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Title: Iterated medial subdivision in surfaces of constant curvature and applications to acute triangulations of hyperbolic and spherical simplicial complexes.

Abstract: Consider a triangle in the Euclidean plane and subdivide it recursively into 4 sub-triangles by joining its midpoints. Each generation of this iterated subdivision yields triangles which are all similar to the original one and exactly twice as small as the triangle(s) of the previous generation. What happens when we perform this iterated medial triangle subdivision on a geodesic triangle when the underlying space is not Euclidean? I will first produce examples of various unfamiliar and unexpected behaviours of this subdivision in non-Euclidean geometries. I will then show that this iterated subdivision nevertheless "stabilizes in the limit" (in a sense that will be made precise) when the underlying space is of constant non-zero curvature. My aim is to combine this result with a forthcoming result of Christopher Bishop on conforming triangulations of PSLGs to construct acute triangulations of hyperbolic and spherical simplicial complexes.

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