Groups of uniform homeomorphisms of covering spaces
Tatsuhiko Yagasaki†Kyoto Institute of Technology, JapanMotivation
The uniform topology is one of basic topologies on function spaces. We study local andglobal topological properties of spaces of uniform embeddings and groups of uniform homeomorphisms in metric manifolds endowed with the uniform topology. Since the notions ofuniform continuity and uniform topology depend on the choice of metrics, we are concernedwith dependence of those properties on the behavior of metrics in neighborhoods of ends ofmanifolds.

Basic Definitions and Notations
A map f : (X, d) → (Y, %) between metric spaces is said to be uniformly continuous if∀ ε > 0 ∃ δ > 0 such that if x, x0 ∈ X and d(x, x0) < δ then %(f (x), f (x0)) < ε.
The map f is called a uniform homeomorphism if
f is bijective and both f and f −1 are uniformly continuous.
A uniform embedding is a uniform homeomorphism onto its image.
For subsets A, B of a metric space (X, d) we say that
B is a uniform neighborhood of A in (X, d) and write A ⊂u Bif B contains the ε-neighborhood Oε(A) of A in X for some ε > 0.
A subset A of X is said to be uniformly discrete if there exists an ε > 0 such that d(x, y) ≥ ε
for any distinct points x, y ∈ A.
Suppose (M, d) is a metric n-manifold (i.e., a separable topological manifold possibly withboundary assigned a fixed metric). For subsets X and C of M , let
E∗u(X, M ; C) = the space of proper uniform embeddingsf : (X, d|X ) → (M, d) with f = id on X ∩ C.
This space is endowed with the uniform topology induced from the sup-metricd(f, g) = sup d(f (x), g(x)) | x ∈ X ∈ [0, ∞](f, g ∈ E∗u(X, M ; C)).
For a subset A of M letu
(M, d) = the group of uniform homeomorphisms h : (M, d) → (M, d)
HA
with h|A = idA, endowed with the uniform topology,uu(M, d) | d(h, idM ) < ∞}(M, d)b = {h ∈ HA
HA
(the subgroup of bounded uniform homeomorphisms),uu(M, d).
(M, d)0 = the connected component of idM in HAHAuuuu(M, d)b.
(M, d)0 ⊂ HA(M, d) and so HA(M, d)b is an open subgroup of HA
Note that HALocal deformation property of uniform embeddings
The newt definition is motivated by the Edwards-Kirby local deformation theorem for embeddings of compact spaces [1].
Definition 1. For a subset A of M we say that
“A has the local deformation property for uniform embeddings in (M, d)”and write A : (LD)M if the following holds:(∗) for any tuple (X, W 0, W, Z, Y ) of subsets of M withX ⊂ A, X ⊂u W 0 ⊂ W ⊂ M and Z ⊂u Y ⊂ M
there exists a neighborhood W of the inclusion map iW : W ⊂ M in E∗u(W, M ; Y ) anda homotopy ϕ : W × [0, 1] −→ E∗u(W, M ; Z) such that(1) for each f ∈ W(i) ϕ0(f ) = f , (ii) ϕ1(f ) = id on X,(iii) ϕt(f ) = f on W − W 0 and ϕt(f )(W ) = f (W ) (t ∈ [0, 1]),(iv) if f = id on W ∩ ∂M , then ϕt(f ) = id on W ∩ ∂M (t ∈ [0, 1]),
(2) ϕt(iW ) = iW (t ∈ [0, 1]).

Definition 2. A map π : (X, d) → (Y, %) is called a metric covering projection if it satisfies
the following conditions:(\)1 There exists an open cover U of Y such that for each U ∈ U the inverse π −1(U ) is
the disjoint union of open subsets of X each of which is mapped isometrically onto Uby π.
(\)2 For each y ∈ Y the fiber π −1(y) is uniformly discrete in X.
(\)3 %(π(x), π(x0)) ≤ d(x, x0) for any x, x0 ∈ X.
Theorem 1. π : (M, d) → (N, %) : a metric covering projection
N : a compact manifold=⇒ (M, d) : (LD)This theorem can be deduced from the Edwards-Kirby local deformation theorem andthe classical Arzela-Ascoli theorem for equi-continuous families of maps[2] Metric manifolds with geometric group actions
In term of covering transformations, Theorem 1 corresponds to the case of free group actions. For the non-free case, we have the following generalization ([4]).
Definition 3. Φ : G × X → X : an action of a discrete group G on a metric space (X, d)(1) Φ : locally isometric⇐⇒ ∀ x ∈ X ∃ ε > 0 such that each g ∈ G maps Oε(x) isometrically onto Oε(gx).
(2) Φ : locally geometric ⇐⇒ Φ : proper, cocompact and locally isometric.
Corollary 1. (M, d) : a metric manifold(M, d) admits a locally geometric group action =⇒ (M, d) : (LD)This follows from Theorem 1 and Additivity of (LD)M .
[3] Typical Model spaces
The following model spaces and their ends have the property (LD).
(1) Euclidean n-space Rn (with the standard Euclidean metric)◦ Rn admits the canonical geometric action of Zn(i) the Euclidean ends Rnr = Rn − Or (0) (r > 0)(ii) the half space Rn≥0 = {x ∈ Rn | xn ≥ 0}(2) Hyperbolic n-space Hn◦ Hn admits a Riemannian covering projection onto a closed hyperbolic n-manifold.
(3) Cylinder (N × R, d)p( (N, %) : Compact metric n-manifold, d((x, t), (y, s)) = %(x, y)2 + |t − s|2 )◦ Z y N × R : m · (x, t) = (x, t + m) : a geometric action(i) Cylindrical end (Product end) : (N × [0, ∞), d)

Global deformation property of uniform homeomorphisms
Euclidean n-space Rn has the similarity transformations
kγ : Rn ≈ Rn : kγ (x) = γx (γ > 0).
Conjugation with these similarity transformations enables us to deduce a global deformation
property for uniform embeddings in the Euclidean ends Rnr = Rn − Or (0) (r > 0) from the local one. Since this global deformation property is preserved by bi-Lipschitz equivalences, we
can consider a more general setting of metric spaces with finitely many bi-Lipschitz Euclideanends ([3]).
Definition 4. A bi-Lipschitz n-dimensional Euclidean end of a metric space (X, d) means aclosed subset L of X which admits a bi-Lipschitz homeomorphism θ : Rn1 ≈ (L, d|L) suchthat θ(∂Rn1 ) = FrX L. We say that L is isolated if d(X − L, θ(Rnr)) → ∞ as r → ∞. Let
L0 = θ(Rn2 ), L00 = θ(Rn3 ).

We omit the subscript M in the symbol (LD)M when A = M .
The Edwards-Kirby deformation theorem ([1]) can be restated in the next form.
Edwards-Kirby deformation theorem.
K ⊂ M : Relatively compact =⇒ K : (LD)M

Theorem 2. Suppose X is a metric space and L1, · · · , Lm are mutually disjoint isolated biLipschitz Euclidean ends of X. Let L0 = L01 ∪ · · · ∪ L0m and L00 = L001 ∪ · · · ∪ L00m. Then there
exists a strong deformation retraction ϕ of Hu(X)b onto HLu 00 (X)b such that

Baisc Properties of (LD). ([4])

Example 2.
[1] Hu(Rn)b ' ∗(1) (Invariance under Uniform Homeo’s)M∼
M : (LD) =⇒ N : (LD)
= N (uniform homeo)(2) (Restriction) A ⊂ B ⊂ M

B : (LD)M =⇒ A : (LD)M(3) (Additivity) A ⊂u U ⊂ M , B ⊂ M

U , B : (LD)M =⇒ A ∪ B : (LD)M(4) (Relatively compact subsets) K ⊂ M : Relatively compact, A ⊂ M
A : (LD)M ⇐⇒ A ∪ K : (LD)M(5) (Neighborhoods of Ends)
K ⊂ M : Relatively compact
M = K ∪ ∪m
i=1Li
Li ⊂ M : Closed, Li : n-manifold, d(Li, Lj ) > 0 (i 6= j)
M : (LD) ⇐⇒ Li : (LD) (i = 1, · · · , m)u(6) (M, d) : (LD) =⇒ H (M, d) : locally contractible
Example 1.
[1] Metric covering spaces over compact manifolds
The following notion is a natural metric version of Riemannian covering projections.

ϕt(h) = h on h−1(X − L0) − L0 for any (h, t) ∈ Hu(X)b × [0, 1].
( (∵) Hu(Rn)b & HRu n3 (Rn) ' ∗)[2] N : a compact connected 2-manifold with a nonempty boundaryC = ∪mi=1Ci : a nonempty union of some boundary circles of N
M := N − C (a noncompact connected 2-manifold)
Li : the end of M corresponding to the boundary circle Cid : a metic on M such that each Li is an isolated bi-Lipschitz Euclidean end of (M, d)=⇒ Hu(M, d)0 ' ∗( (∵) Hu(M, d)0 ' HLu 00 (M )0 ≈ HC (N )0 ' ∗)In [2] we showed that Hu(R)b is homeomorphic to `∞.
Conjecture. Hu(Rn)b is homeomorphic to `∞ for any n ≥ 1.

References[1] R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63–88.
[2] K. Mine, K. Sakai, T. Yagasaki and A. Yamashita, Topological type of the group of uniform homeomorphisms of the real lines,
Topology Appl., 158 (2011), 572–581.
[3] T. Yagasaki, Groups of uniform homeomorphisms of covering spaces, to appear in J. Math. Soc. Japan (Article in Press
JMSJ 6460), (arXiv:1203.4028).
[4] T. Yagasaki, On local deformation property for uniform embeddings, preprint, (arXiv:1301.3265).

†Address for correspondence: Graduate School of Science and Technology, Kyoto Institute of Technology, Matsugasaki, Sakyoku, Kyoto, 606-8585, Japan