Lifting Möbius groups (Q1866412)

Let \(\pi: G\to H\) be a surjective group homomorphism. We say that a subgroup \(K\) of \(H\) {can be lifted to \(G\) by \(\pi\)} if there exists a subgroup \(K'\) of \(G\) such that \(\pi\) induces an isomorphism between \(K'\) and \(K\). A classical case of study of liftings is the case of the natural map \(\pi: SL(2,\mathbb C)\to PSL(2,\mathbb C)\). It is easily seen that no group of \(PSL(2,\mathbb C)\) with an element of order two can be lifted. It is known from a result of \textit{M. Culler} that a discrete subgroup of \(PSL(2,C)\) with no element of order two can be lifted [Adv. Math. 59, 64--70 (1986; Zbl 0582.57001)]. In the paper under review, the author considers the case of indiscrete subgroups. He shows that in general these cannot be lifted. He also finds torsion free subgroups of \(PSL(2,\mathbb R)\) and \(PSU(2)\) that cannot be lifted. The constructions involve considering a class of subgroups of \(PSL(2,\mathbb C)\) which are generated by four elements and the arguments involve techniques of transcendental number theory. Finally, the author considers the question of the difference between lifts of subgroups and lifts of representations.