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Hyperbolic geometry
 In Flavors of geometry
, 1997
"... 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65 ..."
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3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65
Spaces Which Are Not Negatively Curved
 Comm. in Anal. and Geom
, 1997
"... this paper will be 2dimensional. Definition of a lamination ..."
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this paper will be 2dimensional. Definition of a lamination
Expansion complexes for finite subdivision rules
 I, Conform. Geom. Dyn
"... Abstract. This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a onetile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has an ..."
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Abstract. This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a onetile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has an invariant partial conformal structure, and hence is conformal. The paper next considers onetile single valence finite subdivision rules. It is shown that an expansion map for such a finite subdivision rule can be conjugated to a linear map, and that the finite subdivision rule is conformal exactly when this linear map is either a dilation or has eigenvalues that are not real. Finally, an example is given of an irreducible finite subdivision rule that has a parabolic expansion complex and a hyperbolic expansion complex. 1.
Boundary curves of surfaces with the 4plane property
 Geom. Topol
"... Let M be an orientable and irreducible 3–manifold whose boundary is an incompressible torus. Suppose that M does not contain any closed nonperipheral embedded incompressible surfaces. We will show in this paper that the immersed surfaces in M with the 4–plane property can realize only finitely many ..."
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Let M be an orientable and irreducible 3–manifold whose boundary is an incompressible torus. Suppose that M does not contain any closed nonperipheral embedded incompressible surfaces. We will show in this paper that the immersed surfaces in M with the 4–plane property can realize only finitely many boundary slopes. Moreover, we will show that only finitely many Dehn fillings of M can yield 3–manifolds with nonpositive cubings. This gives the first examples of hyperbolic 3–manifolds that cannot admit any nonpositive cubings.
Problems around 3–manifolds J
"... This is a personal view of some problems on minimal surfaces, Ricci flow, polyhedral ..."
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This is a personal view of some problems on minimal surfaces, Ricci flow, polyhedral
arXiv version: fonts, pagination and layout may vary from AGT published version Onesided Heegaard splittings of RP 3
"... Using basic properties of onesided Heegaard splittings, a direct proof that geometrically compressible onesided splittings of RP 3 are stabilised is given. The argument is modelled on that used by Waldhausen to show that twosided splittings of S 3 are standard. 57M27; 57N10 1 ..."
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Using basic properties of onesided Heegaard splittings, a direct proof that geometrically compressible onesided splittings of RP 3 are stabilised is given. The argument is modelled on that used by Waldhausen to show that twosided splittings of S 3 are standard. 57M27; 57N10 1