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I'm reading Francis M. Choucroun, "Analyse harmonique des groupes d'automorphismes d'arbres de Bruhat-Tits", Mémoires de la S. M. F., tome 58 (1994), p. 1 - 166, and he speaks of a weakly doubly transitive or weakly triply transitive group action (in French). My first thought was that maybe this means that the group acts transitively on the unordered pairs or unordered tripes, but when you actually look at the arguments involving the notion that interpretation seems not to be supported. I was wondering if anyone could clarify what the meaning of "weakly doubly transitive" is in this paper.

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It's defined in Définition 1.4.1. Weakly doubly transitive means for him that the action is transitive on pairs of vertices $(x,y)$ and $(x',y')$ such that $d(x,y)=d(x',y')$. Weakly $k$-transitive means the same with all the pairwise distances.

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    $\begingroup$ In that case weakly doubly transitive is the same as distance-transitive in other parts of the literature... $\endgroup$
    – Nick Gill
    Commented May 22, 2013 at 8:38

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