Group properties invariant under Bohr-homeomorphisms
by
Dikran Dikranjan
University of Udine, Italy

We discuss the Bohr topology G^# of discrete abelian groups G. For a bounded group G the essential order eo(G) of G is the smallest positive integer m such that mG is finite. Call G almost homogeneous if G has at most one infinite Ulm-Kaplansky invariant for every prime p (note that bounded groups of square-free essential order are almost homogeneous). Call a pair G, H of infinite abelian groups:

(a) Bohr-equivalent if G^# is homeomorphic to H^#;

(b) strongly Bohr-equivalent if G^a and H^a are Bohr-equivalent for every cardinal a;

(c) weakly Bohr-equivalent if there exist embeddings G^# --> H^# and H^# --> G^#;

(d) almost isomorphic if G and H have isomorphic finite index subgroups ([5]).

(e) weakly isomorphic if |mG|=|mH| for every natural m such that at least one of these cardinals is infinite.

Almost isomorphic abelian groups need not be strongly Bohr-equivalent, but they are always Bohr-equivalent ([4], according to [1] this implication is not reversible for non-bounded groups). The study of Bohr-equivalence was motived by van Douwen's question of whether abelian groups of the same cardinality are necessarily Bohr-equivalent. Kunen [5] proved that the direct sum V_p^a of a copies of Z_p and V_q^a are not weakly Bohr-equivalent for distinct primes p, q and a=\omega (the counterpart for V₂^a, V_q^a, q > 2 and a > 2^2c was proved in [2]). Recently Givens and Kunen [3] proved that the weak Bohr-equivalence preserves boundedness. Moreover, eo(G)=eo(H) for weakly Bohr-equivalent bounded groups G, H such that one of them is either countable or has a prime exponent.

We show that

(1) ``weakly isomorphic" ===> ``weakly Bohr-equivalent" ===> eo(G)=eo(H) for arbitrary bounded abelian groups G, H.

(2) ``weakly Bohr-equivalent" ===> ``weakly isomorphic" ===> ``almost isomorphic" for almost homogeneous abelian groups G, H (so these three properties, along with "Bohr-equivalent", coincide for almost homogeneous abelian groups).

(3) eo(G)=eo(H) ===> ``weakly isomorphic" for countable bounded abelian groups (so all three properties in (1) coincide in the case of countable groups).

(4) if G and H are strongly Bohr-equivalent, then they are simultaneously torsion-free (resp. p-torsion-free, for any prime p).

(5) if G and H are weakly isomorphic bounded abelian groups (in particular, if G, H are Bohr-equivalent and almost homogeneous), then G admits a pseudocompact group topology iff H does.

One can replace in (5) ``pseudocompact" by ``countably compact and hereditarily separable" (or just ``countably compact" when the groups have size at most 2<sup>c</sup>) in appropropriate forcing model of ZFC where 2<sup>c</sup> is ``arbitrarily large''. In case the groups have size at most c, this can be done also in any model of ZFC satisfying MA.

The following specific questions remain open:

(i) ([5]) are the weakly isomorphic (hence weakly Bohr-equivalent) groups V₄^\omega and V₂^\omega×V₄^\omega also Bohr-equivalent?

(ii) are the groups V₄^\omega₁ and V₂^\omega₁×V₄^\omega weakly Bohr-equivalent?

[1] W. Comfort, S. Hernández and F. Javier Trigos-Arrieta, Cross Sections and Homeomorphism Classes of Abelian Groups Equipped with the Bohr Topology, Topology Appl.115, n. 2 (2001) 215-233.

[2] D. Dikranjan and S. Watson, A Solution to van Douwen's Problem on Bohr Topologies, Jour. Pure Appl. Algebra 163, n. 2 (2001), 147-158.

[3] B. Givens and K. Kunen, Chromatic Numbers and Bohr Topologies, Topology Appl. to appear.

[4] J. Hart and K. Kunen, Bohr compactifications of discrete structures, Fund. Math. 160 (1999), no. 2, 101-151.

[5] K. Kunen, Bohr topology and partition theorems for vector spaces, Topology Appl. 90 (1998) 97-107.

Date received: May 29, 2003