Universal valued Abelian groups

Abstract

The counterparts of the Urysohn universal space in category of metric spaces and the Gurarii space in category of Banach spaces are constructed for separable valued Abelian groups of fixed (finite) exponents (and for valued groups of similar type) and their uniqueness is established. Geometry of these groups, denoted by G_r(N), is investigated and it is shown that each of G_r(N)'s is homeomorphic to the Hilbert space l^2. Those of G_r(N)'s which are Urysohn as metric spaces are recognized. `Linear-like' structures on G_r(N) are studied and it is proved that every separable metrizable topological vector space may be enlarged to G_r(0) with a `linear-like' structure which extends the linear structure of the given space.Comment: 60 page

Generic norms and metrics on countable abelian groups

“…It is the countable Boolean group (see [11]). The case of exponent 2 is obviously special and it is open whether other bounded countable abelian groups admit such a norm (see the open problems in [10], where it is proved that groups of exponent 3 do not admit such a norm). We conjecture that they do not.…”

“…If (G, λ) is a normed abelian group and g ∈ G then lim n→∞ λ(n·g) n exists and is equal to inf n λ(n·g) n . Following Niemiec in [10], by O 0 we denote the class of those abelian normed groups (G, λ) such that for all g ∈ G, lim n λ(n·g) n = 0. The next lemma shows that if there is a generic norm λ on G, then necessarily (G, λ) ∈ O 0 .…”

“…}, from [10], are the only generic abelian Polish groups. We remark here that G ∞ (0), constructed in [10], is the completion of the Fraïssé limit of all finite abelian groups with rational norms. For N ∈ {2, 3, 4, .…”

“…}, G ∞ (N ) is the completion of the Fraïssé limit of all finite abelian groups of exponent N with rational norms. We refer the reader to [10] for more details.…”

“…In particular, the completion is isometric to the Urysohn space and thus the Urysohn space has a structure of a monothetic abelian group. Niemiec in [10] proves that the Shkarin's universal metric abelian Polish group is isometric to the Urysohn space and its canonical countable dense subgroup (which is N Q/Z) is isometric to the rational Urysohn space. Here we generalize these results by proving that actually every unbounded countable abelian group, i.e.…”