A radius 1 irreducibility criterion for lattices in products of trees
Annales Henri Lebesgue, Volume 5 (2022), pp. 643-675.


Keywords Irreducible lattice, groups acting on trees, just-infinite group, $2$-transitive group


Let T 1 ,T 2 be regular trees of degrees d 1 ,d 2 3. Let also ΓAut(T 1 )×Aut(T 2 ) be a group acting freely and transitively on VT 1 ×VT 2 . For i=1 and 2, assume that the local action of Γ on T i is 2-transitive; if moreover d i 7, assume that the local action contains Alt(d i ). We show that Γ is irreducible, unless (d 1 ,d 2 ) belongs to an explicit small set of exceptional values. This yields an irreducibility criterion for Γ that can be checked purely in terms of its local action on a ball of radius 1 in T 1 and T 2 . Under the same hypotheses, we show moreover that if Γ is irreducible, then it is hereditarily just-infinite, provided the local action on T i is not the affine group F 5 F 5 * . The proof of irreducibility relies, in several ways, on the Classification of the Finite Simple Groups.


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