Stable isoperimetric ratios and hyperbolic geometry

Rudd, Cameron Gates

Permalink

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

Description

Title

Stable isoperimetric ratios and hyperbolic geometry

Author(s)

Rudd, Cameron Gates

Issue Date

2022-04-19

Director of Research (if dissertation) or Advisor (if thesis)

Dunfield, Nathan

Doctoral Committee Chair(s)

Albin, Pierre

Committee Member(s)

• Hirani, Anil
• Samperton, Eric

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Language

eng

Abstract

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.

Graduation Semester

2022-05

Type of Resource

Thesis

Handle URL

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

Copyright and License Information

© 2022 Cameron Gates Rudd

Owning Collections