Generic elements in isometry groups of Polish ultrametric spaces

Malicki, Maciej

doi:10.1007/s11856-014-1128-6

Cited by 5 publications

(3 citation statements)

References 7 publications

(14 reference statements)

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“…We should mention that recent work of Malicki [13] shows ample generics for the isometry groups of Polish ultrametric Urysohn spaces.…”

Section: Example 14mentioning

confidence: 92%

Extending partial isometries of generalized metric spaces

Conant

2019

Fund. Math.

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We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class K of finite generalized metric spaces satisfies the Hrushovski extension property: for any A ∈ K there is some B ∈ K such that A is a subspace of B and any partial isometry of A extends to a total isometry of B. We prove the Hrushovski property for the class of finite generalized metric spaces over a semi-archimedean monoid R. When R is also countable, we use this to show that the isometry group of the Urysohn space over R has ample generics. Finally, we prove the Hrushovski property for classes of integer distance metric spaces omitting metric triangles of uniformly bounded odd perimeter. As a corollary, given odd n ≥ 3, we obtain ample generics for the automorphism group of the universal, existentially closed graph omitting cycles of odd length bounded by n.

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“…We should mention that recent work of Malicki [13] shows ample generics for the isometry groups of Polish ultrametric Urysohn spaces.…”

Section: Example 14mentioning

confidence: 92%

Extending partial isometries of generalized metric spaces

Conant

2019

Fund. Math.

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“…They naturally appear in a variety of places, including general topology, mathematical logic, theoretic computer science, p-adic dynamics and theoretical biology. For example, results regarding Lipschitz and uniformly continuous and Borel reducibilities are studied in [32,3], generic elements in isometry groups are described in [24], Polish ultrametric spaces on which each Baire one function is first return recoverable are characterized in [7] and locally contractive maps on perfect Polish ultrametric spaces are studied in [11].…”

Section: Introductionmentioning

confidence: 99%

Shadowing, Finite Order Shifts and Ultrametric Spaces

Darji¹,

Sobottka²

2020

Preprint

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Inspired by recent novel work of Good and Meddaugh, we establish fundamental connections between shadowing, finite order shifts and ultrametric Polish spaces. We develop a theory of shifts of finite type for infinite alphabets. We call them shifts of finite order. We develop the basic theory of the shadowing property in general Polish spaces, exhibiting similarities and differences with the theory in compact spaces. We connect these two theories in the setting of zero dimensional Polish spaces, showing that a surjective uniformly continuous map of an ultrametric Polish space has the shadowing property if, and only if, it is the inverse limit of shifts of finite order. As corollaries we obtain that a variety of maps in ultrametric Polish spaces have the shadowing property, such as similarities and, more generally, maps which themselves, or their inverses, have Lipschitz constant 1. Finally, we apply our results to the dynamics of p-adic integers and p-adic rationals. 1

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“…These include the automorphism groups of the following countable structures: the random graph (Hrushovski [9]), the free group on countably many generators (Bryant and Evans [4]), arithmetically saturated models of true arithmetic (Schmerl [20]) or the automorphism group of the rational Urysohn space (Solecki [21].) Also, Malicki [14] characterized all Polish ultrametric spaces whose isometry group contains an open subgroup with ample generics. Moreover, very recently, the first examples Polish groups with ample generics that are not isomorphic to the automorphism group of some countable structure have been found (see [15], and [11].…”

Section: Introductionmentioning

confidence: 99%

Consequences of the Existence of Ample Generics and Automorphism Groups of Homogeneous Metric Structures

Malicki¹

2016

J. symb. log.

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Abstract. We define a criterion for a homogeneous, complete metric structure X that implies that the automorphism group Aut(X) satisfies all the main consequences of the existence of ample generics: it has the automatic continuity property, the small index property, and uncountable cofinality for non-open subgroups. Then we verify it for the Urysohn space Í, the Lebesgue probability measure algebra MALG, and the Hilbert space ℓ2, thus proving that Iso(Í), Aut(MALG), U (ℓ2), and O(ℓ2) share these properties. We also formulate a condition for X which implies that every homomorphism of Aut(X) into a separable group K with a left-invariant, complete metric, is trivial, and we verify it for Í, and ℓ2.

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Generic elements in isometry groups of Polish ultrametric spaces

Cited by 5 publications

References 7 publications

Extending partial isometries of generalized metric spaces

Extending partial isometries of generalized metric spaces

Shadowing, Finite Order Shifts and Ultrametric Spaces

Consequences of the Existence of Ample Generics and Automorphism Groups of Homogeneous Metric Structures

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