Abstract

This is a semisurvey paper, where we start by advertising Tits’ synthetic construction (1953) of the hyperbolic plane $H^2(\mathrm{Cay}<!--FUNCTION\; APPLICATION-->)$ over the Cayley numbers $\mathrm{Cay}<!--FUNCTION\; APPLICATION-->$ and of its automorphism group which is the exceptional simple Lie group $G=F_{4(-20)}$. Let $G=\mathrm{KAN}<!--FUNCTION\; APPLICATION-->$ be the Iwasawa decomposition. Our contributions are:

• Writing down explicitly the action of $N$ on $H^2(\mathrm{Cay}<!--FUNCTION\; APPLICATION-->)$ in Tits’ model, facing the lack of associativity of $\mathrm{Cay}<!--FUNCTION\; APPLICATION-->$.

• If $\mathrm{MAN}<!--FUNCTION\; APPLICATION-->$ denotes the minimal parabolic subgroup of $G$, characterizing $M$ geometrically.

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