Notes on balanced groups (Q2767752)

Continuing earlier papers by Protasov and Saryev [\textit{I. V. Protasov}, Math. Notes 49, 614-616 (1991); translation from Mat. Zametki 49, 87-91 (1991; Zbl 0737.22001); \textit{I. V. Protasov} and \textit{A. Saryev}, Izv. Akad. Nauk Turkm. SSR, Ser. Fiz.-Tekh. Nauk 3, 21-25 (1988; Zbl 0686.22001)], this paper compares the concepts ``balanced'', ``functionally balanced'' and ``discretely balanced'' for topological groups. G is balanced if the right and left uniformities coincide; the other concepts refer to left-right invariance of uniform continuity for real valued functions, and of uniform discreteness for subsets. Among other results, it is shown that all three concepts coincide for locally connected groups. The paper also characterizes certain classes of groups which fail to satisfy one or the other of the balancedness conditions.