Locally projective graphs and their densely embedded subgraphs

Abstract

AbstractThe article contributes to the classification project of locally projective graphs and their locally projective groups of automorphisms outlined in Chapter 10 of Ivanov (The Mathieu Groups, Cambridge University Press, Cambridge, 2018). We prove that a simply connected locally projective graph $$\Gamma $$ Γ of type (n, 3) for $$n \ge 3$$ n ≥ 3 contains a densely embedded subtree provided (a) it contains a (simply connected) geometric subgraph at level 2 whose stabiliser acts on this subgraph as the universal completion of the Goldschmidt amalgam $$G_3^1\cong \{S_4 \times 2,S_4 \times 2\}$$ G 3 1 ≅ { S 4 × 2 , S 4 × 2 } having $$S_6$$ S 6 as another completion, (b) for a vertex x of $$\Gamma $$ Γ the group $$G_{\frac{1}{2}}(x)$$ G 1 2 ( x ) which stabilizes every line passing through x induces on the neighbourhood $$\Gamma (x)$$ Γ ( x ) of x the (dual) natural module $$2^n$$ 2 n of $$G(x)/G_{\frac{1}{2}}(x) \cong L_n(2)$$ G ( x ) / G 1 2 ( x ) ≅ L n ( 2 ) , (c) G(x) splits over $$G_{\frac{1}{2}}(x)$$ G 1 2 ( x ) , (d) the vertex-wise stabilizer $$G_1(x)$$ G 1 ( x ) of the neighbourhood of x is a non-trivial group, and (e) $$n \ne 4$$ n ≠ 4 .