Regular near polygons

Abstract

In this chapter $\mathcal{S} = (\mathcal{P},\mathcal{L},I)$ denotes a finite regular near 2d-gon, d≥2, with parameters s, t, t _i (i∈{0, ..., d}, i.e. $\mathcal{S}$ has order (s, t) and for every two points x and y at distance i there are t _i +1 lines through y containing a (necessarily unique) point of Γ_i+1(x). Notice that t₀=−1, t₁=0 and t _d =t. The number |Γ _i (x)|, i∈ {0, ..., d}, is independent from the chosen point x and equal to

$$ k_i = \frac{{s^i \prod _{j = 0}^{i - 1} (t - t_j )}} {{\prod _{j = 0}^{i - 1} (t - t_j )}}. $$

The total number of points is equal to

$$ v = k_0 + k_1 + \cdots + k_d . $$

If x, y and z are points such that d(x, y)=i, d(y, z)=1 and d(x, z)=i+1, then Γ₁(x)∩Γ_i−1(y)⊆Γ₁(x)∩Γ₁(z). As a consequence, t _i ≤ t _i +1 for every i ∈ {0, ..., d−1}. Since Γ _d (x)≠0 for at least one (and hence all) point(s) of $\mathcal{S}$, we must have that t _d −1 ≠ t.