Stabilizers of actions of lattices in products of groups

We prove that any ergodic non-atomic probability-preserving action of an irreducible lattice in a semisimple group, with at least one factor being connected and of higher-rank, is essentially free. This generalizes the result of Stuck and Zimmer [Stabilizers for ergodic actions of higher rank semisimple groups. Ann. of Math. (2)139(3) (1994), 723–747], who found that the same statement holds when the ambient group is a semisimple real Lie group and every simple factor is of higher-rank. We also prove a generalization of a result of Bader and Shalom [Factor and normal subgroup theorems for lattices in products of groups. Invent. Math.163(2) (2006), 415–454] by showing that any probability-preserving action of a product of simple groups, with at least one having property $(T)$, which is ergodic for each simple subgroup, is either essentially free or essentially transitive. Our method involves the study of relatively contractive maps and the Howe–Moore property, rather than relying on algebraic properties of semisimple groups and Poisson boundaries, and introduces a generalization of the ergodic decomposition to invariant random subgroups, which is of independent interest.

Transitive simple subgroups of wreath products in product action

A transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a definition and detailed examination of ‘Cartesian decompositions’ of the permuted set, relating them to certain ‘Cartesian systems of subgroups’. These concepts, and the bijective connections between them, are explored in greater generality, with specific future applications in mind.

Groups with many characteristically simple subgroups

1. A group G is called characteristically simple if it has no proper non-trivial subgroups which are invariant under all automorphisms of G. It is known that if G is characteristically simple then each countable subgroup lies in a countable characteristically simple subgroup of G. A similar assertion holds for simple groups. These results were proved by Philip Hall in lectures in 1966, and further proofs appear in (4) and (6). For simple groups there is a well known and elementary result in the other direction: if every two-generator subgroup of a group G lies in a simple subgroup, then G is simple. These considerations prompt the question (first raised, I believe, by Philip Hall) whether a group G is necessarily characteristically simple if each countable subgroup lies in a characteristically simple subgroup.

On the Simple Group of J. Tits

In the series of simple groups 2F4(q),q =2 2m+1, discovered by Ree, Tits [4] showed that the group 2F4(2) was not simple but contained a simple subgroup of index 2. In this note we extend the characterization of obtained by the author in [3].

A Characterization of the Tits' Simple Group

In [6], J. Tits has shown that the Ree group 2F4(2) is not simple but possesses a simple subgroup of index 2. In this paper we prove the following theorem:THEOREM. Let G be a finite group of even order and let z be an involution contained in G. Suppose H = CG(z) has the following properties:(i) J = O2(H) has order 29and is of class at least 3.(ii) H/J is isomorphic to the Frobenius group of order 20.(iii) If P is a Sylow 5-subgroup of H, then Cj(P) ⊆ Z(J).Then G = H • O(G) or G ≊ , the simple group of Tits, as defined in [6].For the remainder of the paper, G will denote a finite group which satisfies the hypotheses of the theorem as well as G ≠ H • O(G). Thus Glauberman's theorem [1] can be applied to G and we have that 〈z〉 is not weakly closed in H (with respect to G). The other notation is standard (see [2], for example).