Mathematical Research Letters

Volume 22 (2015)

Number 6

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

Pages: 1679 – 1698



Nathan M. Dunfield (Department of Mathematics, University of Illinois, Urbana, Il., U.S.A.)

Neil R. Hoffman (School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, Australia)

Joan E. Licata (Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia)


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.

Accepted 17 June 2015

Published 23 May 2016