Geometric Bordisms of the Accola-Maclachlan, Kulkarni and Wiman Type II Surfaces

Abstract

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.