong\,}{\rm SL}_{3}{{R}}$, and ${\Delta}$ is a Lie group by 2.9.

(c) According to [16] 2.1,2, the induced group ${\Delta}^{\mskip-3.0mu*}{=\,}{\Delta}|_{\cal E}$ is a motion group of the classical quaternion plane or ${\Delta}^{\mskip-3.0mu*}{\,\cong\,}{\rm SL}_{3}{{C}}$ acts on a Hughes plane and fixes a proper subplane of ${\cal E}$. We will have to deal with this group in the case $\dim{\cal E}{\,=\,}4$. If ${\Delta}^{\mskip-3.0mu*}{\,\cong\,}{\rm PU}_{3}{(}{H},r)$, then $\dim{\Delta}^{\mskip-3.0mu*}{=\,}21$. Recall that ${\Delta}$ is semi-simple. The kernel ${K}$ is compact and ${\Lambda}{\,=\,}{K}^{1}$ is semi-simple; ${\Delta}/{\Lambda}$ is locally isomorphic to ${\rm U}_{3}{(}{H},r)$. By [10] only the cases $\dim{\Delta}{\,<\,}27$ must be considered. Then $\dim{\Lambda}{\,\mid\,}6$ and ${\Lambda}$ is a compact Lie group. For ${\Delta}^{\mskip-3.0mu*}{\,\cong\,}{\rm PSU}_{4}{(}{C},1)$ as a motion group of the quaternion plane cf. [19] 18.32. A maximal compact subgroup of ${\Delta}^{\mskip-3.0mu*}$ is isomorphic to ${\rm U}_{3}{{C}}$ and may be the image of a non-Lie group. Again ${K}$ is compact by stiffness, it is a finite extension of the semi-simple group ${\Lambda}{\,=\,}{K}^{1}$, and ${\Lambda}$ is a Lie group, see [19] 93.11. Thus ${\Delta}$ is a Lie group.

Note that in steps (a–c) the smaller bound $\dim{\Delta}{\,\geq\,}19$ suffices; the assumption $\dim{\Delta}{\,\geq\,}20$ will only be needed in the next step.

(d) Finally, let $\dim{\cal E}{\,=\,}4$. Suppose that $\dim{\Delta}|_{\cal E}{\,>\,}3$. According to [19] 71.8 the induced group ${\Delta}|_{\cal E}$ is an almost simple Lie group, and ${\Delta}{:}{K}{\,\leq\,}8$ or ${\Delta}{:}{K}{\,=\,}16$. In this step, the first case will be considered. Then $\dim{K}{\,\geq\,}12$ and ${\Lambda}{\,=\,}{K}^{1}{\,\cong\,}{\rm G}_{2}$. Therefore ${\Delta}{:}{K}{\,\geq\,}6$, and 2.9 shows that ${\Delta}$ is a Lie group.

(e) In the second case, ${\Delta}|_{\cal E}{\,\cong\,}{\rm PSL}_{3}{{C}}$ by [19] 71.8. This includes the possibility that there is a Hughes plane ${\cal H}^{\Delta}{\,=\,}{\cal H}$ such that {\cal E}{\,<\,}{\cal H}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}\,}{\cal P}. The plane ${\cal E}$ is the classical complex plane  ([19] 72.8), and $4{\,\leq}\dim{K}{\,\leq\,}8$. The normal subgroup ${\Lambda}{\,=\,}{K}^{1}$ is semi-simple and hence $\dim{\Lambda}{\,\in\,}\{6,8\}$. Consequently $\dim{\Delta}{\,\geq\,}22$. Choose a line $L$ of ${\cal E}$, a point z{\,\in\,}L\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt{\cal E}, and two point $a,b$ of ${\cal E}$,  $a,b{\,\notin\,}L$. If $\langle{\cal E},z\rangle{\,<\,}{\cal P}$, then ${\Lambda}$ would be a compact Lie group by 2.5(c). Hence ${\Delta}_{z}$ acts effectively on ${\cal E}$. The stabilizer ${\Gamma}{\,=\,}{\Delta}_{z,a,b}$ fixes each point of ${\cal E}$ on the line $ab$, and $\dim{\Gamma}{\,\geq\,}3$ because of 2.6. Let ${N}$ be a compact central subgroup of ${\Lambda}$, so that $z^{N}{\,\subset\,}{\cal F}{\,=\,}{\cal F}_{\Gamma}$. As ${N}$ is not a Lie group and ${N}_{z}{\,=\,}1\kern-3.0pt{\rm l}$, it follows from [19] 71.2 that {\cal F}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. Now ${\Gamma}$ is compact by 2.5(b). On the other hand, ${\Gamma}$ is a group of axial collineations in ${\Delta}_{z}$. These form a group ${C}^{2}{\times}{C}^{\times}$ and do not contain a compact subgroup of dimension $3$, a contradiction. $\square$

3.2 Theorem. If ${\cal F}_{\Delta}{\,=\,}\emptyset$ and $\dim{\Delta}{\,\geq\,}22$, then ${\Delta}$ is a Lie group.

Proof. (a) Suppose that ${\Delta}$ is not a Lie group. Then $\dim{\Delta}{\,<\,}27$ by [10], and there are arbitrarily small compact, [math]-dimensional central subgroups ${N}$ such that ${\Delta}/{N}$ is a Lie group, cf. [19] 93.8. Choose a fixed point $x$ of some element \zeta{\,\in\,}{N}\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt\{1\kern-3.0pt{\rm l}\}. From $\zeta|_{x^{\Delta}}{\,=\,}1\kern-3.0pt{\rm l}$ and ${\cal F}_{\Delta}{\,=\,}\emptyset$ it follows that ${\cal D}{\,=\,}\langle p^{\Delta}\rangle$ is a proper connected subplane. Put ${\Delta}|_{\cal D}{\,=\,}{\Delta}/{K}$. If ${\cal D}$ is flat, then ${\Delta}{:}{K}{\,=\,}8$ and ${\Lambda}{\,=\,}{K}^{1}{\,\cong\,}{\rm G}_{2}$ by stiffness, so that ${\Delta}|_{\cal D}{\,\cong\,}{\rm SL}_{3}{{R}}$ and 2.9 applies.

If $\dim{\cal D}{\,=\,}4$, then ${\Delta}/{K}$ is a Lie group by [19] 71.2, and stiffness 2.5(e) shows that $\dim{K}{\,<\,}8$ or ${\Lambda}{\,=\,}{K}^{1}{\,\cong\,}{\rm SU}_{3}{{C}}$. Hence ${\Delta}{:}{K}{\,\geq\,}14$, and then ${\Delta}/{K}{\,\cong\,}{\rm PSL}_{3}{{C}}$ by [11] 5.6. Now $6{\,\leq\,}\dim{\Lambda}{\,\leq\,}8$, and the same arguments as at the end of step (e) of the previous proof show that this is impossible. Consequently $\dim{\cal D}{\,\,=}8$.

(b) Let {\cal D}{\;\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P} and note that ${\Delta}$ acts on ${\cal D}$ without fixed elements. By [16] 2.1 the following holds for ${\Delta}|_{\cal D}{\,=\,}{\Delta}/{K}$: either ${\cal D}$ is the classical quaternion plane, ${\Delta}/{K}$ is a motion group of ${\cal D}$ and ${\Delta}{:}{K}{\,\in\,}\{21,15\}$, or ${\cal D}$ is a Hughes plane and ${\Delta}^{\prime}|_{\cal D}{\,\cong\,}{\rm SL}_{3}{{C}}$. Moreover ${\Delta}/{K}$ is a Lie group and we may assume that ${N}|_{\cal D}{\,=\,}1\kern-3.0pt{\rm l}$. The kernel ${K}$ is compact of dimension at most $7$.

(c) In the first case, the motion group ${\Delta}/{K}$ is covered by a subgroup ${\Upsilon}{\hskip 1.0pt\circeq\,}{\rm U}_{3}{(}{H},r)$ of ${\Delta}$  (see 2.9), and $\dim{K}{\,\leq\,}5$. The representation of ${\Upsilon}$ on ${K}/{N}$ shows that ${K}{\,\leq\,}{\rm Cs}_{\hskip 1.0pt}{\hskip 0.5pt{\Upsilon}}$. We may assume that ${K}$ is connected, since $\dim{K}^{1}{\Upsilon}{\,=\,}\dim{\Delta}$ and ${\Delta}$ is connected. Moreover, the center ${Z}$ of ${K}$ has positive dimension: because ${K}$ is not a Lie group, the claim follows from the structure of compact groups as described in [19] 93.11. The stabilizer ${\Pi}{\,=\,}{\Delta}_{p}$ of a non-absolute point $p{\,\in\,}{\cal D}$ fixes also the polar $L$ of $p$, and ${\Pi}/{K}{\,\circeq\,}{\rm U}_{2}{{H}}{\times}{\rm U}_{1}{{H}}$. In particular, ${\Pi}$ is compact and $\dim{\Pi}{\,\geq\,}14$.

Choose z{\,\in\,}L\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt{\cal D} and note that ${\Pi}_{z}\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{K}{\,=\,}1\kern-3.0pt{\rm l}$ because $\langle{\cal D},z\rangle{\,=\,}{\cal P}$. Hence ${\Pi}_{z}{\,=\,}{\Delta}_{z}$ is a Lie group. From [19] 96.11 it follows that ${\Pi}{\hskip 1.0pt:\hskip 1.0pt}{\Pi}_{z}{\,\leq\,}7$ and $\dim{\Delta}_{z}{\,\geq\,}7$. The group ${\Delta}_{z}$ acts faithfully on ${\cal D}$, it is isomorphic to a subgroup of ${\rm U}_{2}{{H}}{\times}{\rm U}_{1}{{H}}$. We apply Richardson’s theorem [19] 96.34 to the action of ${\Delta}_{z}$ on $S{\,:=\,}L\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\cal D}{\,\approx\,}{S}_{4}$  (cf.[19] 96.22). Put ${\Phi}{\,=\,}{\Delta}_{z}|_{S}{\,=\,}{\Delta}_{z}/{H}$ and note that ${H}$ is isomorphic to a subgroup of ${\rm U}_{1}{{H}}$, so that $\dim{\Phi}{\,\geq\,}4$.

(d) There are the following possibilities:

1. ${\Phi}{\,\cong\,}{\rm SO}_{3}{{R}}{\times}{\rm SO}_{2}{{R}}$ and ${\Phi}$ has a one-dimensional orbit,  2) ${\Phi}$ has two fixed points $u,v{\,\in\,}S$,

2. ${\Phi}{\,\cong\,}{\rm SO}_{5}{{R}}$ is transitive on $S$ and $\dim{\Phi}{\,\geq\,}10$. In the first two cases, there are points $u,v,w{\,\in\,}S$ and $q{\,\in\,}pv$ such that ${\Delta}_{z,u,v,w,q}$ has dimension ${>\,}0$, but this group fixes $S$ pointwise and is trivial on ${\cal D}$. In the third case, ${\Delta}_{z,v}$ fixes a second point $u{\,\in\,}S$. Let $p,v{\,\neq\,}q{\,\in\,}pv\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\cal D}{\,:=\,}J$. Then ${\Gamma}{=\,}{\Delta}_{z,q}$ fixes each point of $J{\,\approx\,}{S}_{4}$, and $\dim{\Gamma}{\,\geq\,}3$. Hence {\cal F}_{\Gamma}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}, and $J$ is contained in a line of ${\cal F}_{\Gamma}$. By the open mapping theorem [19] 51.19,20 we conclude that $J$ is in fact a line of ${\cal F}_{\Gamma}$. Therefore $M{\,=\,}L\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\cal F}_{\Gamma}{\,\approx\,}{S}_{4}$, and Richardson’s theorem [19] 96.34 shows that ${\Gamma}{K}$ induces a Lie group on $M$. As ${K}_{z}{\,=\,}1\kern-3.0pt{\rm l}$, both ${K}$ and ${\Delta}$ would be Lie groups.

(e) If ${\Delta}/{K}{\,\cong\,}{\rm PSU}_{4}{(}{C},1)$ is the $15$-dimensional motion group of the planar polarity, then ${\Delta}$ is doubly transitive on the absolute $5$-sphere $U$ in ${\cal D}$ (cf. [19] 18.32), and $\dim{K}{\,=\,}7$. In homogeneous coordinates over the quaternion field ${H}$, the planar polarity can be described by the form $\overline{x}ix{+}\overline{y}iy{+}\overline{z}iz$, see [19] 13.18. The line ‘‘at infinity" $W$, given by $z{\,=\,}0$, intersects $U$ in the circle consisting of all points $(1,h,0)$ with $h{\hskip 1.0pt\in\hskip 1.0pt}{H},\,h\overline{h}{=}1,\;hi{=-}ih$. Denote the polar lines of two points $p,p^{\prime}{\,\in\,}U$ by $L$ and $L^{\prime}$, and choose points $q{\,\in\,}pp^{\prime}\kern 2.0pt{\scriptstyle\cap}\kern 2.0ptU$ and z{\,\in\,}L\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt{\cal D}. As ${\Delta}_{z}\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{K}{\,=\,}1\kern-3.0pt{\rm l}$, we have ${\Delta}_{z}{\cong\,}{\Delta}/{K}$ , and representation of ${\Delta}_{z}$ on the compact group ${K}$ shows ${\Delta}{\,=\,}{\Delta}_{z}{\times}{K}$. The connected component ${A}$ of the center of ${K}$ is in the center of ${\Delta}$; it is not a Lie group. The stabilizer ${\Gamma}{=\,}{\Delta}_{z,p,p^{\prime},q}$ fixes also the point $s{\,=\,}L\kern 2.0pt{\scriptstyle\cap}\kern 2.0ptL^{\prime}{\,\in\,}{\cal D}$, and $\dim{\Gamma}{\,=\,}4$. The fact that ${\Delta}_{z}$ is (doubly) transitive on $U$ implies that ${\Delta}_{z}$ has a subgroup ${\Phi}{\,\cong\,}{\rm SU}_{3}{{C}}$ which is transitive on $U$, and ${\Gamma}^{\prime}{\cong\,}{\rm SU}_{2}{{C}}$. Let $\iota$ be the involution in ${\Gamma}^{\prime}$, obviously it is planar. It is a consequence of the stiffness result [19] 83.11 that the involution $\overline{\iota}{\,=\,}\iota|_{\cal D}$ is not planar. Hence $\iota$ induces a reflection $\overline{\iota}$ on ${\cal D}$. Its axis $pp^{\prime}\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\cal D}{\,\approx\,}{S}_{4}$ is also a line of ${\cal F}{\,=\,}{\cal F}_{\iota}$, and all lines of ${\cal F}$ are homeomorphic to ${S}_{4}$. The central group ${A}$ acts effectively on ${\cal F}$, and $\dim{A}{\,<\,}4$. By the structure theorem for compact groups ([19] 93.11), the semi-simple factor ${K}^{\prime}$ is locally isomorphic to $({\rm SU}_{2}{{C}})^{2}$. Exactly one of the two factors of ${K}^{\prime}$ acts trivially on ${\cal F}_{\Gamma}{\,=\,}{\cal F}$, the other factor induces a $3$-dimensional Lie group on $L\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\cal F}$. By Richardson’s theorem, ${A}$ would be a Lie group.

(f) Of the possibilities listed in step (b), only the following remains: ${\cal D}$ is a Hughes plane (cf. [19] § 86, in particular, 86.33,34) ), $\dim{\Delta}{\,=\,}17$, and there is an invariant subplane ${\cal C}$ isomorphic to the classical complex plane. The commutator group ${\Upsilon}{=\,}{\Delta}^{\prime}{\,\cong\,}{\rm SL}_{3}{{C}}$ induces on ${\cal C}$ the full automorphism group ${\rm PSL}_{3}{{C}}$. The center of ${\Upsilon}$ is generated by an element $\varepsilon$ of order $3$. The kernel ${K}$ is compact by stiffness, and $\dim{K}{\,\geq\,}5$. Moreover, ${\Upsilon}$ acts trivially on ${K}$, and ${K}{\,\leq\,}{\rm Cs}_{\hskip 1.0pt}{\hskip 0.5pt{\Upsilon}}$. Choose a point $z$ which is not incident with any line of ${\cal C}$. As ${\cal D}$ is a Baer subplane, there is a unique line $L$ of ${\cal D}$ with $z{\,\in\,}L$, and ${\cal F}_{\varepsilon}{\,=\,}{\cal C}$ implies $L^{\varepsilon}{\,\neq\,}L$. The stabilizer ${\Upsilon}_{\mskip-3.0muz}$ fixes each point of $z^{K}{\,\subset\,}L$ and of $z^{\varepsilon{K}}{\,\subset\,}L^{\varepsilon}$. Both orbits of $z$ are homeomorphic to ${K}$; their dimension is at least $5$. Hence $\langle z^{K},z^{\varepsilon{K}}\rangle{\,=\,}{\cal P}$ and ${\Upsilon}_{\mskip-3.0muz}{\,=\,}1\kern-3.0pt{\rm l}$. It follows that $z^{\mskip-3.0mu{\Upsilon}}$ is open in $P$, and ${\Delta}$ would be a Lie group by 2.6. $\square$

4. Exactly one fixed element

Throughout this section, ${\Delta}$ fixes a line $W$ but no point.

4.0. If ${\cal F}_{\Delta}{\,=\,}\{W\}$ and if $\dim{\Delta}{\,\geq\,}22$, then ${\Delta}$ is a Lie group.

Proof. Accordimg to [17] the assertion is true for $\dim{\Delta}{\geq\,}23$. Thus we may assume that $\dim{\Delta}{\,=\,}22$ and that ${\Delta}$ is not a Lie group. Steps (a,b) are the same as in [17].

(a) There exist arbitrarily small compact central subgroups ${N}{\,\leq\,}{\Delta}$ of dimension [math] such that ${\Delta}/{N}$ is a Lie group, cf. [19] 93.8. If ${N}$ acts freely on P\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0ptW, then each stabilizer ${\Delta}_{x}$ with $x{\,\notin\,}W$ is a Lie group because ${\Delta}_{x}{\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt}{N}{\,=\,}1\kern-3.0pt{\rm l}$. By [19] 51. 6 and 8 and 52.12, the one-point compactification $X$ of P\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0ptW is homeomorphic to the quotient space $P/W$, and $X$ is a Peano continuum (i.e., a continuous image of the unit interval); moreover, $X$ is homotopy equivalent to ${S}_{16}$, and $X$ has Euler characteristic $\chi(X){\,=\,}2$. According to a theorem of Löwen [7], these properties suffice for ${\Delta}$ to be a Lie group.

(b) Suppose now that $x^{\zeta}{\,=\,}x$ for some $x{\,\notin\,}W$ and some \zeta{\,\in\,}{N}\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt1\kern-3.0pt{\rm l}. By assumption, $x^{\Delta}$ is not contained in a line and hence generates a ${\Delta}$-invariant subplane ${\cal D}{\,=\,}\langle x^{\Delta},W\rangle{\,\leq\,}{\cal F}_{\zeta}{\,<\,}{\cal P}$. Put ${\Delta}^{\mskip-3.0mu*}{\,=\,}{\Delta}|_{\cal D}$ and write ${\Delta}|_{\cal D}{\,=\,}{\Delta}/{K}$. If ${\cal D}$ is flat, then $\dim{\Delta}{\leq\,}6{+}14$ by Stiffness; similarly, $\dim{\cal D}{\,=\,}4$ implies $\dim{\Delta}{\,\leq\,}12{+}8$. Hence {\cal D}{\;\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P},  ${K}$ is compact, $\dim{K}{\,<\,}8$,  $\dim{\Delta}^{\mskip-3.0mu*}{\,\geq\,}15$,  ${\Delta}^{\mskip-3.0mu*}$ is a Lie group  ([9]), and we may assume that $1\kern-3.0pt{\rm l}{\,\neq\,}{N}{\,\leq\,}{K}$. If even $\dim{\Delta}^{\mskip-3.0mu*}{\,>\,}16$, then \hrefhttp://arxiv.org/abs/1402.0304 [16] 1.10 shows that ${\cal D}$ is the classical quaternion plane, because ${\Delta}^{\mskip-3.0mu*}$ does not fix a flag.

(c) If ${\Delta}$ is not transitive on $S{\,=\,}W\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\cal D}$, then there are points $u,v,w{\,\in\,}S$ such that ${\Delta}{\hskip 1.0pt:\hskip 1.0pt}{\Delta}_{u,v,w}{\,\leq\,}9$. Choose a line $L$ of ${\cal D}$ in the pencil {{L}}_{v}\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt\{W\} and a point z{\,\in\,}L\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt{\cal D}, so that $\langle{\cal D},z\rangle{\,=\,}{\cal P}$,

${\Delta}_{z}\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{K}{\,=\,}1\kern-3.0pt{\rm l}$, and ${\Delta}_{z}$ acts faithfully on ${\cal D}$. In particular, ${\Delta}_{z}$ fixes $L$ and ${\Delta}_{z}$ is a Lie group. Moreover, $\dim z^{{\Delta}_{L}}{\,<\,}8$  (or else ${K}$ would be a Lie group by 2.6 above. Therefore ${\Delta}{\hskip 1.0pt:\hskip 1.0pt}{\Delta}_{z}{\,\leq\,}3{+}4{+}7{\,=\,}14$ and ${\Lambda}{\,=\,}({\Delta}_{z,u,w})^{1}$ satisfies $\dim{\Lambda}{\,\geq\,}2$. The stiffness result 2.5(b̂) shows that ${\Lambda}{\,\cong\,}{\rm Spin}_{3}{{R}}$. From [19] 55.32 and the structure of compact groups ([19] 93.11) it follows that ${K}^{1}$ is commutative or an almost direct product of ${\rm Spin}_{3}{{R}}$ with a connected commutative group. By [19] 93.19 the group ${\Lambda}$ acts trivially on the commutative factor ${A}$ of ${K}^{1}$. As ${K}/{N}{K}^{1}$ is finite and ${K}\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\Lambda}{\,=\,}1\kern-3.0pt{\rm l}$, the group ${\Lambda}$ centralizes also a non-commutative factor of ${K}^{1}$, and hence ${\Lambda}{\,\leq\,}{\rm Cs}_{\hskip 1.0pt}{\hskip 0.5pt{K}}$. In the cases $\dim{K}{\,\leq\,}5$, $\dim{\Delta}_{\cal D}{\,\geq\,}17$, the plane ${\cal D}$ is classical and ${\Lambda}$ fixes all points of a circle in $S$ containing $u,v,w$. If, on the other hand, $\dim{K}{\,>\,}0$, then $z^{K}{\,\subseteq\,}{\cal F}_{\Lambda}$ has positive dimension. In any case, ${\cal F}_{\Lambda}$ is connected. Because ${N}$ acts freely on L\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt{\cal D} and groups of planes of dimension ${\leq\,}4$ are Lie groups (cf. [19] 32.21 and 71.2), it follows that {\cal F}_{\Lambda}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. We have ${K}_{z}{\,=\,}1\kern-3.0pt{\rm l}$,  $\dim z^{K}{\,=\,}\dim{K}$,  $z^{K}{\,\subseteq\,}{\cal F}_{\Lambda}$, and therefore $\dim{K}{\,\leq\,}4$, in fact even $\dim{K}{\,<\,}4$ because of 2.6. Now $\dim{\Delta}^{\mskip-3.0mu*}{\,\geq\,}19$, and ${\cal D}$ is the classical quaternion plane, see [19] 84.28 or [16] 1.10. In particular, $S{\,\approx\,}{S}_{4}$. Recall that $\dim u^{\Delta}{\,+\,}\dim v^{\Delta}{\,+\,}\dim w^{{\Delta}_{u,v}}{\,\geq\,}8$. It follows that ${\Delta}$ is doubly transitive on the $3$-dimensional orbit $V{\,=\,}v^{\Delta}$. Similarly, ${\Delta}_{z}$ is doubly transitive on $V$, and $\dim w^{{\Delta}_{u,v}}{\,\geq\,}2$. The doubly transitive groups have been determined by Tits, see [19] 96.16. The large orbit of $w$ excludes the possibiliy that $V{\,\approx\,}{R}^{3}$. Hence $V$ is compact and ${\Delta}_{z}|_{V}$ is simple, in fact one of the groups ${\rm PU}_{2}{(}{H},1)$ or ${\rm PSU}_{3}{(}{C},1)$. In the first case, ${\Lambda}$ would be $4$-dimensional. In the second case, we obtain $\dim{\Lambda}{\,=\,}2$, contradicting stiffness.

(d) The previous step shows that ${\Delta}$ is transitive on $S{\,=\,}W\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\cal D}$. Hence $S{\,\approx\,}{S}_{4}$, and ${\Delta}$ has a subgroup Yu${\,\cong\,}{\rm U}_{2}{{H}}{\,\cong\,}{\rm Spin}_{5}{{R}}$, see [19] 53.2, 96.19–22, 55.40, and 94.27. The central involution $\sigma$ of Yu induces on ${\cal D}$ a reflection with axis $S$. As its center is not fixed by ${\Delta}$, it follows from [19] 61.13 that ${\Delta}^{\hskip-1.5pt*}$ contains a transitive translation group with axis $S$. Therefore ${\Delta}{\hskip 1.0pt:\hskip 1.0pt}{K}{\,\geq\,}18$ and ${\cal D}$ is classical (cf. [19] p. 500, Theorem or [16] 1.10). By [19] 55.40 the group ${\rm SO}_{5}{{R}}$ does not act on any compact plane. Hence the involution $\sigma$ is even a reflection of ${\cal P}$, and $\sigma^{\Delta}\sigma{\,=\,}{T}{\,\cong\,}{R}^{8}$ is normal in ${\Delta}$. The group ${\Delta}|_{\cal D}$ is contained in ${\rm SL}_{2}{{H}}{\hskip 1.0pt\cdot\hskip 1.0pt}{H}^{\times}{\ltimes}{R}^{8}$. Either ${\Delta}|_{S}$ is compact, or ${\Delta}$ has a subgroup ${\Upsilon}{\,\cong\,}{\rm SL}_{2}{{H}}$  (use [19] 94.34). In the first case, ${\Delta}|_{S}{\,=\,}$Yu$|_{S}{\,\cong\,}{\rm SO}_{5}{{R}}$. For $v,w{\,\in\,}S$, the stabilizer ${\Delta}_{v}$ fixes a second point $u{\,\in\,}S$, and ${\Delta}{\hskip 1.0pt:\hskip 1.0pt}{\Delta}_{u,v,w}{\,=\,}4{+}3$. Choose a line $L$ of ${\cal D}$ in the pencil {{L}}_{v}\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt\{W\}, a point z{\,\in\,}L\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt{\cal D}, and put ${\Gamma}{\,=\,}{\Delta}_{L,w}$. Again we may assume that ${N}{\,\leq\,}{K}$. By 2.6 we have $\dim z^{\Gamma}{<\,}8$. The dimension formula yields ${\Delta}{\hskip 1.0pt:\hskip 1.0pt}{\Gamma}{\,\leq\,}11$. Let ${\Lambda}{\,=\,}({\Gamma}_{\hskip-2.0ptz})^{1}$. Then $\dim{\Lambda}{\,\geq\,}4$. Again ${\Lambda}$ is a Lie group since ${N}$ acts freely on L\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt{\cal D}. As ${\cal D}$ is the classical quaternion plane, any collineation which fixes $3$ collinear points of ${\cal D}$ even fixes all points of a circle. Hence ${\cal F}_{\Lambda}$ is connected. By [19] 32.21 and 71.2, we have z^{N}{\,\subseteq\,}{\cal F}_{\Lambda}{\;\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. Now Stiffness would imply $\dim{\Lambda}{\,\leq\,}3$ contrary to what has been stated before.

(e) If ${\Delta}$ has a subgroup ${\Upsilon}{\,\cong\,}{\rm SL}_{2}{{H}}$, then the arguments in step (d) imply ${\Delta}{:}{K}{\,\geq\,}15{+}8$. A maximal compact subgroup of ${\Delta}$ is connected and properly contains Yu. Consequently $\dim{\Delta}{\,\geq\,}24$, and ${\Delta}$ is a Lie group by [17] 4.0. $\square$

For $8$-dimensional planes the following has been proved in [16] 3.1:

4.1. Suppose that ${\cal F}_{\Delta}$ consists of a unique line in an $8$-dimensional plane and that ${\Delta}$ is a semi-simple group of dimension $>\mskip-3.0mu3$. Then ${\Delta}$ is a Lie group and $\dim{\Delta}{\,\leq\,}10$. In the case of equality, ${\Delta}$ is isomorphic to ${\rm O}^{\prime}_{5}{(}{R},1)$ or to some covering group of ${\rm O}^{\prime}_{5}{(}{R},2)$.

4.2. Let ${\cal F}_{\Delta}{\,=\,}\{W\}$. If ${\Delta}$ is semi-simple and if $\dim{\Delta}{\,\geq\,}11$, then ${\Delta}$ is a Lie group.

Proof. As in steps (a,b) of the previous proof we may consider a subplane ${\cal D}{\,=\,}\langle x^{\Delta},W\rangle<{\cal P}$. Both groups ${\Delta}^{\mskip-3.0mu*}{\,=\,}{\Delta}|_{\cal D}{\,=\,}{\Delta}/{K}$ and ${\Lambda}{\,=\,}{K}^{1}$ are semi-simple. If ${\cal D}$ is flat and ${\Delta}^{\mskip-3.0mu*}{\,\neq\,}1\kern-3.0pt{\rm l}$, then ${\Delta}^{\mskip-3.0mu*}{\,\circeq\,}{\rm SL}_{2}{{R}}$ and either contains a central reflection or acts as hyperbolic motion group without fixed element. Hence ${\Delta}^{\mskip-3.0mu*}{\,=\,}1\kern-3.0pt{\rm l}$ and ${\Delta}{\,=\,}{\Lambda}{\,=\,}{\rm G}_{2}$ or $\dim{\Delta}{\,\leq\,}10$ by stiffness. If $\dim{\cal D}{\,=\,}4$, then [17] 2.14 shows that $\dim{\Delta}^{\mskip-3.0mu*}{\,\leq\,}3$ and ${\Lambda}{\,\cong\,}{\rm SU}_{3}{{C}}$ or $\dim{\Lambda}{\,<\,}8$ by stiffness. Thus ${\Delta}$ is a Lie group by 2.9 or $\dim{\Delta}{\,\leq\,}10$. In the case {\cal D}{\,\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P} the kernel ${K}$ is compact and then ${\Lambda}{\,\cong\,}{\rm Spin}_{3}{{R}}$ and $\dim{\Delta}^{\mskip-3.0mu*}{\geq\,}8$. By the result 4.1 the group ${\Delta}^{\mskip-3.0mu*}$ is a Lie group and $\dim{\Delta}^{\mskip-3.0mu*}{\leq\,}10$. From 2.9 it follows that ${\Delta}$ is a Lie group. $\square$

5. Fixed flag

For $8$-dimensional planes, the following results have been proved in [15] Theorem 1.3 and in [18] 5.1:

5.0. If the semi-simple group ${\Delta}$ of an $8$-dimensional plane fixes exactly one line and possibly some points on this line, and if $\dim{\Delta}{\,>\,}3$, then ${\Delta}$ is a Lie group.

5.1. Let ${\cal F}_{\Delta}{\,=\,}\{v,W\}$ be a flag in an $8$-dimensional plane. If $\dim{\Delta}{\,\geq\,}10$, then ${\Delta}$ is a Lie group.

5.2. If ${\Delta}$ fixes a flag $\{v,W\}$ and possibly one or two further points on $W$, but no other line, if ${\Delta}$ is semi-simple, and if $\dim{\Delta}{\,\geq\,}11$, then ${\Delta}$ is a Lie group.

Proof. The first arguments are the same as in steps (a,b) of the proof of 4.0 above. Thus there is a connected proper subplane ${\cal D}{\,=\,}\langle x^{\Delta},v,W\rangle$ of ${\cal P}$. Put ${\Delta}^{\mskip-3.0mu*}{\,=\,}{\Delta}|_{\cal D}{\,=\,}{\Delta}/{K}$, and note that both ${\Delta}^{\mskip-3.0mu*}$ and ${\Lambda}{\,=\,}{K}^{1}$ are semi-simple. If ${\cal D}$ is flat, then ${\Delta}^{\mskip-3.0mu*}$ is trivial by [19] 33.8, the kernel ${K}$ is isomorphic to the Lie group ${\rm G}_{2}$ or $\dim{K}{\,\leq\,}10$ by stiffness 2.5(d). If $\dim{\cal D}{\,=\,}4$, then $\dim{\Delta}^{\mskip-3.0mu*}{\,\leq\,}3$  (see [17] 2.14 or [19] 71. 8,10 and 72, 1–4), and stiffness 2.5(a,e) shows that $\dim{\Lambda}{\,<\,}8$ or ${\Lambda}{\,\cong\,}{\rm SU}_{3}{{C}}$ and 2.9 applies.

Finally, let {\cal D}{\,\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. Then $\dim{\Delta}^{\mskip-3.0mu*}{\leq\,}3$ or ${\Delta}^{\mskip-3.0mu*}$ is a Lie group by 5.0 above. Moreover, ${\Lambda}$ is compact and semi-simple, hence a Lie group, and then ${\Lambda}{\,\cong\,}{\rm Spin}_{3}{{R}}$ by 2.5(b̂) and $\dim{\Delta}^{\mskip-3.0mu*}{>\,}3$. Therefore ${\Delta}$ is a Lie group by 2.9. $\square$

6. Two non-incident fixed elements

It has been stated in 2.3 above that ${\Delta}$ is a Lie group whenever $\dim{\Delta}{\,\geq\,}27$. In the case ${\cal F}_{\Delta}{\,=\,}\{a,W\},\ a{\,\notin\,}W$, no improvement of this result has been found in general; for semi-simple groups it has been proved in [10] (a) that $\dim{\Delta}{\,\geq\,}26$ suffices.

Suppose that ${Z}{\,=\,}{\rm Cs}_{\hskip 1.0pt}{\hskip 0.5pt{\Delta}}$ is not a Lie group. If ${Z}|_{W}{\,\neq\,}1\kern-3.0pt{\rm l}$, then there is a point $p$ such that ${\Lambda}{\,=\,}({\Delta}_{p})^{1}$ fixes a quadrangle, and stiffness implies $\dim{\Lambda}{\,\leq\,}11$. On the other hand ${\Delta}{:}{\Delta}_{p}{\,<\,}16$ by 2.6. Thus $\dim{\Delta}{\,\leq\,}26$.

Throughout section $6$, assume that ${\Delta}$ fixes a line $W$ and a point $a{\,\notin\,}W$.

6.1 Proposition. If ${\Delta}$ is semi-simple, and if there is some planar element $\zeta{\,\in\,}{Z}$ (so that ${\cal F}_{\zeta}$ is a subplane of ${\cal P}$), then $\dim{\Delta}{\,\leq\,}19$.

Proof. Assume that $\dim{\Delta}{\,>\,}19$, and let $x^{\zeta}{\,=\,}x{\,\in\,}{\cal F}_{\zeta}$. Then $x^{\Delta}{\,\neq\,}x$ by assumption, $x^{\Delta}{\,\subseteq\,}{\cal F}_{\zeta}$, and ${\cal F}_{\zeta}$ is connected. Note that ${\Delta}|_{{\cal F}_{\zeta}}{\,=\,}{\Delta}/{K}$ and ${K}^{1}$ are semi-simple. If ${\cal F}_{\zeta}$ is flat or if $\dim{\cal F}_{\zeta}{\,=\,}4$, then stiffness yields $\dim{K}{\,\leq\,}14$, and ${\Delta}{:}{K}{\,\leq\,}3$  (see [17] 2.14 or [19] 71.8–72.4). If {\cal F}_{\zeta}{\,\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}, then ${K}^{1}$ is compact and semi-simple, hence a Lie group, and ${K}^{1}{\,\cong\,}{\rm Spin}_{3}{{R}}$ by 2.5(b̂). The classification of all planes such that ${\Delta}{:}{K}{\,>\,}16$ as summarized in [16] 1.10 shows that ${\cal F}_{\zeta}$ is the classical quaternion plane, and then ${\Delta}/{K}{\,\cong\,}{\rm SL}_{2}{{H}}{\cdot}{\rm Spin}_{3}{{R}}$. It follows that a maximal compact subgroup of ${\Delta}$ is locally isomorphic to ${\rm Spin}_{5}{{R}}{\cdot}({\rm Spin}_{3}{{R}})^{2}$, and ${\Delta}$ would be a Lie group. Thus $\dim{\Delta}{\,\leq\,}16{+}3$. $\square$

6.2 Proposition. Suppose that ${\Delta}$ is semi-simple and that $\dim{\Delta}{\,\geq\,}25$. If the center ${Z}$ of ${\Delta}$ acts non-trivially on $W$, then ${\Delta}$ is a Lie group.

Proof. Because of [10] (a) we may assume that $\dim{\Delta}{\,=\,}25$. Assume that ${Z}$ is not a Lie group. There are several possibilities:

(a) ${\Delta}$ has an almost simple factor ${\Psi}$ of dimension $16$. Then ${\Psi}$ is locally isomorphic to ${\rm SL}_{3}{{C}}$ and ${\Psi}$ is a Lie group. Denote the product of the other factors of ${\Delta}$ by ${\Gamma}$. Let $v^{Z}{\,\neq\,}v{\,\in\,}W$ and p{\,\in\,}K{:=\,}av\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt\{a,v\}. Then ${\Lambda}{\,=\,}({\Psi}_{p})^{1}$ has positive dimension by 2.6, and ${\cal D}{\,=\,}{\cal F}_{\Lambda}$ is a proper subplane. In particular, $\dim{\cal D}{\,\in\,}\{0,2,4,8\}$. Put ${\Xi}{\,=\,}({\Gamma}_{\mskip-3.0mup})^{1}$ and note that $p^{\Gamma}$ is contained in ${\cal D}$. It follows that $\dim{\Xi}{\,>\,}0$. Hence ${\cal E}{\,=\,}{\cal F}_{\Xi}$ is also a proper subplane. Now ${\cal E}^{\Psi}{\,=\,}{\cal E}$ and $\dim{\Lambda}{\,\geq\,}8$. Stiffness implies $\dim{\cal D}{\,\leq\,}4$. If ${\cal D}$ is connected, then ${Z}$ induces a Lie group on ${\cal D}$ and there is some $\delta{\,\in\,}{Z}$ such that ${\cal D}{\,\leq\,}{\cal F}_{\delta}{\,=\,}{\cal F}{\,=\,}{\cal F}^{\Delta}{\,<\,}{\cal P}$. The factor ${\Psi}$ acts almost effectively on ${\cal F}$. Hence {\cal F}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}  (use [19] 71.8 and 72.8). Let ${\Delta}|_{\cal F}{\,=\,}{\Delta}/{K}$. As ${K}^{1}$ is semi-simple, stiffness shows $\dim{K}{\,\leq\,}6$ and ${\Delta}{:}{K}{\,=\,}19$. By the classification of $8$-dimensional planes as summarized in [16] 1.10 (cf. also [19] 84.28), the plane ${\cal F}$ is the classical quaternion plane and ${\Delta}|_{\cal F}{\,\cong\,}{\rm SL}_{2}{{H}}{\cdot}{H}^{\times}$ in contradiction to the action of ${\Psi}$ on ${\cal F}$. Consequently $\dim{\cal D}{\,=\,}0$,  $\dim{\Xi}{\,=\,}9$, and $\dim{\cal E}{\,\leq\,}4$ by stiffness, but then ${\Psi}|_{\cal E}{\,=\,}1\kern-3.0pt{\rm l}$, which is impossible.

(b) ${\Delta}$ has an almost simple factor ${\Psi}$ of dimension $15$, and ${\Psi}$ is locally isomorphic to one of the groups ${\rm SU}_{4}{{C}}$, ${\rm SL}_{2}{{H}}$, ${\rm SL}_{4}{{R}}$, or ${\Psi}$ maps onto a group ${\rm PSU}_{4}{(}{C},r)$, $r{\,=\,}1,2$. Then there is a second almost simple factor ${\Gamma}$ of dimension $10$. This situation can be treated in a similar way as case (a), but a few more arguments are needed.

($i$) ${\Delta}$* is not transitive on $W$*. In fact, transitivity implies $W{\,\approx\,}{S}_{8}$ by 2.6, a maximal compact subgroup ${\Phi}$ of ${\Delta}|_{W}$ is also transitive ([19] 96,19), and ${\Phi}{\,\cong\,}{\rm SO}_{9}{{R}}$ would be too big (cf. [19] 96.23).

($ii$) There are points $v{\,\in\,}W$ and p{\,\in\,}K{\,:=\,}av\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt\{a,v\} such that $\dim v^{\Delta}{\,<\,}8$ and ${\Psi}_{p}{\,\neq\,}1\kern-3.0pt{\rm l}$.

The first claim follows from ($i$), and $\dim{\Psi}_{v}{\,\geq\,}8$. If ${\Psi}_{v}$ acts freely on $K$, then ${\Psi}_{v}$ is transitive on $K$ by [19] 96.11, lines are homeomorphic to ${S}_{8}$ by 2.6, and ${\Psi}_{v}{\,\approx\,}K{\,\simeq\,}{S}_{7}$ (see [19] 52.5). A maximal compact subgroup ${\Phi}$ of ${\Psi}_{v}$ is at most $7$-dimensional and has the homotopy of ${S}_{7}$, but $\pi_{\nu}{S}_{7}{\,=\,}0$ for $\nu{\,<\,}7$ and $\pi_{\nu}{\Phi}{\,\neq\,}0$ for $\nu{\,=\,}{\rm 1\ or\ 3}$, see [19].

($iii$) One of ${\cal D}{\,=\,}{\cal F}_{{\Psi}_{p}}$ or ${\cal E}{\,=\,}{\cal F}_{{\Gamma}_{\mskip-3.0mup}}$ is a connected proper subplane. Note that $p^{\Gamma}{\subseteq\,}{\cal D}$. If $v^{\Gamma}{\,\neq\,}v$, then ${\cal D}$ contains a quadrangle and a connected orbit of ${\Gamma}$; thus ${\cal D}{\,<\,}{\cal P}$. If $v^{\Gamma}{\,=\,}v$, however, then $v^{\Psi}{\,\neq\,}v$, and ${\Gamma}$ fixes each point of the connected orbit $v^{\Psi}{\,\subseteq\,}W$; moreover, $\dim{\Gamma}_{\mskip-3.0mup}{\,>\,}1$, and ${\cal E}{\,<\,}{\cal P}$.

($iv$) If $v^{\Gamma}{\neq\,}v$, then $\dim{\Gamma}|_{\cal D}{\,=\,}10$,  {\cal D}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}, and $\dim{\Gamma}_{\mskip-3.0mup}{\geq\,}2$. We have ${\cal E}^{\Psi}{\,=\,}{\cal E}$. Either ${\cal E}{\,<\,}{\cal P}$ or $v^{\Psi}{\,=\,}v$. In the latter case, ${\Psi}_{p}|_{\cal D}{\,=\,}1\kern-3.0pt{\rm l}$,  $\dim{\Psi}_{p}{\,=\,}7$ by stiffness, and ${\Psi}$ is transitive on $K$. Hence ${\Gamma}_{\mskip-3.0mup}{\,\leq\,}{\Delta}_{[av]}$, and ${\Gamma}_{\mskip-3.0mup}{\,\triangleleft\,}{\Gamma}$ would be a proper normal subgroup. Consequently $\dim{\Psi}|_{\cal E}{\,=\,}15$, and {\cal E}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}.

($v$) If $v^{\Gamma}{=\,}v$, then $v^{\Psi}{\,\neq\,}v$,  ${\cal E}{\,<\,}{\cal P}$,  $\dim{\Psi}|_{\cal E}{\,=\,}15$, and {\cal E}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. From [15] 3.3 and $\dim{\Psi}{\,>\,}10$ it follows that ${\cal E}$ is the classical quaternion plane. In particular, ${\Psi}{\,\cong\,}{\rm SL}_{2}{{H}}$. Moreover ${Z}|_{\cal E}$ is a Lie group and there is some $\zeta{\,\in\,}{Z}$ such that ${\cal E}{\,=\,}{\cal F}_{\zeta}$ is ${\Delta}$-invariant and $\dim{\Delta}|_{\cal E}{\,=\,}25$. This is impossible.

($vi$) Corollary. Both ${\cal D}$ and ${\cal E}$ are Baer subplanes.

Again ${\cal E}$ is classical by 2.11. As in the previous step, ${\cal E}^{\Delta}{\,=\,}{\cal E}$ and $\dim{\Delta}|_{\cal E}{\,=\,}25$, a final contradiction.

(c) ${\Delta}$ has an almost simple factor ${\Psi}$ of dimension $14$. In this case, ${\Psi}$ is a Lie group, there are two other factors ${B}$ and ${A}$ of dimension $8$ and $3$, respectively, and ${B}$ is a a Lie group or maps onto ${\rm PSU}_{3}{(}{C},1)$.

($i$) If ${\Psi}{\,\cong\,}{\rm G}_{2}$ is compact or, more generally, whenever ${\Psi}$ has a subgroup ${\Phi}{\,\cong\,}{\rm SO}_{3}{{R}}$, then any two conjugate commuting involutions $\iota,\kappa{\,\in\,}{\Phi}$ are planar (use [19] 55.35), and ${\cal F}_{\iota}$ is a Baer subplane. (For involutions in ${\rm G}_{2}$ see [19] 31-35 or [17] 2.5.) Choose a point $x$ in ${\cal F}_{\iota},\ a{\,\neq\,}x{\,\notin\,}W$, and put $v{\,=\,}ax\kern 2.0pt{\scriptstyle\cap}\kern 2.0ptW$ and ${\Gamma}{\,=\,}{B}{A}$. Either ${\cal F}_{\iota,\kappa}$ is a $4$-dimensional subplane or $\kappa$ induces on ${\cal F}_{\iota}$ a reflection with center $v$, say. In the first case, ${Z}|_{{\cal F}_{\iota,\kappa}}$ is a Lie group by [19] 71.2, and there is some $\zeta{\,\in\,}{Z}$ such that ${\cal F}_{\zeta}$ is a connected proper subplane, but then ${\Delta}|_{{\cal F}_{\zeta}}$ is too big. In the second case, $v^{\Gamma}{\,=\,}v$,  $V{\,=\,}v^{\Psi}{\,\neq\,}v$, and ${\Gamma}|_{V}{\,=\,}1\kern-3.0pt{\rm l}$. It follows that $\dim{\Gamma}_{\mskip-3.0mux}{\geq\,}3$ and that ${\cal E}{\,=\,}{\cal F}_{{\Gamma}_{\mskip-3.0mux}}{\,=\,}{\cal E}^{\Psi}$ is a connected proper subplane. The simple group ${\Psi}$ acts effectively on ${\cal E}$. Consequently $\iota|_{\cal E}$ is planar and $\dim({\cal E}\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\cal F}_{\iota}){\,=\,}4$. Again this is impossible, cf. also [19] 84.8.

($ii$) Similar arguments apply if ${B}$ has a subgroup ${\Phi}{\,\cong\,}{\rm SO}_{3}{{R}}$, in particular, if ${B}$ is compact. Again involutions $\iota,\kappa{\,\in\,}{\Phi}$ are planar and ${\cal F}_{\iota,\kappa}$ is a triangle. The factor ${\Psi}$ fixes all points of some ${B}$-orbit $V{\,=\,}v^{B}{\,\subseteq\,}W$, and $\dim{\Psi}_{x}{\,\geq\,}6$. The center ${Z}$ acts effectively on ${\cal D}{\,=\,}{\cal F}_{{\Psi}_{x}}$, and {\cal D}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}\,}{\cal P}. Now stiffness 2.5(b̂) shows $\dim{\Psi}_{x}{\,\leq\,}3$ because ${\Psi}$ is Lie group, a contradiction. Therefore ${B}$ has no subgroup ${\rm SO}_{3}{{R}}$.

($iii$) Therefore ${\Psi}$ is isomorphic to the twofold coverring $\widetilde{\rm G}$ of ${\rm G}_{2}(2)$; a maximal compact subgroup ${\Phi}$ of ${\Psi}$ is isomorphic to ${\rm Spin}_{4}{{R}}$. The center of ${\Psi}$ is generated by a reflection $\omega$ with axis $W$. There are two other central involutions $\iota,\kappa$ in ${\Phi}$, these may be planar or they are reflections with centers $u,v{\,\in\,}W$. In the first case the argumentation is the same as in step ($i$). If $\iota{\,\in\,}{\Delta}_{[u,av]}$, then $v^{\Gamma}{\,=\,}v{\,\neq\,}v^{\Psi}{\,=\,}V$ and ${\Gamma}|_{V}{\,=\,}1\kern-3.0pt{\rm l}$. For the same reasons as in step (b)($ii$) the group ${B}$ does not act freely on K{\,=\,}av\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt\{a,v\} and ${B}_{x}{\,\neq\,}1\kern-3.0pt{\rm l}$ for a suitable $x{\,\in\,}K$. It follows that ${\cal E}{\,=\,}{\cal F}_{{B}_{\mskip-3.0mux}}{\,=\,}{\cal E}^{{\Psi}{A}}$ is a proper connected sub plane. Because $\omega$ is a reflection of ${\cal P}$, the group ${\Psi}$ acts effectively on ${\cal E}$. The arguments can now be repeated for ${\Psi}|_{\cal E}$. Hence $\iota$ and $\kappa$ induce reflections on ${\cal E}$, and ${\Gamma}$ fixes all points of $V{\,\subseteq\,}W$. Moreover, [19] 96.13, applied to ${\Phi}|_{V}$, shows that $\dim V{\,\geq\,}3$, so that {\cal E}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. If $\dim({\Psi}{A})|_{\cal E}{\,=\,}17$, the classification as summarized in [19] 84.28 or [16] 1.10 would imply that ${\cal E}$ is a translation plane. Therefore ${A}|_{\cal E}{\,=\,}1\kern-3.0pt{\rm l}$ and ${\cal E}{\,=\,}{\cal F}_{A}$ is ${\Psi}{B}$-invariant. By stiffness $\dim({\Psi}{B})|_{\cal E}{\,=\,}22$ is too big. Thus case (c) is impossible.

(d) An almost simple factor ${\Psi}{\,\triangleleft\,}{\Delta}$ of maximal dimension satisfies $\dim{\Psi}{\,=\,}10$; all other factors have dimension $3$ or $6$. Write ${\Delta}$ as an almost direct product ${B}{A}$ with $\dim{A}{\,=\,}3$, and choose $v{\,\in\,}W$ such that $v^{Z}{\,\neq\,}v$. Let $a,v{\,\neq\,}x{\,\in\,}av$. Then $\dim{B}_{x}{>\,}6$  (by 2.6), and ${\cal E}{\,=\,}{\cal F}_{{B}_{x}}{\,=\,}{\cal E}^{A}$ is a proper subplane. If $\dim{\cal E}$ is $2$ or $4$, then ${Z}|_{\cal E}$ is a Lie group, and there is some $\zeta{\,\in\,}{Z}$ such that {\cal E}{\,\leq\,}{\cal F}{\,=\,}{\cal F}_{\hskip-2.0pt\zeta}{\,=\,}{\cal F}^{\Delta}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. By stiffness $\dim{\Delta}|_{\cal F}{\,=\,}19$,  ${\cal F}$ is the classical quaternion plane (cf. [19] 84.28 or [16] 1.10), and then ${\Delta}|_{\cal F}{\,\cong\,}{\rm SL}_{2}{{H}}{\cdot}{H}^{\times}$ is not semi-simple. Therefore $\dim{\cal E}{\,\in\,}\{0,8\}$. If ${\cal E}$ is not connected, then ${A}|_{\cal E}{\,=\,}1\kern-3.0pt{\rm l}$ and ${\cal D}{\,=\,}{\cal F}_{A}{\,=\,}{\cal D}^{B}{\,=\,}{\cal D}^{\Delta}$. By stiffness, the semi-simple kernel of ${\Delta}|_{\cal D}$ has dimension at most $6$ and contains ${A}$. Hence $\dim{\Delta}|_{\cal D}{\,\geq\,}19$, but again this is impossible. Thus {\cal E}{\,=\,}{\cal E}^{Z}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}, and ${Z}$ acts effectively on ${\cal E}$  (or else ${\cal E}{\,=\,}{\cal F}_{\zeta}$ for some $\zeta{\,\in\,}{Z}$,  ${\cal E}^{\Delta}{\,=\,}{\cal E}$,  $\dim{\Delta}|_{\cal E}{\,=\,}19$, a contradiction.) Now ${B}_{x}\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{Z}{\,=\,}1\kern-3.0pt{\rm l}$,  ${B}_{x}$ is a Lie group, and $\dim{B}_{x}{\,\leq\,}3$ by stiffness, in contrast to what has been stated above.

(e) ${\Delta}$ has exactly two factors of dimension $8$; all other factors have dimension $3$ or $6$. This is the last possibility, it can be treated exactly in the same way as (d).

Remark. The cases (a) and (c) can also be exclude as in step (d). $\square$

6.3 Proposition. Suppose that ${\Delta}$ is semi-simple and that $\dim{\Delta}{\,\geq\,}25$. If the center ${Z}$ of ${\Delta}$ acts trivially on $W$, then ${\Delta}$ is a Lie group.

Proof. Again we may assume that $\dim{\Delta}{\,=\,}25$. Suppose that ${Z}$ is not a Lie group. There are the same possibilities as in 6.2.

(a) ${\Delta}$ has an almost simple factor ${\Psi}$ of dimension $16$. Then ${\Psi}$ is locally isomorphic to ${\rm SL}_{3}{{C}}$ and ${\Psi}$ is a Lie group. Denote the product of the other factors of ${\Delta}$ by ${\Gamma}$. The factor ${\Psi}$ has a subgroup ${\Phi}{\,\cong\,}{\rm SO}_{3}{{R}}$. As before, the involutions in ${\Phi}$ are necessarily planer. Let $\iota,\kappa$ be two commuting involutions in ${\Phi}$. If $\kappa$ induces a planar involution on ${\cal F}_{\iota}$, then ${Z}$ acts freely on the $4$-dimensional plane ${\cal F}_{\iota,\kappa}$, and ${Z}$ would be a Lie group. Therefore $\kappa|_{{\cal F}_{\iota}}$ is a reflection, and $\kappa$ fixes points $u,v{\,\in\,}W\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\cal F}_{\iota}$. Choose $x{\,\in\,}{\cal F}_{\iota}$ not incident with the three fixed lines. Then $\dim x^{\Gamma}{<\,}8$ because of 2.6, and $\dim{\Gamma}_{\mskip-3.0mux}{\geq\,}2$. The group ${\Gamma}_{\mskip-3.0mux}$ fixes a quadrangle and ${\cal E}{\,=\,}{\cal F}_{{\Gamma}_{\mskip-3.0mux}}{\,=\,}{\cal E}^{\Psi}$ is a proper subplane. We have $\dim{\Psi}|_{\cal E}{\,=\,}16$ and {\cal E}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. Hence ${\cal E}$ is the classical quaternion plane by 2.11 . But then ${\rm SL}_{3}{{C}}$ does not act on ${\cal E}$ and the group ${Z}$ would be a Lie group.

Remark. The proof of (a) shows that a factor ${\Psi}$ of dimension ${>\,}10$ does not have a subgroup ${\rm SO}_{3}{{C}}$.

(b) ${\Delta}$ is an almost direct product ${\Psi}{\Gamma}$ of two factors of dimension $15$ and $10$, respectively. As in 6.2(b)($i$,$ii$), ${\Delta}$ is not transitive on $W$, and there is a point $x$ on a line $av$ such that ${\Psi}_{x}{\,\neq\,}1\kern-3.0pt{\rm l}$.

If $V{\,=\,}v^{\Gamma}{\,\neq\,}v$, then ${\cal D}{\,=\,}{\cal F}_{{\Psi}_{x}}{=\,}{\cal D}^{\Gamma}$ is a connected proper subplane, ${Z}$ acts freely on ${\cal D}$, and {\cal D}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. Because of 2.6 it follows that $\dim x^{\Gamma}{<\,}8$ and $\dim{\Gamma}_{\mskip-3.0mux}{\geq\,}3$. As ${\Gamma}$ acts almost effectively on $V$ and ${\Gamma}$ has a compact subgroup of dimension at least $4$, Mann’s theorem [19] 96.13 shows that $\dim V{\,\geq\,}3$. Suppose that $v^{\Psi}{=\,}v$. Then ${\Psi}_{x}|_{V}{\,=\,}1\kern-3.0pt{\rm l}$, $\dim x^{\Psi}{\,<\,}8$ by 2.6, and $\dim{\Psi}_{x}{\geq\,}8$, but {\cal F}_{{\Psi}_{x}}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}, and $\dim{\Psi}_{x}{\,\leq\,}7$ by stiffness.Therefore $v^{\Psi}{\neq\,}v$ and {\cal E}{\,=\,}{\cal F}_{{\Gamma}_{\mskip-3.0mux}}{=\,}{\cal E}^{\Psi}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. In fact, ${\cal E}$ is classical by 2.11. As a group of homologies ${Z}$ acts freely on ${\cal E}$, and ${Z}$ would be a Lie group. Consequently $v^{\Gamma}{=\,}v$ whenever $\dim v^{\Delta}{\,<\,}8$. Moreover $\dim{\Gamma}_{\mskip-3.0mux}{\,>\,}1$ and {\cal E}{\,=\,}{\cal E}^{\Psi}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. Again ${\cal E}$ is classical by 2.11, but then ${Z}$ would be a Lie group, a contradiction.

(c) ${\Delta}$ has an almost simple factor ${\Psi}$ of dimension $14$. In this case, ${\Psi}$ is a Lie group, there are two other factors ${B}$ and ${A}$ of dimension $8$ and $3$, respectively, and ${B}$ is a a Lie group or maps onto ${\rm PSU}_{3}{(}{C},1)$. This case is similar to the previous one. Put ${\Upsilon}{\,=\,}{\Psi}{A}$. Again ${\Delta}$ is not transitive on $W$, and $\dim v^{\Delta}{\,<\,}8$ for somme $v{\,\in\,}W$. Let $a,v{\,\neq\,}x{\,\in\,}av$. Then $\dim{\Upsilon}_{\mskip-3.0mux}{>\,}1$. If $v^{B}{\,\neq\,}v$, then {\cal D}{\,=\,}{\cal F}_{{\Upsilon}_{\mskip-3.0mux}}{=\,}{\cal D}^{B}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}, and $\dim{B}_{x}{>\,}0$ because of 2.6. It follows that $v^{\Upsilon}{\neq\,}v$  (or else $\dim{\Upsilon}_{\mskip-3.0mux}{>\,}8$ and ${\Upsilon}_{\mskip-3.0mux}|_{\cal D}{\,=\,}1\kern-3.0pt{\rm l}$, which contradicts stiffness). Hence {\cal E}{\,=\,}{\cal F}_{{B}_{x}}{=\,}{\cal E}^{\Upsilon}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. The kernel of ${\Upsilon}|_{\cal E}$ has dimension at most $3$, and $\dim{\Psi}|_{\cal E}{\,=\,}14$. Now ${\cal E}$ is classical by 2.11, and ${Z}$ would be a Lie group. If $v^{B}{\,=\,}v$, however, then ${B}|_{v^{\Delta}}{\,=\,}1\kern-3.0pt{\rm l}$, and $\dim x^{B}{\,<\,}8$  (use 2.6). Again $\dim{B}_{x}{>\,}0$, and ${\cal E}$ is the classical quaternion plane by 2.11. As before, ${Z}$ is a Lie group in contrast to the assumption.

(d) An almost simple factor ${\Psi}{\,\triangleleft\,}{\Delta}$ of maximal dimension satisfies $\dim{\Psi}{\,=\,}10$; all other factors have dimension $3$ or $6$. Write ${\Delta}$ as an almost direct product ${\Upsilon}{\Gamma}$ with $\dim{\Upsilon}{\,=\,}16$. The arguments in 6.2(b)($i$,$ii$) show that ${\Delta}$ is not transitive on $W$. Choose $v$ and $x$ as in step (c). By 2.6 we conclude that $\dim{\Upsilon}_{\mskip-3.0mux}{>\,}1$. If $v^{\Gamma}{\,\neq\,}v$, then {\cal D}{\,=\,}{\cal F}_{{\Upsilon}_{\mskip-3.0mux}}{=\,}{\cal D}^{\Gamma}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P} and ${\Upsilon}_{\mskip-3.0mux}|_{\cal D}{\,=\,}1\kern-3.0pt{\rm l}$. Stiffness implies $\dim{\Upsilon}_{\mskip-3.0mux}{\leq\,}7$ and $\dim x^{\Upsilon}{\geq\,}9$. As $x^{\Gamma}$ is contained in ${\cal D}$, we have $\dim{\Gamma}_{\mskip-3.0mux}{\,>\,}0$. Therefore {\cal E}{\,=\,}{\cal F}_{{\Gamma}_{\mskip-3.0mux}}{\,=\,}{\cal E}^{\Upsilon}{\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P} and $\dim x^{\Upsilon}{\leq\,}8$. This contradiction shows that $v^{\Gamma}{=\,}v$ and ${\Gamma}|_{v^{\Delta}}{\,=\,}1\kern-3.0pt{\rm l}$. Consequently again {\cal E}{\,=\,}{\cal F}_{{\Gamma}_{\mskip-3.0mux}}{\,=\,}{\cal E}^{\Upsilon}{\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. Let ${\Upsilon}|_{\cal E}{\,=\,}{\Upsilon}/{K}$. Then ${\Lambda}{\,=\,}{K}^{1}$ is compact and semi-simple, hence a Lie group, and stiffness 2.5(b̂) shows  $\dim{\Lambda}{\,\leq\,}3$. As ${Z}$ acts effectively on ${\cal E}$, the plane ${\cal E}$ is not classical. We have $\dim{\Upsilon}|_{\cal E}{\,\geq\,}13$, and 2.11 implies that ${\Upsilon}/{K}{\,\cong\,}{\rm U}_{2}{(}{H},r){\cdot}{\rm SU}_{2}{{C}}$. The complement of ${\Psi}$ in ${\Upsilon}$ can be chosen arbitrarily among the factors of dimension $3$ or $6$. Hence all these factors are compact and $3$-dimensional. Therefore ${\Delta}$ is a Lie group.

(e) ${\Delta}$ has exactly two factors of dimension $8$; all other factors have dimension $3$ or $6$. The first part of the proof is identical to the previous one, and there is a Baer subplane ${\cal E}{\,=\,}{\cal F}_{{\Gamma}_{\mskip-3.0mux}}{\,=\,}{\cal E}^{\Upsilon}$. As ${\Upsilon}$ has two $8$-dimensional factors, $\dim{\Upsilon}|_{\cal E}{\,=\,}16$, and 2.11 implies that ${\cal E}$ is the classical quaternion plane, but then ${Z}$ is a Lie group contrary to the assumption. $\square$

Combined, Propositions 6.2 and 6.3 yield

6.4 Theorem. If ${\cal F}_{\Delta}{\,=\,}\{a,W\},\,a{\notin}W$, if ${\Delta}$ is semi-simple, and if $\dim{\Delta}{\,\geq\,}25$, then ${\Delta}$ is a Lie group.

7. Fixed double flag

7.1. Let ${\cal F}_{\Delta}$ be a double flag $\{u,v,uv,ov\}$. If ${\Delta}$ is semi-simple and if $\dim{\Delta}{\,\geq\,}25$, then ${\Delta}$ is a Lie group.

Proof. For $\dim{\Delta}{\,\geq\,}26$, this has been proved in [17] 8.0. It suffices, therefore, to consider the case $\dim{\Delta}{\,=\,}25$. The arguments are similar to those in [17]. Let ${\Psi}$ be an almost simple factor of ${\Delta}$ of maximal dimension. Then $\dim{\Psi}{\,\in\,}\{16,15,14,10,8\}$. The product of the other factors of ${\Delta}$ will be denoted by ${\Gamma}$, the center of ${\Delta}$ by ${Z}$.

(a) If $p{\,\notin\,}uv\kern 2.0pt{\scriptstyle\cup}\kern 2.0ptov$, then ${Z}_{p}{\,=\,}1\kern-3.0pt{\rm l}$: suppose that $p^{\zeta}{=\,}p$ for some \zeta{\,\in\,}{Z}\hskip-1.0pt$$\scriptstyle\setminus$$\hskip-1.0pt\{1\kern-3.0pt{\rm l}\}. Then $\zeta|_{p^{\Delta}}{\,=\,}1\kern-3.0pt{\rm l}$ and {\cal D}{\,=\,}\langle p^{\Delta}\rangle{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}. With ${\Delta}|_{\cal D}{\,=\,}{\Delta}/{K}$ we get ${\Delta}{:}{K}{\,\leq\,}$ and $\dim{K}{\,\leq\,}7$, a contradiction.

(b) Each reflection $\sigma{\,\in\,}{\Psi}$ is in ${\Psi}_{[u,ov]}$, and each non-central involution $\beta{\,\in\,}{\Psi}$ is planar. In fact, $\sigma{\,\in\,}{\Psi}_{[uv]}$ would imply that $\sigma^{\Delta}\sigma{\,\cong\,}{R}^{k}$ is normal in ${\Psi}$. If $\beta$ is a reflection, then $\langle\beta^{\Delta}\rangle{\,=\,}{\Psi}$ is a compact group of homologies by [19] 61.2, and $\dim{\Psi}{\,=\,}3$.

(c) Let $\beta$ be a planar involution in ${\Psi}$, and let ${\Upsilon}$ be a semi-simple subgroup of the centralizer ${\rm Cs}_{\hskip 1.0pt{\Delta}}{\hskip 0.5pt\beta}$. Then $\dim{\Upsilon}{\,\leq\,}13$: Write ${\Upsilon}|_{{\cal F}_{\beta}}{\,=\,}{\Upsilon}/{K}$ and ${\Lambda}{\,=\,}{K}^{1}$. The kernel ${\Lambda}$ is compact and semi-simple, hence a Lie group, and then ${\Lambda}$ is contained in ${\rm Spin}_{3}{{R}}$ by stiffness. In [16] 6.1 it has been shown that ${\Upsilon}{:}{K}{\,\leq\,}10$.

(d) If $\dim{\Psi}{\,=\,}16$, then ${\Psi}{\,\cong\,}{\rm(P)}{\rm SL}_{3}{{C}}$. The centralizer of the involution $\beta{\,=\,}{\rm diag}(1,-1,-1)$ contains a group ${\rm SL}_{2}{{C}}{\cdot}{\Gamma}$ of dimension $6{+}9$ and is too large.

(e) Each group ${\Psi}$ of type ${\rm A}_{3}$ contains a planar involution $\beta$ centralizing a $6$-dimensional semi-simple subgroup of ${\Psi}$. This is easy in the cases ${\rm SU}_{4}{{C}}$,  ${\rm SL}_{2}{{H}}$,  ${\rm SL}_{4}{{R}}$ and its universal covering group; it is less obvious for the groups related to ${\rm SU}_{4}{(}{C},r)$, $r{\,=\,}1,2$, which are not necessarily Lie groups. In both cases, a suitable involution $\beta$ is determined by ${\rm diag}(-1,-1,1,1)$; then ${\rm Cs}_{\hskip 1.0pt{\Psi}}{\hskip 0.5pt\beta}$ contains a group locally isomorphic to ${\rm SU}_{2}{{C}}{\times}{\rm SL}_{2}{{R}}$ or to $({\rm SU}_{2}{{C}})^{2}$, respectively, and ${\rm Cs}_{\hskip 1.0pt{\Delta}}{\hskip 0.5pt\beta}$ is too large.

(f) Each involution of the compact group ${\rm G}_{2}$ is centralized by ${\rm SO}_{4}{{R}}$, see [19] 11.31. The non-compact groups of type ${\rm G}_{2}$ contain either ${\rm SO}_{4}{{R}}$ or ${\rm Spin}_{4}{{R}}$. In any case, the centralizer of a non-central involution is too large.

(g) If ${\Psi}$ is locally isomorphic to a group ${\rm U}_{2}{(}{H},r)$ of dimension $10$, then $\beta{\,=\,}{\rm diag(1,-1)}$ describes a planar involution centralized by a semi-simple group of dimension $6{+}15$, which is too large.

(h) If $\dim{\Psi}{\,=\,}10$, then ${\Psi}$ maps onto ${\rm Sp}_{4}{{R}}$ or ${\rm O}^{\prime}_{5}{(}{R},2)$. In the latter case, the diagonal element ${\rm diag(-1\kern-3.0pt{\rm l},1)}$ corresponds to a planar involution $\beta$ such that ${\rm Cs}_{\hskip 1.0pt{\Psi}}{\hskip 0.5pt\beta}{\,\circeq\,}{\rm O}^{\prime}_{4}{(}{R},1){\,\cong\,}{\rm SO}_{3}{{C}}$, again a contradiction.

(i) Finally, let ${\Psi}/({Z}\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\Psi}){\,\cong\,}{\rm Sp}_{4}{{R}}$. Conceivably, ${\Psi}$ is not a Lie group. Again ${\Psi}$ cannot contain a planar involution. Hence the unique involution in ${\Psi}$ is a reflection $\sigma{\,\in\,}{\Psi}_{[u,av]}{\,:=\,}{H}$. As $\dim{H}{\,\leq\,}8$, there is a point $p$ such that ${\cal G}{\,=\,}\langle p^{{\Psi}}\rangle$ is a connected subplane, and {\cal G}{\hskip 2.0pt\leq\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 2.0pt\hbox{\scriptscriptstyle\bullet}\mskip-3.0mu\,}{\cal P}, or else $\dim{\Psi}|_{\cal G}{\,\leq\,}6$. We have ${\Gamma}_{\hskip-2.0ptp}|_{\cal G}{\,=\,}1\kern-3.0pt{\rm l}$, and ${\Gamma}_{\hskip-2.0ptp}$ is a compact Lie group by step (a). Stiffness 2.5(b̂) shows $\dim{\Gamma}_{\hskip-2.0ptp}{\,\leq\,}3$ and $\dim p^{\Gamma}{\,\geq\,}12$. Consequently $\langle p^{\Gamma}\rangle{\,=\,}{\cal P}$ and ${\Psi}_{\hskip-3.0ptp}{\,=\,}1\kern-3.0pt{\rm l}$. It follows that ${\cal G}{\,=\,}{\cal P}$ and ${\Gamma}_{\hskip-2.0ptp}{\,=\,}1\kern-3.0pt{\rm l}$. Split ${\Gamma}$ into an almost direct product ${A}{B}$ with $\dim{A}{\,=\,}6$. Then $\dim p^{B}{>\,}8$,  $\langle p^{B}\rangle{=\,}{\cal P}$, and $({\Psi}{A})_{p}{\,=\,}1\kern-3.0pt{\rm l}$. Hence $p^{{\Psi}{A}}$ is open in $P$ by [19] 53.1), and $p^{\Delta}$ is also open. This implies that ${\Delta}$ is a Lie group, see 2.6.

(j) The cases $\dim{\Psi}{\,=\,}8$ can be treated similarly. Again there is a point $p$ such that {\cal G}{\,=\,}\langle p^{\Psi}\rangle{\,\leq\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 2.0pt\hbox{\scriptscriptstyle\bullet}\mskip-3.0mu\,}{\cal P}, ${\Gamma}_{\hskip-2.0ptp}|_{\cal G}{\,=\,}1\kern-3.0pt{\rm l}$,  $\dim p^{\Gamma}{\geq\,}14$,  $\langle p^{\Gamma}\rangle{\,=\,}{\cal P}$, and ${\Psi}_{p}{\,=\,}1\kern-3.0pt{\rm l}$. Let ${B}$ be the other $8$-dimensional factor of ${\Delta}$ and denote the product of the remaining factors by ${\Upsilon}$. Then $\langle p^{{B}{\Psi}}\rangle{\,=\,}{\cal P}$, or else an action of ${B}{\Psi}$ on ${\cal G}$ would have a kernel of dimension [math]. Now ${\Upsilon}_{\hskip-3.0ptp}{\,=\,}1\kern-3.0pt{\rm l}$,  $\langle p^{\Upsilon}\rangle{\,=\,}{\cal P}$,  $({B}{\Psi})_{p}{\,=\,}1\kern-3.0pt{\rm l}$,  $p^{\Delta}$ is open in $P$, and ${\Delta}$ is a Lie group by 2.6. $\square$

8. Fixed triangle

8.1. If ${\cal F}_{\Delta}{\,=\,}\langle o,u,v\rangle$ is a triangle, if ${\Delta}$ is semi-simple, and if $\dim{\Delta}{\,\geq\,}21$, then ${\Delta}$ is a Lie group.

Proof. For groups of dimension ${\geq\,}22$, the claim is true by [17] 9.2 Corollary. Hence we may suppose that $\dim{\Delta}{\,=\,}21$. Then ${\Delta}$ is almost simple or one of the following holds: (i) ${\Delta}$ has a $15$-dimensional factor, (ii) ${\Delta}$ is a product of $3$ factors of dimensions $10,8,3$, (iii) each factor of ${\Delta}$ has dimension $3$ or $6$. Let again ${Z}{\,=\,}{\rm Cs}_{\hskip 1.0pt}{\hskip 0.5pt{\Delta}}$ be the center of ${\Delta}$ and suppose that ${Z}$ is not a Lie group. Note that ${Z}$ acts freely on the complement $D$ of $ou{\,\kern 2.0pt{\scriptstyle\cup}\kern 2.0pt\,}ov{\,\kern 2.0pt{\scriptstyle\cup}\kern 2.0pt\,}uv$; in fact, $p^{\zeta}{=\,}p$,  $1\kern-3.0pt{\rm l}{\,\neq\,}\zeta{\hskip 1.0pt\in\hskip 1.0pt}{Z}$ implies ${\cal D}{\,=\,}\langle p^{\Delta},o,u,v\rangle{\,<\,}{\cal P}$. Put ${\Delta}|_{\cal D}{\,=\,}{\Delta}/{K}$. Then ${\Delta}{:}{K}{\,\leq\,}11$  (see [19] 83.26),  $\dim{K}{\,\geq\,}10$,  ${\cal D}$ is flat by stiffness, ${\Delta}|_{\cal D}$ is trivial, and ${K}$ is too large.

(a) If ${\Delta}$ is almost simple, then ${\Delta}$ is a Lie group, or $\overline{\Delta}{\,=\,}{\Delta}/{Z}$ is isomorphic to ${\rm O}^{\prime}_{7}{(}{R},2)$ or to ${\rm PSp}_{6}{{R}}$.

(b) In the first case, $\overline{\Delta}$ has a subgroup ${\rm SO}_{5}{{R}}$; it can be lifted to a group Yu${\,\cong\,}{\rm U}_{2}{{H}}{\,\leq\,}{\Delta}$, see [19] 94.27 or 2.9, and note that by 2.4 there is no subgroup ${\rm SO}_{5}{{R}}$ in ${\Delta}$. The central involution $\iota$ of Yu is not planar (or else Yu$|_{{\cal F}_{\iota}}{\,\cong\,}{\rm SO}_{5}{{R}}$ contrary to 2.4). Therefore $\iota$ is a reflection; it is in the kernel of the action of ${\Delta}$ on one side of the fixed triangle and hence $\iota{\,\in\,}{Z}$; but then $\iota{\,=\,}\overline{\iota}$ would be trivial.

(c) In the second case, the subgroup ${\rm SU}_{3}{{C}}$ of $\overline{\Delta}$ can be lifted to an isomorphic copy ${\Omega}{\,<\,}{\Delta}$, and ${\Omega}$ is contained in a maximal compact subgroup ${\Xi}{\,<\,}{\Delta}$; moreover $\dim{\Xi}{\,=\,}7$ and ${\Xi}{\,=\,}{\Theta}{\Omega}$ is connected, ${\Theta}$ is commutative and ${\Theta}{\,\leq\,}{\rm Cs}_{\hskip 1.0pt}{\hskip 0.5pt{\Omega}}$  ([19] 93.10,11). Let $x$ be an arbitrary point in $D$  (the complement of the fixed lines). Then ${Z}$ acts freely on the subplane ${\cal D}{\,=\,}{\cal F}_{\hskip-2.0pt{\Delta}_{x}}$, and ${\Delta}_{x}$ is a Lie group because ${\Delta}_{x}\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{Z}{\,=\,}1\kern-3.0pt{\rm l}$. If ${\Delta}$ has an $8$-dimensional orbit on two sides of the fixed triangle, then ${\Delta}$ induces a Lie group on these orbits (see 2.6), and ${\Delta}$ itself would be a Lie group. Consequently $\dim x^{\Delta}{\,\leq\,}14$ and $\dim{\Delta}_{x}{\geq\,}7$. As ${Z}$ is not a Lie group, 2.12 implies $\dim{\cal D}{\,\in\,}\{0,8\}$. In the second case, Stiffness 2.5(b̂) shows that ${\Delta}_{x}{\,\cong\,}{\rm Spin}_{3}{{R}}$. Hence $\dim{\cal D}{\,=\,}0$ for each $x{\,\in\,}D$. There are $3$ pairwise commuting conjugate involutions in ${\Omega}$; these are planar because axes of reflections are fixed by ${\Delta}$. An involution $\iota{\,\in\,}{\Omega}$ is centralized in ${\Omega}$ by a group ${\Phi}{\,\cong\,}{\rm SU}_{2}{{C}}$. In the case ${\Phi}|_{{\cal F}_{\iota}}{\,\neq\,}1\kern-3.0pt{\rm l}$ there would exist planar involutions on ${\cal F}_{\iota}$, and ${Z}$ would be a Lie group. Hence ${\cal F}_{\iota}{\,=\,}{\cal F}_{\Phi}$. All involutions $\kappa{\,\in\,}{\Omega}$ such that $\iota\kappa{\,=\,}\kappa\iota$ generate a subgroup ${\Psi}{\,\cong\,}{\rm SU}_{2}{{C}}$. As there are no $4$-dimensional subplanes, $\kappa$ and its conjugates induce reflections on ${\cal F}_{\iota}$. Therfore ${\Psi}|_{{\cal F}_{\iota}}$ is a group of homologies, say with center $o$.

On the other hand, there is a product ${\Gamma}{\Upsilon}{\,<\,}{\Delta}$ corresponding to ${\rm Sp}_{2}{{R}}{\times}{\rm Sp}_{4}{{R}}$ in a double covering of $\overline{\Delta}$  (use [19] 94.27). Note that ${\rm Sp}_{2}{{R}}{\,=\,}{\rm SL}_{2}{{R}}$ and that ${\rm Sp}_{4}{{R}}$ has a subgroup ${\rm SU}_{2}{{C}}$. Suppose that the latter group is mapped onto a group ${\rm SO}_{3}{{R}}$ in ${\Upsilon}$. The arguments above show that ${\Upsilon}$ contains a circle group of homologies, but this is impossible, since ${\Upsilon}$ is almost simple. Therefore ${\Upsilon}$ contains a central involution. Up to conjugacy, $\iota{\,\in\,}{\rm Cs}_{\hskip 1.0pt}{\hskip 0.5pt\hskip-3.0pt{\Upsilon}}$, and $\dim({\Theta}{\Upsilon})|_{{\cal F}_{\iota}}{\,\geq\,}10$. According to [18] 8.0, it follows that $({\rm Cs}_{\hskip 1.0pt{\Delta}}{\hskip 0.5pt\iota})^{1}|_{{\cal F}_{\iota}}$ is a Lie group. Recall that ${Z}$ acts freely on $D$. Hence ${Z}$ is a Lie group, and so is ${\Delta}$

(i) A $15$-dimensional factor ${\Upsilon}$ is a Lie group or ${\Upsilon}/{Z}$ is locally isomorphic to ${\rm SU}_{4}{(}{C},r),r=1,2$. The group ${\Delta}$ is an almost direct product ${\Upsilon}{\Gamma}$. By stiffness, $p^{\hskip-1.5pt{\Upsilon}}{\neq\,}p$ for each $p{\,\in\,}D$, and then $\langle p^{\hskip-2.0pt{\Upsilon}\mskip-3.0mu},o,u,v\rangle{\,=\,}{\cal P}$. Consequently, ${\Gamma}$ acts freely on $D$ and \langle p^{\Gamma}\mskip-3.0mu,o,u,v\rangle{\,\leq\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 2.0pt\hbox{\scriptscriptstyle\bullet}\mskip-3.0mu\,}{\cal P} by 2.12.

(d) Suppose in case (i) that $r{\,=\,}1$. Then ${\Upsilon}/{Z}$ has a subgroup ${\rm SU}_{3}{{C}}$. As in step (c) there is a group ${\Omega}{\,\cong\,}{\rm SU}_{3}{{C}}$ in ${\Upsilon}$, and all involutions in ${\Omega}$ are planar. Choose some involution $\iota{\,\in\,}{\Omega}$. Using the arguments and notation of step (c), we obtain the following: ${\rm SU}_{2}{{C}}{\,\cong\,}{\Phi}{\,\leq\,}{\rm Cs}_{\hskip 1.0pt{\Omega}}{\hskip 0.5pt\iota}$, ${\cal F}_{\hskip-2.0pt\iota}{=\,}{\cal F}_{\Phi}$, ${\rm SU}_{2}{{C}}{\,\cong\,}{\Psi}{\,\leq\,}{\rm Cs}_{\hskip 1.0pt{\Omega}}{\hskip 0.5pt\iota}$, and ${\Psi}|_{{\cal F}_{\iota}}$ is a group of homologies. The center ${Z}$ acts freely on $D\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\cal F}_{\iota}$. Hence the kernel of ${\Gamma}|_{{\cal F}_{\Phi}}$ is a compact Lie group; by stiffness $\dim{\Gamma}|_{{\cal F}_{\iota}}{\,=\,}6$, and ${\Theta}{\Psi}{\Gamma}$ induces a $10$-dimensional group on ${\cal F}_{\iota}$. According to [18] 8.0 the induced group is a Lie group. Therefore ${Z}$ and ${\Delta}$ are Lie groups.

(e) If $r{\,=\,}2$, then ${\Upsilon}/{Z}{\,\cong\,}{\rm O}^{\prime}_{6}{(}{R},2)$ has a subgroup ${\rm SO}_{4}{{R}}$. By 2.9 there is a subgroup ${X}{\,<\,}{\Upsilon}$ which is isomorphic to ${\rm SO}_{4}{{R}}$ or to ${\rm Spin}_{4}{{R}}$. If ${X}$ contains a reflection, say $\iota{\,\in\,}{\Delta}_{[o,uv]}$, then $\iota{\,\in\,}{Z}$  (or else $\langle\iota^{\hskip-2.0pt{\Upsilon}}\rangle{\,=\,}{\Upsilon}{\,\leq\,}{\Delta}_{[o,uv]}$, which is impossible). Recall that ${\Gamma}$ acts freely on $D$. Suppose first that ${X}{\,\cong\,}{\rm SO}_{4}{{R}}$. Then all involutions in ${X}$ are planar. Let $\beta$ denote the central involution in ${X}$. There is a subgroup ${\Phi}{\,\cong\,}{\rm SO}_{3}{{R}}$ in ${X}$. The kernel of ${X}|_{{\cal F}_{\beta}}$ is a compact Lie group; by stiffness it is contained in ${\rm Spin}_{3}{{R}}$. Therefore ${\Phi}$ acts faithfully on ${\cal F}_{\beta}$, and {\cal C}{\,=\,}{\cal F}_{\alpha,\beta}{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal F}_{\beta} for each involution $\alpha{\,\in\,}{\Phi}$, but ${\cal C}^{\Gamma}{=\,}{\cal C}$, and orbits of ${\Gamma}$ on $D\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{\cal C}$ are $6$-dimensional, a contradiction. Consequently ${X}{\,\cong\,}{\rm Spin}_{4}{{R}}{\,=\,}({\rm Spin}_{3}{{R}})^{2}$. Only one of the $3$ involutions in ${X}$ can be contained in the center ${Z}$. Let $\beta$ be a planar involutin in ${X}$ such that ${X}/\langle\beta\rangle{\,\cong\,}{\rm SO}_{3}{{R}}{\times}{\rm Spin}_{3}{{R}}$. If $\iota$ is any involution in the facor ${\Phi}{\,\cong\,}{\rm SO}_{3}{{R}}$, then $\iota|_{{\cal F}_{\beta}}{\,=\,}1\kern-3.0pt{\rm l}$ and, as ${\Phi}$ is simple, even ${\Phi}|_{{\cal F}_{\beta}}{\,=\,}1\kern-3.0pt{\rm l}$, but this contradicts stiffness 2.5(b̂).

(ii) ${\Delta}$ has $3$ almost simple factors ${\Upsilon},{\Psi},{\Gamma}$ of dimensions $10,8,3$, respectively. Recall that a semi-simple group of an $8$-dimensional plane with a triangle of fixed elements has dimension at most $9$  ([15] 6.1 = ($\dagger$) ). By stiffness it follows that $\langle p^{\Delta},o,u,v\rangle{\,=\,}{\cal P}$. If $p^{\Upsilon}{\,=\,}p$, then $\langle p^{{\Psi}{\Gamma}}\rangle{\,=\,}{\cal P}$, ${\Upsilon}_{\hskip-2.0ptp}{\,=\,}1\kern-3.0pt{\rm l}$, and $p^{\hskip-2.0pt{\Upsilon}}{\neq\,}p$. By ($\dagger$) we have $\langle p^{\Upsilon}\rangle{\,=\,}{\cal P}$. If {\cal D}{\,=\,}\langle p^{{\Psi}{\Gamma}}\rangle{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}, then ($\dagger$) implies ${\Gamma}|_{\cal D}{\,=\,}1\kern-3.0pt{\rm l}$, and ${\Gamma}$ is a compact Lie group. Stiffness shows ${\Gamma}{\,\cong\,}{\rm Spin}_{3}{{R}}$. The involution $\iota{\,\in\,}{\Gamma}$ is planar, and ${\cal F}_{\hskip-2.0pt\iota}^{\Delta}{\,=\,}{\cal F}_{\iota}$, a contradiction. Therefore $\langle p^{{\Psi}{\Gamma}}\rangle{\,=\,}{\cal P}$, and hence ${\Upsilon}_{\hskip-3.0ptp}{\,=\,}{\Psi}_{\hskip-2.0ptp}{\,=\,}1\kern-3.0pt{\rm l}$. The different possibilities will be treated separately.

(f) If ${\Upsilon}$ is the compact group ${\rm U}_{2}{{H}}$, then each non-central involution $\iota$ in ${\Upsilon}$ is planar, since $\langle\iota^{\mskip-3.0mu{\Upsilon}}\rangle{\,=\,}{\Upsilon}$ is not axial. But then $\iota{\,\in\,}{\Upsilon}_{\mskip-3.0mup}$ for each fixed point $p$ of $\iota$. This contradicts the statements at the end of (ii). Analogous arguments show that ${\Psi}$ is not compact.

(g) If ${\Upsilon}$ is locally isomorphic to ${\rm U}_{2}{(}{H},1)$, then ${\Upsilon}$ is simply connected, because ${\rm O}^{\prime}_{5}{(}{R},1)$ has a subgroup ${\Phi}{\,\cong\,}{\rm SO}_{3}{{R}}$ and ${\cal F}_{\Phi}$ is flat. Therefore ${\Upsilon}$ has a maximal compact subgroup ${X}{\,\cong\,}{\rm Spin}_{4}{{R}}$. Again a non-central involution $\beta$ is planar, which contradicts (ii).

(h) Only the possibility ${\Upsilon}/{Z}{\,\cong\,}{\rm(P)}{\rm Sp}_{4}{{R}}$ remains. Again ${\Delta}$ has no subgroup ${\rm SO}_{3}{{R}}$. Either ${\Psi}/{Z}$ is locally isomorphic to ${\rm SU}_{3}{(}{C},1)$ or ${\Psi}$ is a double covering ${\Pi}$ of ${\rm SL}_{3}{{R}}$. A subgroup ${\rm SU}_{2}{{C}}$ of the unitary group can be lifted to an isomorphic copy ${\Phi}{\,<\,}{\Psi}$; the involution $\iota{\,\in\,}{\Phi}$ is not central and hence it is planar, which contradicts (ii). Thus ${\Psi}{\,\cong\,}{\Pi}$. Both factors ${\Upsilon}$ and ${\Psi}$ have a subgroup ${\rm Spin}_{3}{{R}}$. The central involutions of these subgroups are reflections (because ${\Upsilon}_{\hskip-3.0ptp}{\,=\,}{\Psi}_{\hskip-2.0ptp}{\,=\,}1\kern-3.0pt{\rm l}$), and these are distinct, or else there would exist a subgroup ${\Omega}{\,\cong\,}{\rm SO}_{3}{{R}}$. A maximal compact subgroup of ${\Upsilon}{\Psi}$ is at most $7$-dimensional. It contains a subgroup ${X}{\,\cong\,}{\rm Spin}_{4}{{R}}$ without any planar involution.

(i) {\cal G}{\,=\,}\langle p^{\Gamma}\mskip-3.0mu,o,u,v\rangle{\,\mskip-3.0mu<\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 1.3pt\hbox{\scriptscriptstyle\bullet}}{\cal P}* for any $p{\,\in\,}D$*: if $p^{\Gamma}{=\,}p$, then $p^{\Delta}{\,\subseteq\,}{\cal F}_{\Gamma}{\,<\,}{\cal P}$ and $\dim{\Delta}_{p}{\,\geq\,}13$. Stiffness would imply ${\Delta}_{p}^{1}{\,\cong\,}{\rm G}_{2}$, which is not true. Hence ${\Gamma}$ acts almost effectively on ${\cal G}$ and {\cal G}{\,\leq\mskip-3.0mu\mskip-3.0mu\mskip-3.0mu\raise 2.0pt\hbox{\scriptscriptstyle\bullet}\mskip-3.0mu\,}{\cal P}. Equality is impossible, since $\dim({\Upsilon}{\Psi})_{p}{\,>\,}1$. Denote the connected component of $({\Upsilon}{\Psi})_{p}$ by ${\Lambda}$. By 2.6 we may assume that $\dim{\Lambda}{\,\geq\,}3$. Note that ${\Lambda}|_{\cal G}{\,=\,}1\kern-3.0pt{\rm l}$. Hence ${\Lambda}$ is compact. It follows that $\dim({\Lambda}\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{X}){\,\geq\,}2$. Obviously, each involution in  ${\Lambda}\kern 2.0pt{\scriptstyle\cap}\kern 2.0pt{X}$ is planar. This contradicts the last statement in step (h).

(iii) Each $6$-dimensional factor of ${\Delta}$ is locally and then even globally isomorpic to ${\rm SL}_{2}{{C}}$. There is at least one factor ${\Gamma}$ such that $\dim{\Gamma}{=\,}3$. We may assume that ${\Gamma}$ is not a Lie group. Then a maximal compact subgroup ${\Theta}$ of ${\Gamma}$ is a $1$-dimensional connected subgroup. Let ${\Xi}$ denote the product of the other factors of ${\Delta}$ and apply [17] 9.2 (B) to the group ${H}{\,=\,}{\Xi}{\Theta}$. As ${\Theta}$ is normal in ${H}$, we obtain $\dim{H}{\,<\,}18$ and $\dim{\Delta}{\,\leq\,}19$. $\square$

Summary

If ${\Theta}$ denotes a compact connected $1$-dimensional subgroup and if $\dim{\Delta}{\,\geq\,}k$, the number in the table, then ${\Delta}$ is a Lie group. Bold face entries improve corresponding results in [17].

[TABLE]

[TABLE]