arXiv subjects

Cameron Gates Rudd

Publications and source records attributed to Cameron Gates Rudd.

6 recordsLinked to original sources

arXiv2026

Filling 1-cycles, 2-cycle complexity, and torsion growth

For 2-complexes with trivial first Betti number, we study quantitative connections between filling inequalities for 1-cycles, the complexity of the second homology, and the size of the torsion part of the first homology. For 2-complexes whose fundamental groups have property $(\tau)$ with respect to all finite index normal subgroups, we prove a geometric lower bound on the logarithm of the size of the first homology of finite covers.

math.GT↗

arXiv2026

Magnetic geodesics, Hodge Laplacian eigenvalues, and isoperimetric inequalities

An isoperimetric constant relating length and stable area, or alternatively for hyperbolic manifolds, length and stable commutator length, serves as a Cheeger constant for the smallest eigenvalue of the Hodge Laplacian acting on coexact 1-forms. Using properties of the magnetic geodesic flow associated to the differential of a coexact eigenform, and its behavior at Ma\~n\'e's critical energy level, we give new proofs of these Cheeger-like inequalities, with improved (and explicit) constants and volume dependence. We also make a few observations about the relationship between Ma\~n\'e's critical values and the eigenvalues, when the manifold is hyperbolic.

math.GT↗

arXiv2025

Property (T) and Poincar\'e duality in dimension three

We use a recent result of Bader and Sauer on coboundary expansion to prove residually finite three-dimensional Poincar\'e duality groups never have property (T). This implies such groups are never K\"ahler. The argument applies to fundamental groups of (possibly non-aspherical) compact 3-manifolds, giving a new proof of a theorem of Fujiwara that states if the fundamental group of a compact 3-manifold has property (T), then that group is finite. The only consequence of geometrization needed in the proof is that 3-manifold groups are residually finite.

math.GT↗

arXiv2023

Stretch laminations and hyperbolic Dehn surgery

We study maximal stretch laminations associated to certain best Lipschitz circle valued maps in Dehn surgery families of hyperbolic 3-manifolds. For these maps, we give a criterion based on the Thurston norm and Dehn filling slope length to determine when such a stretch lamination is a union of Dehn filling core curves. We use this to show there exist infinitely many examples where the homotopy class of the circle valued map includes a fibration and where the laminations have only closed leaves. This gives information about non-maximal horospherical orbit closures in the infinite cyclic covers associated to these fibrations.

math.GT↗

arXiv2023

Computing a Link Diagram from its Exterior

A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2,500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.

math.GT↗

arXiv2022

Stable Isoperimetric Ratios and the Hodge Laplacian of Hyperbolic Manifolds

We show that for a closed hyperbolic 3-manifold the size of the first eigenvalue of the Hodge Laplacian acting on coexact 1-forms is comparable to an isoperimetric ratio relating geodesic length and stable commutator length with comparison constants that depend polynomially on the volume and on a lower bound on injectivity radius, refining estimates of Lipnowski and Stern. We use this estimate to show that there exist sequences of closed hyperbolic 3-manifolds with injectivity radius bounded below and volume going to infinity for which the 1-form Laplacian has spectral gap vanishing exponentially fast in the volume.

math.GT↗