Let
be a closed orientable surface of genus at least
. The action of an
automorphism
on
the curve complex of
is an isometry. Via this isometric action on the curve complex, a translation length is defined on
. The geometry of
the mapping torus
depends on
.
As it turns out, the structure of the minimal-genus Heegaard splitting also depends on
: the canonical Heegaard
splitting of
, constructed
from two parallel copies of
,
is sometimes stabilized and sometimes unstabilized. We give an example
of an infinite family of automorphisms for which the canonical Heegaard
splitting of the mapping torus is stabilized. Interestingly, complexity bounds on
provide insight into the stability of the canonical Heegaard splitting of
.
Using combinatorial techniques developed on
–manifolds, we prove that if
the translation length of
is
at least
, then the canonical
Heegaard splitting of
is unstabilized.
