LIFTINGS OF MÖBIUS GROUPS TO MATRIX GROUPS

It is well known that any discrete subgroup G of PSL(2, C) can be lifted to a subgroup of SL(2, C) if and only if G does not have two-torsion ([1], [4]). In this note we show the existence of liftings of certain discrete subgroups of the group M(U) of orientation preserving and reversing Möbius transformations mapping the upper half-plane U onto itself. The orientation preserving Möbius transformations mapping the upper half-plane onto itself form the connected subgroup PSL(2, R) of M(U). The lifting of subgroups of M(U) to SL(2, C) has two complications: first of all, the group M(U) is not connected, and, secondly, there is no natural mapping from PSL(2, C) to M(U). In this note we consider these problems and show that all subgroups G of M(U) for which U/G is a compact Klein surface can be lifted to SL(2, C). This result is closely related with the general considerations of M. Culler ([1]) and is based on the arguments presented in [8]. We also study the uniqueness of liftings of subgroups of PSL(2, C) to SL(2, C). Assume that G ⊂ PSL(2, C) can be lfted to a subgroup of SL(2, C). Consider $\hat{\mathrm{G}}={\cap }_{\tilde{\mathrm{G}}}\tilde{\mathrm{G}}$ where G̃ goes through all liftings of G to subgroups of SL(2, C). Let G# ⊂ G be the projection of Ĝ. The group G# is non-trivial because it contains, e.g., all squares of elements of G. We show that –under rather general conditions – G# is actually generated by squares and commutators of elements of G.