Big Heegaard distance implies finite mapping class group

Abstract

We show that if M is a closed three manifold with a Heegaard splitting with sufficiently big Heegaard distance then the subgroup of the mapping class group of the Heegaard surface, whose elements extend to both handlebodies is finite. As a corollary, this implies that under the same hypothesis, the mapping class group of M is finite.

Subgroups of mapping class groups related to Heegaard splittings and bridge decompositions

“…Once this generalisation is completed, we can also remove this restriction from our result. We note that Johnson [12] proved that the mapping class group MCG(M, S) of the pair (M, S) is finite if the Hempel distance is greater than 3, improving the result of Namazi [28]. In particular, G 1 ∩ G 2 is finite if the Hempel distance is greater than 3.…”

“…In particular, G 1 ∩ G 2 is finite if the Hempel distance is greater than 3. Thus Theorem B below may be regarded as a partial refinement of this consequence of the results of [12] and [28]. We also note that Theorems C and E below may be regarded as a variant of the asymptotical faithfulness of the homomorphism π 1 (H i ) → π 1 (M) established by Namazi The authors would like to thank Brian Bowditch for his essential contribution to the proof of Theorem B, without which they should not have been able to complete the work.…”

Hyperbolic manifolds containing high topological index surfaces

“…We will generalize this result to link and graph complements, with the additional benefit of avoiding many of the technical details of [7] necessary to treat the boundary components. Unfortunately, our bound will be worse than that obtained by Bachman and Schleimer, though it will be sufficient for many applications of this type of bound (e.g., [20], [15], [22], [5], and [21]). We note also that our proof requires a minimal starting position similar to that used by Hartshorn, an assumption the Bachman-Schleimer method was able to avoid.…”

Mapping Class Groups of Heegaard Splittings

“…For genus three and greater, the group Mod(M, Σ) is isomorphic to the group of isometries of C(Σ) that take each set H ± onto itself. Namazi [27] used this fact to show that for sufficiently high distance (depending on the genus) Heegaard splittings, the mapping class group of the Heegaard splitting is finite. The first author later showed using purely topological methods that this is true for any Heegaard splitting with distance four or greater in any genus [17].…”