Uniform universal covers of uniform spaces

Berestovskiî, V. N.; Plaut, Conrad

doi:10.1016/j.topol.2006.12.012

Cited by 43 publications

(139 citation statements)

References 24 publications

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“…This paper is connected with, and was inspired by, our previous paper [8] and a 20-year-old announcement of the first author, cited in [18]. The results from [8] may be used to give an alternative proof of Proposition 18.…”

Section: Question 9 Which (Finitely Presented) Groups Act Freely (Bymentioning

confidence: 88%

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Covering R-trees, R-free groups, and dendrites

Berestovskii

Plaut

2010

Advances in Mathematics

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We prove that every length space X is the orbit space (with the quotient metric) of an R-tree X via a free action of a locally free subgroup Γ (X) of isometries of X. The mapping φ : X → X is a kind of generalized covering map called a URL-map and is universal among URL-maps onto X. X is the unique R-tree admitting a URL-map onto X. When X is a complete Riemannian manifold M n of dimension n 2, the Menger sponge, the Sierpin'ski carpet or gasket, X is isometric to the so-called "universal" R-tree A c , which has valency c = 2 ℵ 0 at each point. In these cases, and when X is the Hawaiian earring H , the action of Γ (X) on X gives examples in addition to those of Dunwoody and Zastrow that negatively answer a question of J.W. Morgan about group actions on R-trees. Indeed, for one length metric on H , we obtain precisely Zastrow's example.

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Section: Question 9 Which (Finitely Presented) Groups Act Freely (Bymentioning

confidence: 88%

“…The results from [8] may be used to give an alternative proof of Proposition 18. In an upcoming paper we will use constructions of [8] to obtain additional examples of URL-maps that are not local isometries.…”

Section: Question 9 Which (Finitely Presented) Groups Act Freely (Bymentioning

confidence: 99%

Covering R-trees, R-free groups, and dendrites

Berestovskii

Plaut

2010

Advances in Mathematics

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“…Suppose V i ∈ S, 1 ≤ i ≤ 2, both intersect V ∈ S and p(V 1 ) = p(V 2 ). That means the edges [V, V 1 ] and [V, V 2 ] in N (S) (1) are mapped to the edge [p(V ), p(V 1 )] in N (U) (1)…”

Section: Chain Liftingmentioning

confidence: 99%

“…Characterization 1 uses ideas of Berestovskii-Plaut [1] later expanded in [2]. In case of surjective maps p : X → Y between connected metrizable spaces we characterize overlays as local isometries: p : X → Y is an overlay if and only if one can metrize X and Y in such a way that p|B(x, 1) : B(x, 1) → B(p(x), 1) is an isometry for each x ∈ X.…”

Section: Introductionmentioning

confidence: 99%

Overlays and group actions

Dydak

2016

Topology and its Applications

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Overlays were introduced by R.H.Fox [6] as a subclass of covering maps. We offer a different view of overlays: it resembles the definition of paracompact spaces via star refinements of open covers. One introduces covering structures for covering maps and p : X → Y is an overlay if it has a covering structure that has a star refinement.We prove two characterizations of overlays: one using existence and uniqueness of lifts of discrete chains, the second as maps inducing simplicial coverings of nerves of certain covers. We use those characterizations to improve results of Eda-Matijević concerning topological group structures on domains of overlays whose range is a compact topological group.In case of surjective maps p : X → Y between connected metrizable spaces we characterize overlays as local isometries: p : X → Y is an overlay if and only if one can metrize X and Y in such a way that p|B(x, 1) : B(x, 1) → B(p(x), 1) is an isometry for each x ∈ X.

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“…Zeeman's example [16, 6.6.14 on p.258] demonstrates difficulty in constructing a theory of coverings by non-locally path-connected spaces (that example amounts to two non-equivalent classical coverings with the same image of the fundamental groups). For coverings in the uniform category see [1] and [3].…”

Section: Introductionmentioning

confidence: 99%

Covering maps for locally path-connected spaces

et al. 2012

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Abstract. We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces.Regular Peano covering maps over path-connected spaces are shown to be identical with generalized regular covering maps introduced by Fischer and Zastrow [15]. If X is path-connected, then every Peano covering map is equivalent to the projection e X/H → X, where H is a subgroup of the fundamental group of X and e X equipped with the topology used in [2], [15] and introduced in [23, p.82]. The projection e X/H → X is a Peano covering map if and only if it has the unique path lifting property. We define a new topology on e X for which one has a characterization of e X/H → X having the unique path lifting property if H is a normal subgroup of π 1 (X). Namely, H must be closed in π 1 (X). Such groups include π(U , x 0 ) (U being an open cover of X) and the kernel of the natural homomorphism π 1 (X, x 0 ) →π 1 (X, x 0 ).

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Uniform universal covers of uniform spaces

Cited by 43 publications

References 24 publications

Covering R-trees, R-free groups, and dendrites

Covering R-trees, R-free groups, and dendrites

Overlays and group actions

Covering maps for locally path-connected spaces

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