ITES2001 Fourth Italian-Spanish conference on General Topology and its applications Bressanone, 27-30 June 2001

On a game in a topological group

Dikran Dikranjan Dipartimento di Matematica e Informatica Università di Udine, Italy

Let G be a topological group. The game 𝔊(G) has two players P 1 and P 2. Player P1 chooses an element x ∈ G and player P2 tries to guess what element x has been chosen by P 1 by asking questions of the following form. For a sequences \underline{m} = \{m_{n}\} of integers the question Q\underline{m} is: does x^{m_{n}} converge to 1 in G when n → ∞ ? The aim of the player P2 is to accumulate as much information about x as possible knowing the answers of all 𝔠 many questions Q\underline{m}, i.e., the set S(x) = {{m_{n}} ∈ ℤ^{ω} : lim x^{m_{n}} = 1}. Since S(x) = S(x^{−1}), P 2 can guess at most the doubleton {x, x^{−1}}, in case x is non-torsion, or the set gen(x) of all generators of the cyclic group ⟨x⟩. In such a case we say that P 2 wins (in other words, P2 wins if S(x) = S(y) always implies ⟨x⟩ = ⟨y⟩ in G). We prove that:

(a) P2 wins the game 𝔊(G) for a non-discrete locally compact group G iff G is isomorphic to the circle group;

(b) P2 wins the game 𝔊(G) for a discrete group G iff G is isomorphic to a

subgroup of the dicrete group ℚ/ℤ. We also discuss the groups G where the game 𝔊(G) is most optimal for player P1, namely those with minimum information for P2 (i.e., S(x) = S(y) whenever x, y are non-trivial elements of G, e.g., the reals ℝ, the group of p-adic numbers, etc.). We give a complete characterization of these groups G in the class of locally compact abelian groups. The proofs are based on the structure theory of locally compact groups and properties of topologically torsion elements (cf. [1]).

References

[1] D. Dikranjan, Answer to a question of Armacost on topologically torsion elements, Second Honolulu Conference on Abelian Groups and Modules, July 25 – August 1, 2001, Hawaii, Preprint.