Continuous cocycle superrigidity for coinduced actions and relative ends

Abstract

We prove that certain coinduced actions for an inclusion of finitely generated commensurated subgroups with relative one end are continuous cocycle superrigid actions. We also show the necessity for the relative end assumption.

Continuous orbit equivalence rigidity for left-right wreath product actions

“…Observe that the generalized full shift Γ X I 0 , where Γ I is a transitive action, may be treated as the coinduced action from the trivial action of the stabilizer subgroup Stab(i) X 0 to Γ, where i ∈ I is any chosen point. In [15], we observed that such a rigidity result indeed holds under certain assumptions on the inclusion Stab(i) < Γ. A crucial condition is that Stab(i) is a commensurable, e.g.…”

“…For case (iii), identify I with Γ/Γ 2 then observe that the generalized full shift Γ X can be regarded as the coinduced action of the trivial action Γ 2 X 0 from the subgroup Γ 2 to the ambient group Γ, see [15, P. 317]. Then we can apply Theorem 1.1 and Proposition 2.5 (8) in [15].…”

“…A crucial condition is that Stab(i) is a commensurable, e.g. a normal subgroup in Γ, see the discussion in the last section in [15]. However, when dealing with the left-right translation action Γ = Γ 1 × Γ 1 I := Γ 1 , this condition is never satisfied when Γ 1 is icc.…”

“…Till now, there are quite a few known rigidity results for continuous orbit equivalence rigidity, e.g. [1,3,5,6,12,15,16,21,36]. Nevertheless, in all these results, either the original actions contain fixed points or assumptions are put on the acting groups for the target actions, and hence not minimal superrigid.…”