Coarse equivalences of functorial constructions

We consider the question of coarse equivalence of some functorial constructions (in particular, symmetric powers, hypersymmetric powers) in the category of metric spaces. Анотація. Груба геометрія займається вивченням властивостей метричних просторів “в нескінченності” (див. [3]). Однією з важливих загальних задач грубої геометрії є класифікація метричних просторів з точністю до грубої еквівалентності. Так в [6], було доведено, що гіперпростори cc (R), exp R та exp R не є грубо еквівалентними. В [7] доведено ізоморфність джойна R  ̊ R+ та півпростору R + в асимптотичній категорії A. В цій статті показано, що гіперпростори exp3 R+, exp3 R, симетричні степені SP 3 R+, SP 3 R та простір R+ ліпшицево еквівалентні (теореми 3.1, 3.2). Крім того доведено, що простори R+ та P2(R) не є грубо еквівалентними. Одним з основних результатів є теорема 3.7, про біліпшицеву еквівалентність гіперпростору exp2 R та R ˆ Cone (RPm ́1). Її можна вважати асимптотичним аналогом одного результату Шорі [4]. В статті [1] означено конус CX і надбудову SX в асимптотичних категоріях для кожного метричного простору. У теоремі 3.12 доведено, що конус CR+ і надбудова SR+ не є грубо еквівалентні. The author is indebted to Mykhailo Zarichnyi for valuable discussions.

One of the main general problems in coarse geometry is that of classification of spaces up to coarse equivalence.
A. Dranishnikov [1] introduced certain functorial constructions in the asymptotic categories. In particular, he considered the products, joins, cones, and spaces of probability measures. Several results concerning geometric properties of these spaces are also obtained in [6,7].
We provide some necessary definitions. For every metric space X we denote by exp X the set of all nonempty compact subsets of a metrizable space X. Any admissiblle metric d on X induces the Hausdorff metric d H on exp X: For every n P N let exp n X be the subspace of exp X consisting of all nonempty sets of cardinality ď n. E. V. Shchepin [5] called the space exp n X the nth hypersymmetric power of X. We will denote by exp c X the subspace of exp X consisting of all subcontinua of X.
It was proved in [6] that the hyperspaces cc (R n ), exp R n and exp c R n are not coarsely equivalent. Also in [7] it was established that the join R n˚R + and half-space R n+1 + are isomorphic in the asymptotic category A. In this note we show that the hyperspaces exp 3 R + , exp 3 R, symmetric powers SP 3 R + , SP 3 R and the space R 3 + are Lipschitz equivalent. In addition, we will prove that the spaces R 3 + and P 2 (R) are not coarsely equivalent. Here P 2 (X) is the subspace of the space of probability measures whose support consists of at most 2 points.
One of our main results is Theorem 3.7 on Lipschitz equivalence of the hypersymmetric power exp 2 R m and R mˆC one (RP m´1 ). This theorem can be considered as an asymptotic counterpart of one Schori's result [4].

PRELIMINARIES
Let " be the equivalence relation on X n defined by the condition: if and only if there is a bijection σ of t1, . . . , nu such that x i = y σ(i) for all i = 1, . . . , n. The orbit space of this relation on X n is called the nth symmetric power of X and is denoted by SP n (X). The equivalence class of " containing (x 1 , . . . , x n ) is denoted by [x 1 , . . . , x n ]. The support of x = [x 1 , . . . , x n ] P SP n (X) is the set supp(x) = tx 1 , . . . , x n u P exp n X.
If (X, d) is a metric space, then it is known that the following function d is a metric on SP n (X): In the case of geodesic metric spaces, every coarse uniform map is asymptotically Lipschitz, [1].

MAIN RESULT
Theorem 3.1. The hyperspace exp 3 R + , the symmetric power SP 3 R + , and the space R 3 + are mutually Lipschitz equivalent. Proof. Every point [a, b, c] P SP 3 R + can be written as (a, b, c), where a ď b ď c. Then, representing SP 3 R + in the Cartesian coordinate system we obtain the cone, Cone (B 2 ). In the spherical coordinate system we see that ( , and the map f : The hyperspace exp 3 R + is the quotient space of the space SP 3 R + with respect to this equivalence relation. This quotient space is nothing but the cone Cone(B 2 ), which is known to be Lipschitz equivalent to R 3 + . □ 72 Mykhailo Romanskyi One can similarly prove that the symmetric power SP k R + is Lipschitz equivalent to R k + .
Theorem 3.2. The hypersymmetric power exp 3 R, symmetric power SP 3 R and R 3 + are Lipschitz equivalent. Proof. Similarly to the proof of Theorem 3.1 we note that for any [a, b, c] P SP 3 R one may assume, without loss of generality, that a ď b ď c, and we then identify this point with (a, b, c). This allows us to represent SP 3 R in the Cartesian coordinate system. The obtained space is easily seen to be Lipschitz equivalent to R 3 + . In order to obtain exp 3 R we have to make the identification в SP 3 R. After this identification, the obtained space is Lipschitz equivalent to R 3 + . □ Theorems 3.1 and 3.2 imply the following.

Corollary 3.3. The hypersymmetric powers exp
The cone Cone(X) over a compact metric space (X, d) is the quotient space of the product (XˆR + )/ ", where the equivalence relation " is given by the condition (x, 0) " (y, 0), x, y P X. If, in addition, diam(X) ď 2, then the metricd on Cone(X) can be given by the formula: Define a bijective map g : (Cone(X),d) Ñ (Cone(Y ),ρ) by the formula g(x, t) = (f (x), t) and prove that this map is bi-Lipschitz. We havê On the other hand, t), (x 2 , s)). Lemma is proved. □ Lemma 3.5. The hemisphere S n + and the cube I n are Lipschitz equivalent. Proof. Define f : I n Ñ S n + as the composition of two bi-Lipschitz maps, f = g˝φ, where g : B n Ñ S n + is the stereographic projection, and the map φ : I n Ñ B n is given by the formula: where (y 1 , y 2 , . . . , y n ) P Bd(I n ) is such that (y 1 , y 2 , . . . , y n ) = k¨(x 1 , x 2 , . . . , x n ) for some k. The Lipschitz constant for φ is λ 1 = 2. Also it is known that the constant λ 2 for the stereographic projection g equals 2. Therefore, f is bi-Lipschitz with constant λ = λ 1¨λ2 = 4. □ By Lemmas 3.4 and 3.5 taking into account that Cone(S n + ) » R n+1 + we obtain the following result.   ; x, y P R m .
We first prove that f´1 is Lipschitz. Indeed, y 2 )).

Theorem is completed. □
The set P (X) of probability measures on a metric space (X, d) is endowed with the Kantorovich-Rubinstein metric d KR , [2]. The Kantorovich-Rubinstein distance between µ = αδ x + (1´α)δ y , and where tx, x 1 , y, y 1 u P R, can be evaluated by the formula Lemma 3.8. Let r ą 0. For any c ą 1 and K ą 0 there exists an r-disjoint set X in the neighborhood O cr (δ 0 ), (X Ă O cr (δ 0 )), of cardinality greater than K, where δ 0 P P 2 (R).

Proof. Take
First we prove that X Ă O cr (δ 0 ). Indeed, Now we show that X is an r-discrete set.
Let j ą i, then α i ą α j and we see that Therefore X is countable. □ Theorem 3.9. The spaces R 3 + and P 2 (R) are not coarse equivalent. Proof. Assume that R 3 + and P 2 (R) are coarse equivalent. Then there exists a coarse uniform map f : P 2 (R) Ñ R 3 + with constant λ ą 0. Consider an r-discrete set X Ă P 2 (R), f The image of the set X is a r λ -discrete set f (X), diam(f (X)) ď λdiam(X) = λr, and |f (X)| = |X| ą K for all K ą 0. However, the cardinality of any bounded r λ -discrete subset in R 3 + is finite. The obtained contradiction proves the theorem. □ Remark 3.10. A similar result can be proved for the superextension λ 3 (R). Recall that λ 3 (R) can be defined as the quotient space of SP 3 (X) by the identification [x, x, y] " [x, x, z]. Note that λ 3 (S 1 ) is homeomorphic to S 3 (see [8]).
Remark also that the spaces R 3 + and P 2 (R) are not homeomorphic. Indeed, there exists an r-disjoint countable set in the cr-neighborhood of the point δ 0 , therefore δ 0 is not a point of local compactness. (Similar arguments cam be applied to every point δ x , where x P R).
Lemma 2 and Proposition 1 from [6] imply the following statement.
Corollary 3.11. There is no Lipschitz embedding of the space P 2 (R) into the space R n for all n P N.
Let C ą 0. A sequence x 1 , x 2 , . . . , x n in a metric space X is called a Cchain connecting x 1 and x n provided d(x i , x i+1 ) ď C for all i = 1, . . . , n´1. A subset A in X is called asymptotically connected if there is C ą 0 such that every two points in X are elements of some C-chain in A.
A subset A in X is called an asymptotic cut in X between two sets M 1 , M 2 Ă X if for every C ą 0 there is r ą 0 such that every C-chain connecting any point from M 1 and any point from M 2 necessarily intersects B r (A).
Let us recall that the reduced cone C(X) and reduced suspension S(X) of a metric space are defined in [1].
Theorem 3.12. The spaces C(R + ) and S(R + ) are not coarsely equivalent.
The metric on C(R + ) and S(R + ) is the quotient metric induced by the euclidean metric on R 2 + . In the sequel we identify any subset in R 2 + with its quotient image image in C(X) and S(X).
Suppose that there exists a coarse equivalence h : C(X) Ñ S(X). Note that the set i + (R + ) is not an asymptotic cut in C(X). Since the property of being an asymptotic cut is a coarse invariant, we conclude that the set h(i + (R + )) is not an asymptotic cut in S(R + ).
However, lim rÑ8 d ( i˘(R + )zB r (0), h(i + (R + ))zB r (0) ) = 8, and the set h(i + (R + )) is asymptotically connected and unbounded. There is a neighborhood of h(i + (R + )) that contains a broken line from the origin to infinity. Simple geometric arguments show that h(i + (R + )) is an asymptotic cut between i + (R + ) and i´(R + ). □