Graphs which are locally Grassmann
Let \(\Gamma\) be an undirected graph and let \(G\) be a subgroup of \(\text{Aut} (\Gamma)\). For each vertex \(x\) of \(\Gamma\), let \(\Gamma_x\) denote the set of vertices adjacent to \(x\) in \(\Gamma\) and let \(G^{\Gamma_x}_x\) denote the permutation group induced on \(\Gamma_x\) by the stabilizer \(G_x\) of \(x\) in \(G\). For each \(i\geq 1\), let \(G^{[i]}_x\) denote the pointwise stabilizer in \(G_x\) of the set of vertices at distance at most \(i\) from \(x\) in \(\Gamma\). Let \[ G_{xy\dots z}=G^{[i]}_x\cap G^{[i]}_y\cap\dots\cap G^{[i]}_x \] for each \(i\geq 1\) and any set of vertices \(x, y,\dots, z\) of \(\Gamma\). An \(s\)-path (or \(s\)-arc) is an \((s+1)\)-tuple \((x_0, x_1,\dots, x_s)\) of vertices such that \(x_i\in\Gamma_{x_{i-1}}\) for \(1\leq i\leq s\) and \(x_i\neq x_{i-2}\) for \(2\leq i\leq s\). Let \(q\) be a prime power and let \(V\) be a vector space of dimension \(n\) over \(\mathbb{F}_q\). Let \({V\brack m}\) denote the set of \(m\)-dimensional subspaces of \(V\) and denote by \(L_n(q)^{{V\brack m}}\) the permutation group induced by the special linear group \(L_n(q)\) on \({V\brack m}\). This paper works with the following two conditions: (*) \(\Gamma\) is connected, \(G\) acts transitively on the vertices of \(\Gamma\) and \(\{x, y\}\) is an edge of \(\Gamma\) such that \(|G_x|<\infty\) and \(G^{[1]}_{xy}\neq 1\). (**) for each vertex \(x\) of \(\Gamma\), \(G^{\Gamma_x}_x\) contains a normal subgroup isomorphic to \(L_n(q)^{{V\brack m}}\) for some \(m\geq 1\), some \(n\geq 2m\) and some power \(q\) of a prime \(p\). The main result of this paper is: (1.1) Theorem Suppose that (*) and (**) hold. If \(n=2m\), then either (i) \(|G^{[1]}_{xy}|=q^{m^2}\) or (ii) \(n=2\), \(p=2\) and \(|G^{[1]}_{xy}|=q^2\) or (iii) \(n=2\), \(p=3\) and \(|G^{[1]}_{xy}|=q^4\). Moreover \(G^{[1]}_{x_0, x_1,\dots,x_k}=1\) for every \(k\)-path in a certain non-empty set (defined in (3.1)) where \(k=2\) in (i), \(k=3\) in (ii) and \(k=5\) in (iii). This result yields: (1.3) Theorem Suppose that \(G\) is transitive on the vertex set of \(\Gamma\) and that \(\{x,y\}\) is an edge of \(\Gamma\) such that \(|G_x|<\infty\) and \(G^{\Gamma_x}_x\) is 2-transitive. Then \(G^{[5]}_{xy}=1\). (In particular, \(|G_x|\) is bounded by a function of the valency \(|\Gamma_x|\) of \(\Gamma\)). The author conjectures that the conclusion of (1.3) holds under the weaker hypothesis that \(G^{\Gamma_x}_x\) is a primite permutation group.
scientific article; zbMATH DE number 537352