A Graph with a Locally Projective Vertex-Transitive Group of Automorphisms Aut($Fi_{22}$) Which Has a~Nontrivial Stabilizer of a Ball of Radius~$2$
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 4, pp. 274-278
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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 $Fi_{22}$.
@article{TIMM_2023_29_4_a21,
author = {V. I. Trofimov},
title = {A {Graph} with a {Locally} {Projective} {Vertex-Transitive} {Group} of {Automorphisms} {Aut(}$Fi_{22}$) {Which} {Has} {a~Nontrivial} {Stabilizer} of a {Ball} of {Radius~}$2$},
journal = {Trudy Instituta matematiki i mehaniki},
pages = {274--278},
publisher = {mathdoc},
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language = {ru},
url = {http://geodesic.mathdoc.fr/item/TIMM_2023_29_4_a21/}
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V. I. Trofimov. A Graph with a Locally Projective Vertex-Transitive Group of Automorphisms Aut($Fi_{22}$) Which Has a~Nontrivial Stabilizer of a Ball of Radius~$2$. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 4, pp. 274-278. http://geodesic.mathdoc.fr/item/TIMM_2023_29_4_a21/