Question: Is it known if the mapping class groups (of surfaces of finite type) are similar to Gromov-hyperbolic groups in the following senses:
Does every finite generating set give us a finite presentation?
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 $S$.