Unpublished Paper

Splitting Tiled Surfaces with Abelian Conformal Tiling Group

Mathematical Sciences Technical Reports (MSTR)
Document Type

Article
Publication Date

9-26-1999
Disciplines

Abstract

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.
Citation Information

Sean A Broughton. "Splitting Tiled Surfaces with Abelian Conformal Tiling Group" (1999) Available at: http://works.bepress.com/allen_broughton/14/

MSTR 99-03