On the hyperbolic unitary geometry
Let \(U_n\) denote an \(n\)-dimensional vector space over \({\mathbb F}_{q^2}\) endowed with a non-degenerate Hermitian form. The hyperbolic line graph \(G(U_n)\) is the graph whose vertices are the hyperbolic lines, i.e. the non-degenerate two-dimensional subspaces of \(U_n\), in which two vertices \(l\) and \(m\) are adjacent if and only if \(l\) is perpendicular to \(m\) with respect to the unitary form. A graph \(\Gamma\) is locally homogeneous if and only if for any pair \(x, y\) of vertices of \(\Gamma\), the induced subgraphs \(\Gamma(x)\) and \(\Gamma(y)\) on the set of neighbors of \(x\) and \(y\) are isomorphic; \(\Gamma\) is said to be locally \(\Delta\), if the \(\Gamma(x)\) are isomorphic to \(\Delta\) for all vertices \(x\) of \(\Gamma\). The main result of this paper is: If \(U_n\) is \(n\)-dimensional vector space over \({\mathbb F}_{q^2}\), \(n\geq 7\), \(q\) a prime power, \(\Gamma\) a connected graph that is locally \(G(U_n)\), then \(\Gamma\) is isomorphic to \(G(U_{n+2})\), unless \((n, q)=(7,2)\). This result implies a theorem in group-theoretic language, which is only stated, given that this kind of translation ``has been thoroughly studied'' and ``is well-understood.'' For \(n\geq 8\) this result has been stated without proof in [the second author, On graphs, geometries, and groups of Lie type. Eindhoven: Technische Univ. Eindhoven (2002; Zbl 1031.51001)]. The case \(n=7\) turned out to be much more complicated and was arrived ``during the preparation of the first author's Ph.D. thesis.''
scientific article; zbMATH DE number 5721558
scientific article