### 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.

Original language | English (US) |
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Pages (from-to) | 1679-1698 |

Number of pages | 20 |

Journal | Mathematical Research Letters |

Volume | 22 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2015 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Mathematical Research Letters*,

*22*(6), 1679-1698. https://doi.org/10.4310/MRL.2015.v22.n6.a7