Geometric structure and Floer theory of three-dimensional manifolds

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Low-dimensional topology continues to draw on groundbreaking ideas of Floer. This has generated a vibrant sub-discipline; new structure is being uncovered and difficult problems are being solved. This activity comes alongside landmark achievements in geometric topology - e.g. Agol and Wise's resolution of the virtual Haken conjecture - highlighting successes of Thurston's program. My proposal is positioned at the nexus of these areas of activity, with a view to bridging between Thurston-style geometric topology and Floer-theoretic invariants in low-dimensions.******This research will draw on the fundamental group, working to uncover the interplay between left-orderable groups, taut foliations, and Floer homology. Interplay between these structures has generated a wealth of new research; the conjectured connection is now established for graph manifolds (see my work with Hanselman, Rasmussen and Rasmussen). This uses novel algebraic tools from bordered Floer homology, a variant of Heegaard Floer homology adapted to manifolds with boundary. I aim to bring these tools to bear on the role of hyperbolic structures in Floer theory.******Understanding the geometric underpinnings of Floer theory builds on questions of Ozsvth-Szab pertaining to relationships with the fundamental group. My work with Boyer and Gordon formulates a conjectural connection that has been a catalyst for new research activity on this problem. Reiterating this, the importance of making connections between geometric 3-manifold topology and Floer homology was singled out by Agol in his Veblen citation. ******My work with Hanselman and Rasmussen recasts bordered invariants for manifolds with torus boundary in terms of immersed curves in the punctured torus. While this requires a mild hypothesis on the 3-manifold in question, our work aligns with that of Haiden-Katzarkov-Kontsevitch on Fukaya categories of surfaces. Our continued research aims to interpret this progress in homological mirror symmetry in our setting in order to establish new results in low-dimensions. This work points to interesting structure both for orderable groups and in foliation theory.******In a related vein, it is conjectured that there do not exist hyperbolic integer homology sphere L-spaces (manifolds with simplest possible Heegaard Floer homology). This is a key instance where an understanding of the relationship between hyperbolic 3-manifolds and Floer theory is required. I propose to approach this problem with mapping class groups of surfaces as a mediating object: bimodules in bordered Floer homology provide a faithful categorical representation of the mapping class group, while geometric limits in hyperbolic geometry suggest a search for stable properties of 3-manifolds associated with iterated mapping classes. This suggests new algebraic structures, and a program towards understanding the paucity of hyperbolic L-space integer homology spheres.