V.I. Trofimov. A graph with a locally projective vertex-transitive group of automorphisms $\mathrm{Aut}(Fi_{22})$ which has a nontrivial stabilizer of a ball of radius 2 ... P. 274-278

Earlier, to confirm that one of the possibilities for the structure of vertex stabilizers of graphs with projective suborbits is realizable, we announced the existence of a connected graph $\Gamma$ admitting a group of automorphisms $G$ which is isomorphic to Aut$(Fi_{22})$ and has the following properties. First, the group $G$ acts transitively on the set of vertices of $\Gamma$, but intransitively on the set of $3$-arcs of $\Gamma$. Second, the stabilizer in $G$ of a vertex of $\Gamma$ induces on the neighborhood of this vertex a group $PSL_3(3)$ in its natural doubly transitive action. Third, the pointwise stabilizer in $G$ of a ball of radius 2 in $\Gamma$ is nontrivial. In this paper, we construct such a graph $\Gamma$ with $G ={\rm Aut}(\Gamma)$.

Keywords: graph, transitive locally projective group of automorphisms, Fischer group $Fi_{22}$

Received September 26, 2023

Revised October 6, 2023

Accepted October 9, 2023

Funding Agency: This work was performed as a part of the research conducted in the Ural Mathematical Center and supported by the Ministry of Education and Science of the Russian Federation (agreement no. 075-02-2023-935).

Vladimir Ivanovich Trofimov, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Prof., Ural Federal University, Yekaterinburg, 620083 Russia, e-mail: trofimov@imm.uran.ru

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Cite this article as: V.I. Trofimov. A graph with a locally projective vertex-transitive group of automorphisms $\mathrm{Aut}(Fi_{22})$ which has a nontrivial stabilizer of a ball of radius 2. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 274–278; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 323, Suppl. 1, pp. S300–S304.