# 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 $S$.

• Maybe I am misunderstanding but isn't 1 true for all finitely presented groups: If you have a finitely presentation, and a different finite generating set you can rewrite all the relations in terms of the new generating set.
– user35370
May 25, 2017 at 23:02
• @Paul Plummer: Yes, of course , the first question has affirmative answer for all groups. May 25, 2017 at 23:50
• As for Q2 "for all" presentations is very unlikely since it fails for some virtually abelian groups and they embed in mapping class groups. May 25, 2017 at 23:52
• @PaulPlummer You're right. Thanks :) May 26, 2017 at 0:59
• @Misha Thanks. How about one generating set? For example Humpheris generating set? Could you elaborate how you relate the cone types of a group and a subgroup? May 26, 2017 at 1:10

Question 2 follows from the work of Lee Mosher.

Mosher, Lee, Mapping class groups are automatic, Ann. Math. (2) 142, No.2, 303-384 (1995). ZBL0867.57004.

• Igor: What Mosher proves is the existence of an automatic structure. He does not prove existence of a finite presentation which has a geodesic combing (satisfying falsification by fellow traveler property). May 25, 2017 at 23:01
• Could you explain what is the relation between 'being automatic' and 'having finitely many cone types'? Does the former imply the later? Under what conditions the former implies the later? May 26, 2017 at 1:15
• @MehdiYazdi: Its complicated, see for instance "A Short course in geometric group theory" by Walter Neumann. Roughly speaking, you are looking for a regular geodesic language on your group (for some generating set). May 26, 2017 at 14:12

It looks like question 1) and 2) has been answered, but in case you're still wondering generally "in what ways are mapping class groups similar to Gromov-hyperbolic groups?", you may be interested in reading about hierarchically hyperbolic spaces, introduced by Jason Behrstock, Mark F. Hagen, Alessandro Sisto in this paper: Hierarchically hyperbolic spaces I: curve complexes for cubical groups.

For a less technical overview you can also check out Sisto's blog post.

(I heard about everything here from Jacob Russel's talk at GSCAGT.)

• Thanks for the references. Question 2 here seems unanswered (automatic does not imply finitely many cone points apparently). Jun 8, 2017 at 12:55
• Cone type I meant. Jun 8, 2017 at 13:48
• Ah, right. So maybe this helps with that question? I'm definitely not an expert though. Jun 8, 2017 at 15:03
• I hope so. I should check it out. Thanks. Jun 8, 2017 at 15:05