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Background NotesThe Background Notes entitled Kaleidoscopic Tilings on Surfaces are a continually evolving set of notes that provide a background of tiling theory and and hyperbolic geometry. They are used on an as needed basis. The first three chapters provide a quick introduction to tiling theory, and a description of some of the main REU research problems. The remaining chapters provide various bit of background theory.
Student Technical Reports (by year in reverse chronological order)Summer 2003When Abelian Groups Split: Robert C. Rhoades and Rachel L. Thomas MSTR 031 (TRrhoadesthomas.pdf)
Summer 2002Tilings of LowGenus Surfaces by Quadrilaterals: John Gregoire and Isabel Averill MSTR 0213 (TRaverillgregoire.pdf)
Descripton of the Limiting Surfaces of Hyperbolic Surfaces Tiled by Quadrilaterals: Michael A. Burr and Katrhyn M. Zuhr (in preparation). Constructing the Moduli Space of Riemann Surfaces with a G (k,l,m,n) Action: Katrhyn M. Zuhr (in preparation). Summer 2001PigeonHoling Monodromy Groups: Niles G. Johnson MSTR 0207 (TRjohnson.pdf).
The Galois Correspondence for Branched Covering Spaces and Its Relationship to Hecke Algebras: Matt Ong, MSTR 0208 (TRong.pdf)
Applications of Graph Theory to Separability, Steve Young MSTR 0209 (TRyoung.pdf).
Towards Finding Fundamental Domains for Hurwitz Groups: Yvonne Lai (in preparation). Summer 2000Hyperbolic Billiard Paths, Rebecca Lehman, Chad White MSTR 0202 (TRlehmanwhite.pdf)
Oval Lengths for Split Metacyclic Groups: Shaun McCance and Sarah Weissman (in prepration) Summer 1999Lengths of Systoles on Tileable Hyperbolic Surfaces: Kevin Woods, MSTR 0009 (TRwoods.pdf).Abstract: The same triangle may tile geometrically distinct surfaces of the same genus, and these tilings may determine isomorphic tiling groups. it is determined if there are geometric differences in the surfaces that can be found using group theoretic methods. Specifically, it is determined if the systole, the shortest closed geodesic on a surface, can distinguish a certain families of tilings. For example, there are three tilings of surfaces of genus 14 by the hyperbolic triangle with angles Pi/2, Pi/3, and Pi/7 whose tiling groups are all PSL(2,7). These tilings can be distinguished by the lengths of their systoles.Separability of Tilings: Nick Baeth, Jason Deblois, Lisa Powell, MSTR 0010 (TRbaethdebloispowell.pdf). Abstract: A tiling by triangles of an orientable surfaces is called kaleidoscopic if the local reflection in any edge of a triangle extends to a global isometry of the surface. Given such a global reflection the fixed point subset of the reflection consists of embedded circles (ovals) whose union is called the mirror of the reflection. The reflection is called separating if removal of the mirror disconnects the surface into two components. We consider surfaces such that the orientation preserving subgroup of the tiling group generated by the reflections is cyclic or abelian. A complete classification of those surfaces with separating reflections is obtained in the cyclic case as well as partial results for abelian, noncyclic groups. Summer 1998Lengths of Geodesics on Klein's Quartic Curve: Ryan DerbyTalbot, MSTR 0003 (TRderbytalbot.pdf).Abstract: A wellknown and much studied Riemann surface is Klein's quartic curve. This surface is interesting since it is the smallest complex curve with maximal symmetry. In addition to this high degree of symmetry, Klein's quartic curve can be tiled by triangles, giving rise to a tiling group generated by reflections. Using the tiling group and the universal cover of the tiling group we are able to compile a list of the lengths of the short, simple, closed geodesics on this surface. In particular, we are able to determine whether the geodesic loops generated by the tiling are the systoles, i.e., the shortest closed geodesics.Quest for Tilings on Riemann Surfaces of Genus Six and Seven: Robert M. Dirks and Maria T. Sloughter, MSTR 0008 (TRdirkssloughter.pdf). Abstract: The problem of kaleidoscopically tiling a surface by congruent triangles is equivalent to finding groups generated in certain ways. In order to admit a tiling, a group must have a specific set of generators as well as an involutary automorphism, $\theta$, that acts to reverse the orientation of the tiles. The purpose of this paper is to explore group theoretic and computational methods for determining the existence of symmetry groups and tiling groups, as well as to classify the symmetry and tiling groups on hyperbolic Riemann surfaces of genus 6 and 7. Summer 1997Symmetry and Tiling Groups for Genus 4 and 5: C. Ryan Vinroot, MSTR 9802, (TRvinroot.pdf).Abstract: All symmetry groups for surfaces of genus 2 and 3 are known. In this paper, we classify symmetry groups and tiling groups with three branch points for surfaces of genus 4 and 5. Also, a class of nontiling symmetry groups is presented, as well as a class of odd order nonabelian tiling groups.Quadrilaterals Subdivided by Triangles in the Hyperbolic Plane: Dawn M. Haney and Lori T. McKeough, MSTR 9804 (TRhaneymckeough.pdf). Abstract: In this paper, we consider trianglequadrilateral pairs in the hyperbolic plane which "kaleidoscopically'' tile the plane simultaneously. These tilings are called divisible tilings }or subdivided tilings. We restrict our attention to the simplest case of divisible tilings, satisfying the corner condition, in which a single triangle occurs at each vertex of the quadrilateral. All possible such divisible tilings are catalogued as well as determining the minimal genus surface which the divisible tiling exists. The tiling groups of these surfaces are also determined.Tilings which Split at a Mirror: Jim Belk, MSTR 9901, (TRbelk.pdf). Abstract: Let r be a reflection on a surface S. The mirror of r is the fixed point subset S_{r} ={x in S : r(x)=x} which is a disjoint union of circles . We say that S splits at the mirror of r if SS_{r} is disconnected. We further assume that the reflection is a symmetry of a tiling of S by triangles. In this paper we investigate a number of conditions on the tiling that guarantee that S splits at a mirror. Summer 1996Ovals Intersections in Tilings on Surfaces: Dennis A. Schmidt, MSTR 9703 (TRschmidt.pdf).Abstract: A tiling is a covering by polygons, without gaps or overlapping, of a compact, orientable surface. We are particularly interested in tilings by triangles that generate a large symmetry group of the surface. An oval of the tiling is a simple, closed curve that is a union of edges of the tiling. We investigate the number of points of intersection of two ovals. We have found that the number of intersections is bounded when the subgroup of orientation preserving symmetries is abelian. However, there is no upper bound on the number of intersections in the nonabelian case. Submitted and Published Papers (latest first)Triangular Surface Tiling Groups for Low Genus: S. Allen Broughton, Robert M. Dirks, Maria T. Sloughter, C. Ryan Vinroot MSTR 0101 ( preprint technical report tileclass.pdf),Abstract: Consider a surface, S, with a kaleidoscopic tiling by nonobtuse triangles (tiles), i.e., each local reflection in a side of a triangle extends to an isometry of the surface, preserving the tiling. The tiling is geodesic if the side of each triangle extends to a closed geodesic on the surface consisting of edges of tiles. The reflection group G^{*}, generated by these reflections, is called the tiling group of the surface. This paper classifies, up to isometry, all geodesic, kaleidoscopic tilings by triangles, of hyperbolic surfaces of genus up to 13. As a part of this classification the tiling groups G^{*} are also classified, up to isometric equivalence. The computer algebra system Magma is used extensively.Symmetry and Tiling Groups for Genus 4 and 5: C. Ryan Vinroot, RoseHulman Institute of Technology Undergraduate Mathematics Journal, 1#1 (2000). (link to journal, link to paper) Abstract: See reference above. Divisible Tilings in the Hyperbolic Plane: Dawn M. Haney, Lori T. McKeough, Brandy M. Smith, Allen Broughton, New York Journal of Mathematics 6 (2000), 237283. (link to journal, link to paper) Abstract: We consider trianglequadrilateral pairs in the hyperbolic plane which "kaleidoscopically'' tile the plane simultaneously. In this case the tiling by quadrilaterals is called a divisible tiling. All possible such divisible tilings are classified. There are a finite number of 1, 2, and 3 parameter families as well as a finite number of exceptional cases Related Papers and Technical ReportsCounting Ovals on a Symmetric Riemann Surface: S. Allen Broughton, MSTR 9704 (ovals.pdf).Abstract: Let S be a compact Riemann surface without boundary. A symmetry, t, of S is an anticonformal, involutary automorphism. The fixed point set of t is a disjoint union of circles, each of which is called an oval of t. A method is presented for counting the ovals of a symmetry when S admits a large group G of automorphisms, normalized by t. The method involves only calculations in G, based on the geometric description of S/G, and the knowledge of the action of t on G. As an illustrative example, the family of generic, symmetric surfaces of genus three with conformal automorphism group S_{4} is constructed as several 1parameter families.Constructing Kaleidoscopic Tiling Polygons in the Hyperbolic Plane, S. Allen Broughton, American Math. Monthly, October, 2000 (link to MAA summary page). Splitting tiled surfaces with abelian conformal tiling group. S. Allen Broughton, MSTR 9903 (septri.pdf) Abstract: Let r be a reflection on a closed Riemann S i.e., an anticonformal involutary isometry of S with a nonempty fixed point subset. Let S_{r} denote the fixed point subset of r, which is also called the mirror of r. If SS_{r} has two components then r is called separating and that S splits at the mirror S_{r}. Otherwise r is called nonseparating. We assume that the system of mirrors, S_{r}, as r 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, lying in Aut^{*}(S), consisting 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. Presentations
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