2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71690We show that if $M$ is a closed three manifold with a Heegaard splitting with sufficiently big "handlebody distance" then the subgroup of the mapping class group of the Heegaard surface, which extend to both handlebodies is finite. As a corollary, this implies that under the same hypothesis, the mapping class group of $M$ is finite.9 pagesGeometric Topologytext