# 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 '17 at 23:02
• @Paul Plummer: Yes, of course , the first question has affirmative answer for all groups. – Misha May 25 '17 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. – Misha May 25 '17 at 23:52
• @PaulPlummer You're right. Thanks :) – Mehdi Yazdi May 26 '17 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? – Mehdi Yazdi May 26 '17 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). – Misha May 25 '17 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? – Mehdi Yazdi May 26 '17 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). – Misha May 26 '17 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). – Mehdi Yazdi Jun 8 '17 at 12:55
• Cone type I meant. – Mehdi Yazdi Jun 8 '17 at 13:48
• Ah, right. So maybe this helps with that question? I'm definitely not an expert though. – Harry Richman Jun 8 '17 at 15:03
• I hope so. I should check it out. Thanks. – Mehdi Yazdi Jun 8 '17 at 15:05