Finite presentations of the mapping class groups of once-stabilized Heegaard splittings

Abstract

Let $g\ge 2$ and assume that we are given a genus $g$ Heegaard splitting of a closed orientable $3$-manifold with the distance greater than $2g+2$. We prove that the mapping class group of the once-stabilization of such a Heegaard splitting is finitely presented.

19 pages, 7 figures, to appear in Algebraic & Geometric Topology