On exceptional homogeneous compact geometries of type \(\mathsf{C}_3\) (Q1691966)

The authors study the exceptional homogeneous compact geometries of type \(C_3\). There are two such geometries and they are associated with the groups \(\mathrm{SU}(3) \times \mathrm{SU}(3)\) and \(\mathrm{SO}(3) \times G_2\) on the octonion plane \(\mathbb{OP}^2\). It follows from \textit{L. Kramer} and \textit{A. Lytchak} [Transform. Groups 19, No. 3, 793--852 (2014; Zbl 1309.51003)] that a homogeneous compact geometry of type \(C_3\) is either covered by a building or isomorphic to one of two exceptional geometries. In the paper under review, both geometries are constructed in an uniform way. Furthermore, it is shown that these geometries are simply connected and their full automorphism groups are determined. In their construction, the authors consider two composition algebras \(\mathbb{A}\) and \(\mathbb{B}\) over \(\mathbb{R}\) that contain either \(\mathbb{R}\) or \(\mathbb{C}\) as a common subfield \(k\) and such that \(\mathbb{A}\) is 4-dimensional over \(k\) and the \(k\)-dimension of \(\mathbb{B}\) is at least 4. By Hurwitz's theorem on composition algebras, there are only three cases for the triple \((k,\mathbb{A},\mathbb{B})\). Points of the geometry \(\Gamma\) are the 1-dimensional subspaces in the \(k\)-vector space \(\mathrm{Pu}_k(\mathbb{A})\) of \(k\)-pure elements of \(\mathbb{A}\). Lines are of the form \([a, b]\) where \(a\in\mathrm{Pu}_k(\mathbb{A})\), \(b\in\mathrm{Pu}_k(\mathbb{B})\) and planes of \(\Gamma\) are formed by the embeddings \(\phi:\mathbb{A}\to\mathbb{B}\) as composition algebras over \(k\). This geometry is flat in the sense that every point is incident with every plane. The group \(G=\mathrm{Aut}_k\mathbb{A}\times \mathrm{Aut}_k\mathbb{B}\) induces a group of automorphisms of \(\Gamma\). It is shown that \(G\) is the full automorphism group in the two cases where \(k=\mathbb{R}\). In the third case, where \(k=\mathbb{C}\) the full automorphism group of \(\Gamma\) is obtained from \(\mathrm{SU}(3) \times\mathrm{SU}(3)\) by factoring out a central cyclic group of order 3 and then extending by a group of order two that arises from complex conjugation. An explicit description of the covering building is provided in case \(\mathbb{A}=\mathbb{B}=\mathbb{H}\) the quaternions with group \(\mathrm{SO}(3) \times\mathrm{SO}(3)\). In the last section, which comprises nearly half the paper, it is verified that the geometry \(\Gamma\) is either simply connected or covered by a building. This is achieved by using the edge-path group of \(\Gamma\) to study the universal cover of \(\Gamma\). The latter admits a compact topology with a flag-transitive continuous group acting on it and thus is known by the work of Kramer and Lytchak [loc. cit.].