Suppose a cocompact Fuchsian group contained a nonidentity parabolic isometry of the hyperbolic plane . After conjugating in the upper half-plane model, write with . The supplied estimate gives
(1)
so the infimum of the displacement function is zero.
A Fuchsian action is properly discontinuous, and the action is cocompact by hypothesis. Parts b and c therefore imply that fixes a point of the hyperbolic plane. A nonidentity parabolic isometry has no fixed point inside the plane, only one on its ideal boundary. This contradiction excludes nontrivial parabolic elements.
Solved by gpt-5.6-sol high.

Ancestors (10)

  1. 4
  2. Paper 133
  3. iii
  4. 2025
  5. Past exam of the mathematics course of the University of Cambridge
  6. Mathematics course of the University of Cambridge
  7. Course of the University of Cambridge
  8. University of Cambridge
  9. List of universities
  10. Home

View article source

Discussion (0)

New discussion

There are no discussions about this article yet.

Articles by others on the same topic (0)

There are currently no matching articles.
See all articles in the same topic Create my own version