Co-induction and invariant random subgroups
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Co-induction and invariant random subgroups. / Kechris, Alexander S.; Quorning, Vibeke.
In: Groups, Geometry, and Dynamics, Vol. 13, No. 4, 2019, p. 1151-1193.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Co-induction and invariant random subgroups
AU - Kechris, Alexander S.
AU - Quorning, Vibeke
PY - 2019
Y1 - 2019
N2 - In this paper we develop a co-induction operation which transforms an invariant random subgroup of a group into an invariant random subgroup of a larger group. We use this operation to construct new continuum size families of non-atomic, weakly mixing invariant random subgroups of certain classes of wreath products, HNN-extensions and free products with amalgamation. By use of small cancellation theory, we also construct a new continuum size family of non-atomic invariant random subgroups of F2 which are all invariant and weakly mixing with respect to the action of Aut(F2). Moreover, for amenable groups Γ ≤ Δ, we obtain that the standard co-induction operation from the space of weak equivalence classes of Δ to the space of weak equivalence classes of Δ is continuous if and only if [Δ : Γ] < ∞ or coreΔ(Γ) is trivial. For general groups we obtain that the co-induction operation is not continuous when [Δ : Γ] = ∞. This answers a question raised by Burton and Kechris in [17]. Independently such an answer was also obtained, using a different method, by Bernshteyn in [8].
AB - In this paper we develop a co-induction operation which transforms an invariant random subgroup of a group into an invariant random subgroup of a larger group. We use this operation to construct new continuum size families of non-atomic, weakly mixing invariant random subgroups of certain classes of wreath products, HNN-extensions and free products with amalgamation. By use of small cancellation theory, we also construct a new continuum size family of non-atomic invariant random subgroups of F2 which are all invariant and weakly mixing with respect to the action of Aut(F2). Moreover, for amenable groups Γ ≤ Δ, we obtain that the standard co-induction operation from the space of weak equivalence classes of Δ to the space of weak equivalence classes of Δ is continuous if and only if [Δ : Γ] < ∞ or coreΔ(Γ) is trivial. For general groups we obtain that the co-induction operation is not continuous when [Δ : Γ] = ∞. This answers a question raised by Burton and Kechris in [17]. Independently such an answer was also obtained, using a different method, by Bernshteyn in [8].
KW - Co-induction
KW - Invariant random subgroups
KW - Small cancellation
KW - Weak mixing
UR - http://www.scopus.com/inward/record.url?scp=85077284680&partnerID=8YFLogxK
U2 - 10.4171/GGD/517
DO - 10.4171/GGD/517
M3 - Journal article
AN - SCOPUS:85077284680
VL - 13
SP - 1151
EP - 1193
JO - Groups, Geometry, and Dynamics
JF - Groups, Geometry, and Dynamics
SN - 1661-7207
IS - 4
ER -
ID: 238960277