Groups with cofinite Zariski topology and potential density (Q6142395)

Let \(\mathscr C\) denote the class of all \(\mathfrak Z\)-cofinite groups, that is, groups \(G\) such that the Zariski topology \(\mathfrak Z_G\) coincides with the cofinite topology of \(G\). For example, \(\mathscr C\) contains all finite groups. \textit{M. Tkachenko} and \textit{I. Yaschenko} [Topology Appl. 122, No. 1--2, 425--451 (2002; Zbl 0997.22003)] proved that for an abelian group \(G\) all proper unconditionally closed subsets of \(G\) are finite precisely when either \(G\) is almost torsion-free or \(G\) has prime exponent. For an abelian group \(G\) the Zariski topology \(\mathfrak Z_G\) coincides with the Markov topology of \(G\), so an abelian group \(G\) is in \(\mathscr C\) precisely when all proper unconditionally closed subsets of \(G\) are finite. Hence, the above theorem provides a characterization of all abelian groups in \(\mathscr C\) as those that are either almost torsion-free or of prime exponent. Another strong motivation for the study of the class \(\mathscr C\) comes from its connection with Markov's notion of potential density and with the problem of complementation of group topologies. This is explained in full detail in the paper. The main aim of this paper is the study of the class \(\mathscr C\) in the general case. Beyond a short alternative proof of the above mentioned characterization of the abelian groups in \(\mathscr C\), it is shown that the statement fails for nilpotent groups of class \(2\). On the other hand, a generalization of that result is provided out of the abelian context using a partial Zariski topology, called \(\mathfrak Z_{mon}\), and an appropriate stronger version of the property almost torsion-free, namely, the weak cancellation law (i.e., for every \(n>0\) the map \(x\mapsto x^n\) in \(G\) is finite-to-one): an infinite group \(G\) is \(\mathfrak Z_{mon}\)-cofinite if and only if either \(G\) satisfies the weak cancellation law or \(G\) has prime exponent. Furthermore, very strong restrictions on the groups in \(\mathscr C\) are provided, which lead to the conjecture that any infinite group in \(\mathscr C\) is necessarily abelian. Several other interesting results are obtained in the paper, moreover related open questions and a detailed background are presented.