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) \ IR and regular continued fractions, established by Series, is extended to a connection between geodesics on Gamma \ H and odd and grotesque continued fractions, where Gamma congruent to Z(3) * Z(3) is the index two subgroup of PSL(2, Z) generated by the order three elements [GRAPHICS] and [GRAPHICS] , and having an ideal quadrilateral as fundamental domain. A similar connection etween geodesics on Theta \ H and even continued fractions is discussed in our framework, where Theta denotes the Theta subgroup of PSL(2, Z) generated by [GRAPHICS] and [GRAPHICS] . (C) 2018 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.