4
$\begingroup$

In theses these notes, Example 5.6, it is said that there is a "symmetric tiling of a genus 2 surface by 8 right-angled hyperbolic pentagons".

Question 1: What does this tiling look like?

Question 2: Is it always possible to tile a genus $n$ surface by $f$ regular $n$-gons with interior angle $\pi/v$ (so that $v$ faces meet at every vertex) as long as the restriction given by the Euler characteristic $$\chi=2-2n= f-nf/2+nf/v$$ is satisfied? Answering this question, Igor Rivin says yes, but it seems his argument only shows that regular hyperbolic $n$-gons of interior angle smaller than the interior angle of a euclidean regular $n$-gon exist.

Improve this question
$\endgroup$
1

2 Answers 2

Reset to default
7
$\begingroup$

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.

Improve this answer
$\endgroup$
6
$\begingroup$

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 $g=2$.

Improve this answer
$\endgroup$
3
  • 1
    $\begingroup$ As you're probably aware, Michael Larsen has addressed how frequently the Hurwitz bound is achieved. mathscinet.ams.org/mathscinet-getitem?mr=1882031 But one could also just ask for a transitive action on the faces (or on the vertices). Bestvina's example desribed by Christopher is of this type. mathoverflow.net/questions/198040/… $\endgroup$
    – Ian Agol
    Commented Jul 11, 2019 at 20:58
  • $\begingroup$ Can you make a face-transitive tiling by heptagons in every genus? (Or various other regular polygons.) $\endgroup$
    – Dylan Thurston
    Commented Jul 12, 2019 at 21:28
  • 1
    $\begingroup$ Any surface is a cyclic cover of genus 2, so maybe a dihedral action? $\endgroup$
    – Ian Agol
    Commented Jul 13, 2019 at 2:53

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .