Thick isotopy property and the mapping class groups of Heegaard splittings.

Let $M$ be a closed orientable $3$-manifold and $S$ a Heegaard surface of $M$. The space of Heegaard surfaces $\mathcal{H}(M,S)$ is defined to be the space of left cosets $\mathrm{Diff}(M)/\mathrm{Diff}(M,S)$. We prove that the fundamental group $\pi_{1}(\mathcal{H}(M,S))$ is finitely generated if and only if any element of $\pi_{1}(\mathcal{H}(M,S))$ can be represented by a "thick" isotopy. As an application, we prove that the mapping class group of a strongly irreducible Heegaard splitting of a closed hyperbolic $3$-manifold is finitely generated.