Conformal inequivalence of annuli and the first-order theory of subgroups of $\textrm {PSL}(2, \textbf {R})$

Abstract:

An algebraic proof is given of the classical fact that two different concentric circular annuli $A(r)$ and $A(s)$ are conformally inequivalent, where $A(r) = \{ z \in {\mathbf {C}}:1 < \left | z \right | < r\}$. Indeed, it is shown that the covering groups of these annuli are not elementarily equivalent in the context of ${\text {PSL}}(2,{\mathbf {R}})$. Considering the universal covering surface as $U$, the upper half-plane, the covering group of a bounded plane domain is naturally contained in ${\text {PSL}}(2,{\mathbf {R}})$ as the group of covering transformations.