We introduce the notion of [Formula: see text]determinacy for [Formula: see text] a pointclass and [Formula: see text] an equivalence relation on a Polish space [Formula: see text]. A case of particular interest is the case when [Formula: see text] is the (left) shiftaction of [Formula: see text] on [Formula: see text] where [Formula: see text] or [Formula: see text]. We show that for all shift actions by countable groups [Formula: see text], and any “reasonable” pointclass [Formula: see text], that [Formula: see text]determinacy implies [Formula: see text]determinacy. We also prove a corresponding result when [Formula: see text] is a subshift of finite type of the shift map on [Formula: see text].
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Extending group actions on metric spaces
We address the following natural extension problem for group actions: Given a group [Formula: see text], a subgroup [Formula: see text], and an action of [Formula: see text] on a metric space, when is it possible to extend it to an action of the whole group [Formula: see text] on a (possibly different) metric space? When does such an extension preserve interesting properties of the original action of [Formula: see text]? We begin by formalizing this problem and present a construction of an induced action which behaves well when [Formula: see text] is hyperbolically embedded in [Formula: see text]. Moreover, we show that induced actions can be used to characterize hyperbolically embedded subgroups. We also obtain some results for elementary amenable groups.
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 Award ID(s):
 1853989
 NSFPAR ID:
 10347319
 Date Published:
 Journal Name:
 Journal of Topology and Analysis
 Volume:
 12
 Issue:
 03
 ISSN:
 17935253
 Page Range / eLocation ID:
 625 to 665
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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