Coding of geodesics on some modular surfaces and applications to odd and even continued fractions

The connection between geodesics on the modular surface PSL(2,Z)∖H and regular continued fractions, established by Series, is extended to a connection between geodesics on Γ∖H and odd and grotesque continued fractions, where Γ≅Z3∗Z3 is the index two subgroup of PSL(2,Z) generated by the order three elements 0−111 and 01−11, and having an ideal quadrilateral as fundamental domain.A similar connection between geodesics on Θ∖H and even continued fractions is discussed in our framework, where Θ denotes the Theta subgroup of PSL(2,Z) generated by 0−110 and 1201.