On the mapping class groups of strongly irreducible Heegaard splittings

We show that for any $g \geq 3$ and $n \geq 2$, there exists a genus-$g$ Heegaard splitting of distance $n$ whose mapping class group is the trivial group or $\mathbb{Z} / 2 \mathbb{Z}$. We also show that there exist Heegaard splittings of distance $2$ that have the infinite-order mapping class groups whereas that are not induced from open book decompositions. Explicit computation of those mapping class groups are given.