On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane

In 1987 Brehm and Kühnel showed that any triangulation of a $d$-manifold (without boundary) that is not homeomorphic to a sphere has at least $3d/2+3$ vertices. Moreover, triangulations with exactly $3d/2+3$ vertices can exist only for ‘manifolds like projective planes’, which can have dimension $2$, $4$, $8$ or $16$ only. There is a $6$-vertex triangulation of the real projective plane $\mathbb{RP}^2$, a $9$-vertex triangulation of the complex projective plane $\mathbb{CP}^2$ and $15$-vertex triangulations of the quaternionic projective plane $\mathbb{HP}^2$. Recently the author constructed first examples of $27$-vertex triangulations of manifolds like the octonionic projective plane $\mathbb{OP}^2$. The four most symmetric of them have the symmetry group $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ of order $351$. These triangulations were constructed using specially designed software after the symmetry group had been guessed. However, it remained unclear why exactly this group is realized as a symmetry group and whether $27$-vertex triangulations of manifolds like $\mathbb{OP}^2$ with other (possibly larger) symmetry groups exist. In this paper we find strong restrictions on the symmetry groups of such $27$-vertex triangulations. Namely, we present a list of $26$ subgroups of $\mathrm{S}_{27}$ containing all possible symmetry groups of $27$-vertex triangulations of manifolds like the octonionic projective plane. (We do not know whether all these subgroups can be realized as symmetry groups.) The group $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ is the largest group in this list, and the orders of all other groups do not exceed $52$. A key role in our approach is played by the use of results of Smith and Bredon on the topology of fixed-point sets of finite transformation groups.
Bibliography: 36 titles.