Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling

TL;DR

This paper constructs new hyperbolic L-spaces lacking symmetries, using hyperbolic geometry and Floer theory, and introduces a novel computational technique for certifying symmetries and isometries in hyperbolic 3-manifolds.

Contribution

It provides the first examples of hyperbolic L-spaces without symmetries and develops a new interval arithmetic method for analyzing hyperbolic 3-manifolds.

Findings

Existence of infinitely many hyperbolic L-spaces with no symmetries.

First examples of 1-cusped hyperbolic 3-manifolds of Heegaard genus 3 with multiple lens space fillings.

Multiple Dehn fillings can reduce Heegaard genus by more than one.

Abstract

An L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic L-spaces with no symmetries. In particular, unlike all previously known L-spaces, these manifolds are not double branched covers of links in S^3. We prove the existence of infinitely many such examples (in several distinct families) using a mix of hyperbolic geometry, Floer theory, and verified computer calculations. Of independent interest is our technique for using interval arithmetic to certify symmetry groups and non-existence of isometries of cusped hyperbolic 3-manifolds. In the process, we give examples of 1-cusped hyperbolic 3-manifolds of Heegaard genus 3 with two distinct lens space fillings. These are the first examples where multiple Dehn fillings drop the Heegaard genus by more than one, which answers a question of Gordon.