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We consider an exhaustion of the modular orbifold by compact subsurfaces and show that the growth rate, in terms of word length, of the reciprocal geodesics on such subsurfaces (so named low lying reciprocal geodesics) converge to the growth rate of the full set of reciprocal geodesics on the modular orbifold. We derive a similar result for the low lying geodesics and their growth rate convergence to the growth rate of the full set of closed geodesics.
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We prove that there exists a homeomorphism $chi$ between the connectedness locus $mathcal{M}_{Gamma}$ for the family $mathcal{F}_a$ of $(2:2)$ holomorphic correspondences introduced by Bullett and Penrose, and the parabolic Mandelbrot set $mathcal{M}_1$. The homeomorphism $chi$ is dynamical ($mathcal{F}_a$ is a mating between $PSL(2,mathbb{Z})$ and $P_{chi(a)}$), it is conformal on the interior of $mathcal{M}_{Gamma}$, and it extends to a homeomorphism between suitably defined neighbourhoods in the respective one parameter moduli spaces. Following the recent proof by Petersen and Roesch that $mathcal{M}_1$ is homeomorphic to the classical Mandelbrot set $mathcal{M}$, we deduce that $mathcal{M}_{Gamma}$ is homeomorphic to $mathcal{M}$.
We prove that the Koebe circle domain conjecture is equivalent to the Weyl type problem that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of the complement of a circle domain. It provides a new way to approach the Koebes conjecture using convex geometry. Combining our result with the work of He-Schramm on the Koebe conjecture, one establishes that every simply connected non-compact polyhedral surface is discrete conformal to the complex plane or the open unit disk. The main tool we use is Schramms transboundary extremal lengths.
To enumerate 3-manifold triangulations with a given property, one typically begins with a set of potential face pairing graphs (also known as dual 1-skeletons), and then attempts to flesh each graph out into full triangulations using an exponential-time enumeration. However, asymptotically most graphs do not result in any 3-manifold triangulation, which leads to significant wasted time in topological enumeration algorithms. Here we give a new algorithm to determine whether a given face pairing graph supports any 3-manifold triangulation, and show this to be fixed parameter tractable in the treewidth of the graph. We extend this result to a meta-theorem by defining a broad class of properties of triangulations, each with a corresponding fixed parameter tractable existence algorithm. We explicitly implement this algorithm in the most generic setting, and we identify heuristics that in practice are seen to mitigate the large constants that so often occur in parameterised complexity, highlighting the practicality of our techniques.
We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,mathbb{Z})$-module.
Discrete conformal structure on polyhedral surfaces is a discrete analogue of the smooth conformal structure on surfaces that assigns discrete metrics by scalar functions defined on vertices. In this paper, we introduce combinatorial $alpha$-curvature for discrete conformal structures on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. Then we prove the local and global rigidity of combinatorial $alpha$-curvature with respect to discrete conformal structures on polyhedral surfaces, which confirms parameterized Glickenstein rigidity conjecture. To study the Yamabe problem for combinatorial $alpha$-curvature, we introduce combinatorial $alpha$-Ricci flow for discrete conformal structures on polyhedral surfaces, which is a generalization of Chow-Luos combinatorial Ricci flow for Thurstons circle packings and Luos combinatorial Yamabe flow for vertex scaling on polyhedral surfaces. To handle the potential singularities of the combinatorial $alpha$-Ricci flow, we extend the flow through the singularities by extending the inner angles in triangles by constants. Under the existence of a discrete conformal structure with prescribed combinatorial curvature, the solution of extended combinatorial $alpha$-Ricci flow is proved to exist for all time and converge exponentially fast for any initial value. This confirms a parameterized generalization of another conjecture of Glickenstein on the convergence of combinatorial Ricci flow, gives an almost equivalent characterization of the solvability of Yamabe problem for combinatorial $alpha$-curvature in terms of combinatorial $alpha$-Ricci flow and provides an effective algorithm for finding discrete conformal structures with prescribed combinatorial $alpha$-curvatures.
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