Stable isoperimetric ratios and hyperbolic geometry

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Rudd, Cameron Gates

Contributors dc:contributor

• Dunfield, Nathan
• Albin, Pierre
• Hirani, Anil
• Samperton, Eric

• hyperbolic geometry
• 3-manifolds
• low-dimensional topology
• geometric topology
• geometric analysis
• spectral geometry

Statement dc:rights

• © 2022 Cameron Gates Rudd

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/115382

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F115382