  In this paper, we prove that the Accola-Maclachlan surface of genus $g$
bounds geometrically an orientable compact hyperbolic $3$-manifold for every
genus. For infinitely many genera, this is an explicit example of
non-arithmetic surface that bounds geometrically a non-arithmetic manifold. We
also provide explicit geodesic embeddings to the Wiman type II and Kulkarni
surfaces of every genus, and prove that these surfaces bound geometrically a
compact, orientable manifold for $g\equiv 1 \,( \text{mod} \, 2)$ or $g \equiv
3 \,( \text{mod} \, 8)$, respectively.
