Abstract

For $q=p^n$ with $p$ an odd prime, the projective linear group $\mathrm{PGL}<!--FUNCTION\; APPLICATION-->(2,q)$ can be seen as the stabilizer of a conic $\mathcal{𝒪}$ in a projective plane $\pi =\mathrm{PG}<!--FUNCTION\; APPLICATION-->(2,q)$. In that setting, involutions of $\mathrm{PGL}<!--FUNCTION\; APPLICATION-->(2,q)$ correspond bijectively to points of $\pi$ not in $\mathcal{𝒪}$. Triples of involutions $\{{\alpha }_P,{\alpha }_Q,{\alpha }_R\}$ of $\mathrm{PGL}<!--FUNCTION\; APPLICATION-->(2,q)$ can then be seen also as triples of points $\{P,Q,R\}$ of $\pi$. We investigate the interplay between algebraic properties of the group $H=⟨{\alpha }_P,{\alpha }_Q,{\alpha }_R⟩$ generated by three involutions and geometric properties of the triple of points $\{P,Q,R\}$. In particular, we show that the coset geometry $\Gamma =\Gamma (H,(H_0,H_1,H_2))$, where $H_0=⟨{\alpha }_Q,{\alpha }_R⟩,H_1=⟨{\alpha }_P,{\alpha }_R⟩$ and $H_2=⟨{\alpha }_P,{\alpha }_Q⟩$ is a regular hypertope if and only if $\{P,Q,R\}$ is a strongly non self-polar triangle, a terminology we introduce. This entirely characterizes hypertopes of $\mathrm{rank}<!--FUNCTION\; APPLICATION-->3$ with automorphism group a subgroup of $\mathrm{PGL}<!--FUNCTION\; APPLICATION-->(2,q)$. As a corollary, we prove the existence of hypertopes of $\mathrm{rank}<!--FUNCTION\; APPLICATION-->3$ with nonlinear diagrams and with automorphism group $\mathrm{PGL}<!--FUNCTION\; APPLICATION-->(2,q)$, for any $q=p^n$ with $p$ an odd prime and $n$ a positive integer. We also study in more details the case where the triangle $\{P,Q,R\}$ is tangent to $\mathcal{𝒪}$.

Keywords

Mathematical Subject Classification

Milestones

Authors