In this paper we construct explicit examples of both closed and non-compact finite volume hyperbolic manifolds which provide counterexamples to the conjecture that the co-rank of a 3-manifold group (also known as the cut number) is bounded below by one-third the first Betti number.
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11 results for “co-rank”
Proves Sard conjecture for specific distributions, controlling divergence of vector fields.
problem Proving the Sard conjecture for certain types of distributions.
method Constructs a singular distribution capturing essential abnormal lifts, proving the conjecture for rank 3 distributions in dimension 4 and generic corank 1 distributions.
result Proves the Sard conjecture for generic co-rank one distributions.
The paper examines the co-rank of weakly parafree 3-manifold groups and provides counterexamples.
problem Investigating the co-rank of weakly parafree 3-manifold groups and their relationship to free groups.
method Constructing specific examples of homology handlebodies and analyzing their fundamental groups.
result The fundamental group of W. Thurston's tripus manifold is not very large, showing that weakly parafree groups of rank at least 3 are not necessarily very large.
In the present paper we consider generic Sub-Riemannian structures on the co-rank 1 non-holonomic vector distributions and introduce the associated canonical volume and ''horizontal'' area forms. As in the classical case, the Sub-Riemannian minimal surfaces can be defined as the critical points of the '`horizontal'' ar…
This research bounds the complexity of Reeb graphs and their cycles, improving approximation of metric spaces.
problem Approximating the complexity of metric spaces using Reeb graphs.
method Analyzing the cycle rank of Reeb graphs and relating it to the co-rank of the fundamental group.
result The cycle rank of a Reeb graph is bounded by the co-rank of the fundamental group, improving approximation bounds.
New examples of sub-Riemannian structures satisfying Minimizing Sard conjecture found.
problem Finding complete sub-Riemannian structures satisfying the Minimizing Sard conjecture.
method Techniques from nonsmooth analysis and geometric measure theory.
result Complete sub-Riemannian structures associated with distributions of co-rank 2 or generic distributions of rank ≥ 2 satisfy the Minimizing Sard conjecture.
Abstract: Proves no non-trivial normal orbits for specific Hamiltonians.
problem Analyzing normal singular geodesics in a conformally generic sub-Riemannian metric.
method Proves the absence of non-trivial normal orbits for specific Hamiltonians.
result No non-trivial normal orbits for the specified Hamiltonians.
Analytic proof for minimal rank Sard conjecture.
problem Proving the minimal rank Sard conjecture in the analytic category.
method Using subanalytic abnormal distribution from [4], we establish a proof.
result The set of points accessible through singular horizontal curves of minimal rank has Lebesgue measure zero.
Following on from work of Dunfield, we determine the fibred status of all the unknown hyperbolic 3-manifolds in the cusped census. We then find all the fibred hyperbolic 3-manifolds in the closed census and use this to find over 100 examples each of closed and cusped virtually fibred non-fibred census 3-manifolds, incl…
Minimal surfaces in sub-Riemannian manifolds and structure of their singular sets in the (2,3) casemath.AP
We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called {\it horizontal} area functional associated to the canonical {\it horizontal} area form. We derive the intrinsic equation in the general case…
Improved bounds on -torus actions on positively curved manifolds.
problem Bounding the rank of -tori for fixed point set components.
method Lowered the rank bound and classified cohomology rings.
result Fixed point set components are classified by integral or -cohomology rings.