Topology and convexity in the space of actions modulo weak equivalence
- Creators
- Burton, Peter
Abstract
We analyze the structure of the quotient A∼(Γ,X,μ) of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We introduce a variant of an abstract construction of Fritz which encapsulates the convex combination operation on A∼(Γ,X,μ) . This formalism allows us to define the geometric notion of an extreme point. We also discuss a topology on A∼(Γ,X,μ) due to Abért and Elek in which it is Polish and compact, and show that this topology is equivalent to others defined in the literature. We show that the convex structure of A∼(Γ,X,μ) is compatible with the topology, and as a consequence deduce that A∼(Γ,X,μ) is path connected. Using ideas of Tucker-Drob, we are able to give a complete description of the topological and convex structure of A∼(Γ,X,μ) for amenable Γ by identifying it with the simplex of invariant random subgroups. In particular, we conclude that A∼(Γ,X,μ) can be represented as a compact convex subset of a Banach space if and only if Γ is amenable. In the case of general we prove a Krein–Milman-type theorem asserting that finite convex combinations of the extreme points of A∼(Γ,X,μ) are dense in this space. We also consider the space A∼s(Γ,X,μ) of stable weak equivalence classes and show that it can always be represented as a compact convex subset of a Banach space. In the case of a free group F_N, we show that if one restricts to the compact convex set FR∼s(F_N,X,μ)⊆A∼s(F_N,X,μ) consisting of the stable weak equivalence classes of free actions, then the extreme points are dense in FR∼s(F_N,X,μ).
Additional Information
© 2017 Cambridge University Press. Received 6 April 2015 and accepted in revised form 9 November 2016. Published online: 03 April 2017. This research was partially supported by NSF grant DMS-0968710. We would like to thank Alexander Kechris for introducing us to this topic and for many helpful discussions. We also thank Robin Tucker-Drob for informing us of his result with Bowen that the space of stable weak equivalence classes forms a simplex, and for raising the question of when it forms a Poulsen simplex. We thank the anonymous referee for useful comments.Attached Files
Submitted - 1501.04079.pdf
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Additional details
- Eprint ID
- 89418
- DOI
- 10.1017/etds.2016.133
- Resolver ID
- CaltechAUTHORS:20180906-131231041
- DMS-0968710
- NSF
- Created
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2018-09-07Created from EPrint's datestamp field
- Updated
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2021-11-16Created from EPrint's last_modified field