Orbit Equivalence Rigidity for Product Actions
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Communications in
Mathematical Physics
Orbit Equivalence Rigidity for Product Actions Daniel Drimbe Department of Mathematics, University of Regina, 3737 Wascana Pkwy, Regina, SK S4S 0A2, Canada. E-mail: [email protected] Received: 26 May 2019 / Accepted: 26 August 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract: Let 1 , . . . , n be hyperbolic, property (T) groups, for some n ≥ 1. We prove that if a product 1 × · · · × n X 1 × · · · × X n of measure preserving actions is stably orbit equivalent to a measure preserving action Y , then Y is induced from an action 0 Y0 such that there exists a direct product decomposition 0 = 1 × · · · × n into n infinite groups. Moreover, there exists a measure preserving action i Yi that is stably orbit equivalent to i X i , for any 1 ≤ i ≤ n, and the product action 1 × · · · × n Y1 × · · · × Yn is isomorphic to 0 Y0 . 1. Introduction An important topic in ergodic theory is the classification of probability measure preserving (pmp) actions up to orbit equivalence. Two pmp actions (X, μ) and (Y, ν) are called orbit equivalent (OE) if there exists a measure space isomorphism θ : (X, μ) → (Y, ν) which preserves the orbits, i.e. θ (x) = θ (x), for almost every x ∈ X. The classification of actions up to OE is driven by the following fundamental question: what aspects of the group and of the action (X, μ) are remembered by the orbit equivalence relation RX := {(x, y) ∈ X × X |x = y}? Equivalence relations RX tend to forget a lot of information about the groups and actions they are constructed from. This is best illustrated by H. Dye’s theorem asserting that any two ergodic pmp actions of Z are OE [Dy58]. D.S. Orstein and B. Weiss have extended this result to the class of countable amenable groups [OW80] (see also [CFW81] for a generalization). Consequently, pmp actions of amenable groups manifest a striking lack of rigidity: any algebraic property of the group (e.g. being finitely generated or torsion free) and any dynamical property of the action (e.g. being mixing or weakly mixing) is completely lost in the passage to equivalence relations. In the non-amenable case, the situation is radically different. More precisely, various properties of the group or of the action (X, μ) can be recovered from the D. Drimbe: The author was partially supported by PIMS fellowship.
D. Drimbe
equivalence relation RX . R. Zimmer’s pioneering work led to such OE rigidity results for actions of higher rank lattices in semisimple Lie groups. In particular, he showed that if m, n ≥ 3, then S L m (Z) Tm is OE to S L n (Z) Tn if and only if m = n [Zi84]. Remarkably, by building upon Zimmer’s ideas, A. Furman showed that most pmp ergodic actions (X, μ) of higher rank lattices, including S L n (Z) Tn , for n ≥ 3, are OE superrigid [Fu99a,Fu99b]. Roughly speaking, this means that both the group and the action (X, μ) are completely remembered by the equivalence relation RX . By using his influent
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