Darren Long and Alan Reid defined a pseudo-modular surface to be a hyperbolic surface which has cusp set all of the rationals, but which is not commensurable to the modular surface. We will show how to construct pseudo-modular surfaces by piecing together suitable marked hyperbolic ideal triangle tiles. In particular, we will show that this allows us to construct infinitely non-commensurable Pseudo-modular surfaces, which answers a question of Darren Long and Alan Reid. This is joint work with Beicheng Lou and Vo Anh Duc.

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