Riemann surfaces of finite type can be first classified in terms of the genus, the number of holes and the number of punctures. If this data is given then Teichmüller theory describes the totality of Riemann surfaces with these characteristics. The situation with Riemann surfaces of infinite type is quite different. Here there are very different constructions leading to a vast plethora of types. The purpose of this paper is to study a class of examples called “flute surfaces” which can, from one point of view, be seen as particularly simple. Let \(S=\mathbf c - \{1,2,3,\dots \}\). (Considered as a subset of a cylinder this looks roughly like a flute.) Complex conjugation induces an involution of \(S\). Any Riemann surface homeomorphic to \(S\) is called a flute surface; if the metric is invariant under the involution it is called an untwisted flute surface. A flute surface is conformally equivalent to a domain in the complex plane with just one infinite end, i.e. an end that is neither a puncture or a hole. The main theorem proved here asserts that if \(K\) is any compact subset of the real line then there is a Fuchsian group \(G\) and a point \(p\) of the upper half–plane so that the boundary of the Dirichlet fundamental domain of \(G\) centered at \(p\) consists of \(K\) together with a countable set of isolated cusps and such that the quotient of the upper half–plane by \(G\) is homeomorphic to \(S\). To prove this the authors analyse the class of flute surfaces using the methods of planetopology and hyperbolic geometry, especially the geometry of geodesics.