  In this paper we prove that two-dimensional translating solitons in
$\mathbb{R}^3$ with finite $L$-index are homeomorphic to a plane or a cylinder
and that a two-dimensional self-expander with finite $L$-index and sub
exponential weighted volume growth has finite topology. We also prove that
translating solitons and self-expanders have finite topology, provided the
bottom of the spectrum of the $L$-stability operator is bounded from below and
their weighted volume have subexponential growth.
