Cardinal characteristics for Menger-bounded subgroups
by
Heike Mildenberger
Kurt Goedel Research Center for Mathematical Logic. Universitaet Wien

The Baer-Specker group is Z^w with pointwise addition. Let G ⊆ Z^w be a subgroup. For g : w→ Z, we write g'(n) = max{|g(m)| : m ≤ n}. Let k ∈ w\{0}. We say "G^k is Menger-bounded" or "G has Menger-bounded k-th power" iff

(∃f ∈ [w]w) (∀F ∈ [G]k)( ∃∞ n)(∀g ∈ F) (g'(n) ≤ f(n)).

This is syntactically the simplest of the equivalent characterisations given in [2, Theorem 5]. Menger-boundedness in a broader sense is defined for topological groups and also called o-boundedness. We refer the reader to [3] for more information.

Machura, Shelah and Tsaban showed in [2] that under the condition that a relative d'(P) of the dominating number is at least d there are subgroups of the Baer-Specker group whose k-th power is Menger-bounded and whose (k+1)-st power is not. We show that the sufficient condition implies r ≥ d and indeed can be replaced by r ≥ d. This result includes an affirmative answer to a question by Tsaban on a possibly weaker still sufficient condition. We show that it is consistent relative to ZFC that g ≤ r < d and there are subgroups of the Baer-Specker group whose k-th power is Menger-bounded and whose (k+1)-st power is not.

This talk is mainly a report on [3].

[1] Liljana Babinkostova, Ljubisa D.R. Kocinac, and Marion Scheepers. Combinatorics of open covers (XI): Menger- and Rothberger-bounded groups. Topology and its Applications, 154(7):1269-1280, 2007.

[2] Michal Machura, Saharon Shelah, and Boaz Tsaban. Squares of Menger-bounded groups. To appear in The Transactions of the American Mathematical Society, http://arxiv.org/pdf/math.GN/0611353, 2007.

[3] Heike Mildenberger. Cardinal characteristics for Menger-bounded subgroups of the Baer-Specker group. To appear in Topology and its Applications, 2008.

2000 Mathematics Subject Classification: 03E15, 03E17, 03E35, 22A25.

Heike Mildenberger, Universität Wien, Kurt Gödel Research Center for Mathematical Logic, Währinger Str. 25, 1090 Vienna, Austria

Date received: April 30, 2008