Skip to main content
Added relevant tags.
Link
HJRW
  • 25k
  • 3
  • 68
  • 144
Bumped by Community user
Bumped by Community user
Bumped by Community user
Source Link
Mehdi Yazdi
  • 858
  • 4
  • 12

Finiteness properties of mapping class groups

Question: Is it known if the mapping class groups (of surfaces of finite type) are similar to Gromov-hyperbolic groups in the following senses:

  1. Does every finite generating set give us a finite presentation?

  2. Are there finitely many cone types with respect to any (some) finite presentation?


Definition: A group is called hyperbolic if its Cayley graph is Gromov-hyperbolic, i.e., triangles are $\delta$-thin for some positive $\delta$.

Definition: Let $G$ be a group with a finite generating set $S$ and let $g \in G$. The cone type of $g$ w.r.t. $S$ is the following set:

$ \mathcal{C}(g) = \{ h \in G | \hspace{2mm} d(e,gh) = d(e,g)+d(e,h)\}$ ,where $d(.,.)$ shows the distance in the Cayley graph w.r.t. to the generating set $S$.