0

I am self-studying hyperbolic geometry and am stuck on the following.

Let $\mathbb{D}$ be the Poincare disc model of the hyperbolic plane with $[3^7]$ tiling on it. Let $T$ be a hyperbolic translation which is also a symmetry of the tiling such that $T(p)=q$ where $p$ and $q$ are two vertices. Then is this $T$ unique?

Since $T$ is a translation it has an axis of translation. Is it possible that some other translation has the same axis and maps $p$ to $q$?

Thanks in advance.

KAK
  • 197
  • Yes, this is true. One can prove this analytically --- with coordinates for $\mathbb D$ and matrix coefficients for the isometries of $\mathbb D$ represented as fractional linear transformation. One can also prove this synthetically, with properties of lines and rays and half-planes and orientations and distances all buit up from the axioms, following the exact same outline for the Euclidean plane (in fact, this is a theorem of neutral geometry, meaning planar geometry without the parallel axiom). – Lee Mosher Nov 16 '22 at 12:52
  • @LeeMosher Thank you. That means it is not unique. Right? You are talking about 1st question or the second one? – KAK Nov 16 '22 at 18:44
  • Very confusing. Didn't even notice that you asked two opposite questions. The translation is unique. Its axis is unique. – Lee Mosher Nov 16 '22 at 18:58

0 Answers0