The coarse classification of locally finite and abelian groups
by
Taras Banakh
Ivan Franko National University of Lviv
Coauthors: Ihor Zarichnyy, Jose Higes

A multi-valued map F:X⇒ Y between two metric spaces is called bornologous if for each e there is d such that for each subset A ⊂ X of diam(A) < e the image F(A)=∪_{a ∈ A}F(a) has diam F(A) < d.

Two metric spaces X, Y are called coarsely equivalent (or else asymorphic) if there is a multi-valued map F:X⇒ Y such that F(X)=Y, F^-1(Y)=X and both the maps F, F^-1 are bornologous.

A metric space X is

• proper if each closed bounded subset of X is compact;

• homogeneous if for any points x, y ∈ X there is a bijective isometry f:X→ X with f(x)=y.

Theorem 1 Each unbounded proper homogeneous metric space X of asymptotic dimension zero is coarsely equivalent to the anti-Cantor set Z₂^∞.

The anti-Cantor set is the direct sum Z₂^∞ of countably many copies of the 2-element group Z₂, endowed with the ultrametric

dist((xi), (yi))= max i  2i|xi-yi|.

The anti-Cantor set is the asymptotic counterpart of the Cantor cube Z₂^w endowed with the ultrametric

dist((xi), (yi))= max i  2-i|xi-yi|.

It is known that each countable G group carries a left-invariant metric turning G into a proper metric space. Such a metric is unique up to a bijective coarse equivalence, see [S]. So, each countable group G can be thought as a homogeneous proper metric space. This space is asymptotically zero-dimensional if and only if the group G is loclly finite in the sense that each finite subset of G generates a finite subgroup, see [S].

Now we see that the above Theorem implies

Theorem 2 Each countable loclaly finite group is coarsely equivalent to the anti-Cantor set.

This theorem can be applied to derive the following coarse classification of countable Abelian groups.

Theorem 3 A countable Abelian group G of asymptotic dimension asdim(G)=n is coarsely equivalent

• to Zⁿ if G is finitely generated and

• to Zⁿ×Z₂^∞ if G is infinitely generated.

References:

[BZ] T.Banakh, I.Zarichnyy, The coarse classification of homogeneous ultra-metric spaces // preprint (arXiv:0801.2132)

[BHZ] T.Banakh, J.Higes, I.Zarichnyy, The coarse classification of countable Abelian groups // preprint.

[S] J. Smith, On asymptotic dimension of countable abelian groups // Topology Appl. 153:12 (2006), 2047-2054.

Date received: April 13, 2008