Near polygons having a big sub near polygon isomorphic to $\mathbb G_n$

Abstract

In [7] a new infinite class $\mathbb G_n$, $n \geq 0$, of near polygons was defined. The near $2n$-gon $\mathbb G_n$ has the property that it contains $\mathbb G_{n-1}$ as a big geodetically closed sub near polygon. In this paper, we determine all near $2n$-gons, $n \geq 4$, having $\mathbb G_{n-1}$ as a big geodetically closed sub near $2(n-1)$-gon under the additional assumption that every two points at distance 2 have at least two common neighbours. We will prove that such a near $2n$-gon is isomorphic to either $\mathbb G_n$, $\mathbb G_{n-1} \otimes \mathbb G_2$, or $\mathbb G_{n-1} \times L$ for some line $L$.