We give a method for obtaining a handle decomposition of an -manifold if the manifold is given by isometric side-pairings of a polyhedron in $\en$, $\sn$ or $\hn$. Every cycle of -faces on the polyhedron corresponds to an -handle of the manifold. Two applications of the method are given. One helps recogniz…
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7 results for “side-pairings”
Optimal Farey sequence for with upper bound .
problem Finding an optimal Farey sequence for the congruence subgroup .
method Proving the existence of a Farey sequence with specific properties and uniqueness.
result The upper bound of the Farey sequence is optimal and equals .
Algorithm finds Dirichlet domains for hyperbolic surfaces.
problem Computing explicit Dirichlet domains for hyperbolic surfaces.
method Algorithm based on side pairings of fundamental polygons, using geometric-topological data structures.
result Algorithm runs in polynomial time, dependent on surface perimeter and genus.
The paper examines conditions for Gromov-Hausdorff convergence of metric quotients and provides examples of conic-flat surfaces.
problem Conditions for Gromov-Hausdorff convergence of metric quotients.
method Analyzes sufficient conditions for Gromov-Hausdorff convergence of metric quotients of a metric space.
result Concrete examples of sequences of two-dimensional conic-flat spheres converging to spheres with singularities.
It is known that the volume function for hyperbolic manifolds of dimension is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by constructing a sequence of finite-sided finite-volume polyhedra with side-pairings t…
The article constructs Fuchsian Schottky groups with conformal boundaries.
problem Creating generalized Schottky groups with specific properties.
method Developed Fuchsian Schottky groups by including orientation-reversing isometries.
result Decomposed compact core of conformally compact Riemann surfaces into pairs of pants.
The paper studies how Kleinian groups can be deformed while preserving their peripheral structures.
problem Determining the extent of the island of discrete representations around the identity map in the space of representations.
method By cutting up the conformal boundary of a hyperbolic 3-manifold into a fundamental domain, the paper provides a computable region within which the fundamental domain is valid, ensuring peripheral structures remain similar under small deformations.
result The paper identifies a region in the space of representations of the fundamental group of a geometrically finite manifold, showing that groups in this region have peripheral structures that look coarsely similar.