Mathematical Research Letters
Volume 22 (2015)
Asymmetric hyperbolic $L$-spaces, Heegaard genus, and Dehn filling
Pages: 1679 – 1698
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.