On 10/06/2020 17:19, Dan Asimov wrote:
It's worth noting that a polyhedron need not be a subset of 3-space.
In fact the existence of a polyhedron with its intrinsic properties should be considered a separate issue from whether it can be embedded in 3-space (or n-space, for that matter).
Very true. So, let's return to the case in my original question (well, the question wasn't _originally_ mine, but you know what I mean) and consider the possibility of a genus-2 surface with 8 pentagonal faces, 4 of which meet at each of its 10 vertices. Should it be obvious to me that this thing exists as an abstract polyhedron? One thing that jumps out at me is that you can famously make a genus-2 surface from an _octagon_ by identifying its edges in pairs (more generally, a genus-g surface from a 4g-gon), and maybe it's not a coincidence that our hypothetical polyhedron is meant to have _eight_ faces. But maybe this is just a law-of-small-numbers coincidence. This seems highly relevant: https://mathoverflow.net/questions/335862/tiling-of-genus-2-surface-by-8-pen... and one answer to that question claims that this answer to another question https://mathoverflow.net/a/331408/1345 contains a picture answering the question "what does this tiling look like?", but unfortunately the _picture_ is just a picture of the obvious tiling of the hyperbolic plane by right-angled pentagons, and the _words_ in the answer that address the question don't successfully communicate to me exactly what's going on. (The relevant bit: "a genus 2 surface is an index 8 orbifold cover of the right-angle pentagon orbifold (i.e., an index 8 torsion free subgroup of the reflection group in a right-angled pentagon is a genus 2 surface group)". I understand what that's saying but I don't have a good enough grasp of what these groups look like to identify an index-8 torsion-free subgroup of the reflection group in a right-angled pentagon or to figure out what the fundamental region of the resulting group is.) Aha, I think I have found a picture that answers the question of what the thing has to look like as an abstract polyhedron: https://books.google.co.uk/books?id=NFnmBwAAQBAJ&pg=PA432 So that's something. -- g