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A classification of finite locally 2-transitive generalized quadrangles. (English) Zbl 07313190
A generalized quadrangle is an incidence geometry of points and lines such that every pair of distinct points determines at most one line and every line contains at least two distinct points, satisfying the GQ Axiom: Given a point $$P$$ and a line $$\ell$$ not incident with $$P,$$ there is a unique point on $$\ell$$ collinear with $$P.$$
One of the outstanding open questions in the area is the classification of flag-transitive finite generalized quadrangles, that is, the classification of all finite generalized quadrangles with a group of collineations that is transitive on incident point-line pairs.
It has been conjectured by W.M. Kantor that the only non-classical finite flag-transitive generalized quadrangles are (up to duality) either the unique generalized quadrangle of order $$(3, 5)$$ or the generalized quadrangle of order $$(15, 17)$$ arising from the Lunelli-Sce hyperoval.
We recall that an antiflag is a non-incident point-line pair. Notice that, by the GQ Axiom, antiflag-transitivity implies flag-transitivity for a generalized quadrangle.
In [J. Bamberg et al., Trans. Amer. Math. Soc. 370, 1551–1601 (2018, Zbl 1381.51001)] the authors have proved that, up to duality, the only non-classical antiflag-transitive generalized quadrangle is the unique generalized quadrangle of order $$(3, 5).$$
If $$G$$ is a subgroup of collineations of a finite generalized quadrangle $$Q$$ that is transitive both on pairs of collinear points and on pairs of concurrent lines, then $$Q$$ is said to be a locally $$(G, 2)-$$transitive generalized quadrangle. The authors prove the following theorem:
Theorem 1 If $$Q$$ is a thick finite locally $$(G, 2)$$-transitive generalized quadrangle and $$Q$$ is not a classical generalized quadrangle, then (up to duality) $$Q$$ is the unique generalized quadrangle of order $$(3, 5).$$
An equivalent definition of locally $$2-$$transitive generalized quadrangle is that it has an incidence graph that is locally $$2$$-arc-transitive. Since an equivalent definition of an antiflag-transitive generalized quadrangle is that it has a locally $$3-$$arc-transitive incidence graph, Theorem 1 provides further progress towards Kantor conjecture.

##### MSC:
 51E12 Generalized quadrangles and generalized polygons in finite geometry 20B05 General theory for finite permutation groups 20B15 Primitive groups 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
GAP; Magma
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