Moufang sets related to polarities in exceptional Moufang quadrangles of type \(F_4\) (Q447760)

A Moufang set is a triple \((X, G, (U_x)_{x\in X})\), where \(X\) is a set, \(G\) is a permutation group on \(X\), \(U_x\) is a subgroup of the stabilizer \(G_x\) acting regularly on \(X\backslash\{x\}\) for all \(x\in X\), and \(gU_xg^{-1} = U_{g(x)}\) for all \(x\in X\) and all \(g\in G\). The full projective group of this Moufang set is the group of all elements of \(\mathrm{Sym}(X)\) that leave the set of root groups invariant.NEWLINENEWLINELet \(Q\) be an exceptional Moufang quadrangle of type \(F_4\). \textit{B. Mühlherr} and \textit{H. Van Maldeghem} have given necessary and sufficient conditions for when such quadrangles admit a polarity [Can. J. Math. 51, No. 2, 347--371 (1999; Zbl 0942.51002)]. Let us assume that \(Q\) satisfies these conditions, and let \(\pi\) be a polarity. Then the absolute flags of \(\pi\) form a Moufang set \(\mathcal{M}\), and it is \(\mathcal{M}\) which is the main object of interest here.NEWLINENEWLINEThe author uses techniques that have been used elsewhere in the study of Moufang sets arising from the Suzuki-Tits quadrangle [\textit{H. Van Maldeghem}, Eur. J. Comb. 28, No. 7, 1878--1889 (2007; Zbl 1124.51004)] to construct a rank 3 geometry \(\Omega\), for which he proves the following result: The automorphism group of \(\Omega\) is isomorphic to both the subgroup of the automorphism group of \(Q\) centralizing \(\pi\), and to the full projective group of \(\mathcal{M}\).NEWLINENEWLINEThis result completes a program of study of Moufang sets admitting a polarity, and allows the author to state the following general corollary: Let \(P\) be a Moufang \(n\)-gon admitting a polarity \(\pi\) and let \(\phi\) be an automorphism of \(P\) that stabilizes the set of absolute points of \(\pi\). Then either \(\phi\) stabilizes the set of absolute lines of \(\pi\) and centralizes \(\pi\), or \(P\) is in a known list of exceptions.