Tiling of genus 2 surface by 8 pentagons

A picture answering Question 1 is here: https://mathoverflow.net/a/331408/1345

Question 2 is a duplicate of regular tiling of a surface of genus 2 by heptagons, although as you point out the accepted answer there is unsatisfactory. I've given a link there answering this question.


Ian answered the second question as asked, but in case you meant to ask a different question: there is not always a symmetric tiling by regular polygons of the given type, even if those restrictions hold. For instance, there is no tiling of the genus 2 surface by heptagons meeting 3 at a vertex so that the symmetries permute and rotate the heptagons in all possible ways. Said differently, the Hurwitz bound of $168(g-1)$ on the number of symmetries of a genus $g$ surface is not achieved for $g=2$.

Tags:

Gt.Geometric Topology

Hyperbolic Geometry

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