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2Manifolds
The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
William P. Thurston – One of the best experts on this subject based on the ideXlab platform.

Finite covers of random 3manifolds
Inventiones mathematicae, 2006CoAuthors: Nathan M. Dunfield, William P. ThurstonAbstract:A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are led to consider the action of the mapping class group of a surface Σ on the set of quotients π_1(Σ)→ Q . If Q is a simple group, we show that if the genus of Σ is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic.

Finite covers of random 3manifolds
Inventiones Mathematicae, 2006CoAuthors: Nathan M. Dunfield, William P. ThurstonAbstract:A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface S on the set of quotients pi_1(S) > Q. If Q is a simple group, we show that if the genus of S is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic.
G. Elber – One of the best experts on this subject based on the ideXlab platform.

Contouring 1 and 2Manifolds in arbitrary dimensions
International Conference on Shape Modeling and Applications 2005 (SMI' 05), 2005CoAuthors: Joonkyung Seong, G. ElberAbstract:We propose an algorithm for contouring kmanifolds (k = 1,2) embedded in an arbitrary ndimensional space. We assume (n k) geometric constraints are represented as polynomial equations in n variables. The common zeroset of these (nk) equations is computed as an 1or 2manifold, respectively, for k = 1 or k = 2. In the case of 1manifolds, this framework is a generalization of techniques for contouring regular intersection curves between two implicitlydefined surfaces of the form F(x,y,z) = G(x,y,z) = 0. Moreover, in the case of 2Manifolds, the algorithm is similar to techniques for contouring isosurfaces of the form F(x, y, z) = 0, where n = 3 and only one (=3 2) constraint is provided. By extending the Dual Contouring technique to higher dimensions, we approximate the simultaneous zeroset as a piecewise linear 1or 2manifold. There are numerous applications for this technique in data visualization and modeling, including the processing of various geometric constraints for freeform objects, and the computation of convex hulls, bisectors, blendings and sweeps.

SMI – Contouring 1 and 2Manifolds in arbitrary dimensions
International Conference on Shape Modeling and Applications 2005 (SMI' 05), 2005CoAuthors: Joonkyung Seong, G. ElberAbstract:We propose an algorithm for contouring kmanifolds (k = 1,2) embedded in an arbitrary ndimensional space. We assume (n k) geometric constraints are represented as polynomial equations in n variables. The common zeroset of these (nk) equations is computed as an 1or 2manifold, respectively, for k = 1 or k = 2. In the case of 1manifolds, this framework is a generalization of techniques for contouring regular intersection curves between two implicitlydefined surfaces of the form F(x,y,z) = G(x,y,z) = 0. Moreover, in the case of 2Manifolds, the algorithm is similar to techniques for contouring isosurfaces of the form F(x, y, z) = 0, where n = 3 and only one (=3 2) constraint is provided. By extending the Dual Contouring technique to higher dimensions, we approximate the simultaneous zeroset as a piecewise linear 1or 2manifold. There are numerous applications for this technique in data visualization and modeling, including the processing of various geometric constraints for freeform objects, and the computation of convex hulls, bisectors, blendings and sweeps.
Nathan M. Dunfield – One of the best experts on this subject based on the ideXlab platform.

Finite covers of random 3manifolds
Inventiones mathematicae, 2006CoAuthors: Nathan M. Dunfield, William P. ThurstonAbstract:A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are led to consider the action of the mapping class group of a surface Σ on the set of quotients π_1(Σ)→ Q . If Q is a simple group, we show that if the genus of Σ is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic.

Finite covers of random 3manifolds
Inventiones Mathematicae, 2006CoAuthors: Nathan M. Dunfield, William P. ThurstonAbstract:A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface S on the set of quotients pi_1(S) > Q. If Q is a simple group, we show that if the genus of S is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic.