Atlas: Balleans of bounded geometry and G-spaces by Igor Protasov

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Analysis and Topology, Lviv - 2008
May 26 - June 7, 2008
Ivan Franko National University of Lviv
Lviv, Ukraine

Organizers
M.Zarichnyi, O.Skaskiv, T.Banakh (Lviv University)

Balleans of bounded geometry and G-spaces
by
Igor Protasov
Department of Cybernetics, Kyiv University, UKRAINE

A ball structure is a triplet B=(X, P, B), where X, P are non-empty sets and, for any x ∈ X and a ∈ P, B(x, a) is a subset of X which is called a ball of radius a around x. It is supposed that x ∈ B(x, a) for all x ∈ X, a ∈ P. The set X is called the support of B, P is called the set of radii. Given any x ∈ X and a ∈ P, we put B^*(x, a)={y ∈ X:x ∈ B(y, a)}.

A ball structure X is called a ballean if

for any a, b ∈ P, there exist a', b' ∈ P such that, for every x ∈ X,

B(x, a) ⊆ B^*(x, a'), B^*(x, b) ⊆ B(x, b');
for any a, b ∈ P, there exists g ∈ P such that, for every x ∈ X,

B(B(x, a), b) ⊆ B(x, g).

Let B₁=(X₁, P₁, B₁), B₂=(X₂, P₂, B₂) be balleans. A mapping f:X₁→ X₂ is called a < -mapping if, for every a ∈ P₁, there exists b ∈ P₂ such that, for every x ∈ X₁,

f(B₁(x, a)) ⊆ B₂(f(x), b).

The category of balleans and < -mappings can be considered as an asymptotic reflection of the category of uniform spaces and uniformly continuous mappings (see [1], [2]).

Let B=(X, P, B) be a ballean, a ∈ P. A subset S of X is called a-separated if B(x, a) and B(y, a) are disjoint for all distinct x, y ∈ S. An a-capacity of a subset Y of X is the cardinal

cap_a=sup{|S|:S is an a-separated subset of Y}.

We say that a ballean B=(X, P, B) has bounded geometry if there exist b ∈ P and a function h:P→w such that cap_b(B(x, a)) ≤ h(a) for all x ∈ X, a ∈ P.

Let G be a group, X be a G-space. We denote by F_G the family of all finite subset of G containing the identity of G, and get the ballean B(G, X)=(X, F_G, B), where B(x, F)={fx:f ∈ F} for all x ∈ X, F ∈ F_g. Clearly, (G, X) is of bounded geometry.

Let S be a set, B=(X, P, B) be a ballean, f, g:S→ X. We say that f, g are close if there exists a ∈ X such that f(x) ∈ B(g(x), a) for every x ∈ X. Two balleans B₁, B₂ with the supports X₁, X₂ are called coarsely equivalent if there exist the < -mappings f₁:X₁→ X₂ and f₂:X₂→ X₁ such that f₁○f₂ and f₂○f₁ are close to corresponding identity mappings of X₁ and X₂.

Theorem 1 Every ballean of bounded geometry is coarsely equivalent to a ballean B(G, B) of some G-space X.

[1] I.Protasov, M.Zarichniy, General Asymptology, Math. Stud. Monogr. Ser., Vol. 12, VNTL, Lviv, 2007.

[2] J.Roe, Lectures on coarse geometry, University Lectures Series, 31, Amer. Math. Soc., Providence, R.I., 2003.

Date received: March 26, 2008