Weight \(\mathbf {1/2}\) multiplier systems for the group \(\mathbf {\Gamma _0^+({\varvec{p}})}\) and a geometric formulation

We construct a weight 1/2 multiplier system for the group \(\Gamma _0^+(p)\) ${\mathrm{\Gamma }}_0^{+}(p)$ , the normalizer of the congruence subgroup \(\Gamma _0(p)\) ${\mathrm{\Gamma }}_0(p)$ where p is an odd prime, and we define an analogue of the eta function and Rademacher symbol and relate it to the geometry of edge paths in a triangulation of the upper half-plane.