Let p be a reflection on a closed Riemann Surface S, i.e., an anti-conformal involutary isometry of S with a non-empty fixed point subset. Let Sp denote the fixed point subset of p, which is also called the mirror of p. If S −Sp has two components, then p is called separating and we say that S splits at the mirror Sp. Otherwise p is called non-separating. We assume that the system of mirrors, Sq, as q varies over all reflections in the isometry group Aut*(S) defines a tiling of the surface, consisting of triangles. In turn, the tiling determines a subgroup G of Aut*(S) of conformal automorphisms of S. We give a simple criterion, derived from the geometry of the tiling, for determining whether the reflection is separating by means of equations in the rational group algebra of G. Examples for abelian G, where the computations are especially simple, are presented.
Broughton, Sean A., "Splitting Tiled Surfaces with Abelian Conformal Tiling Group" (1999). Mathematical Sciences Technical Reports (MSTR). 58.