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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 $S$.

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    $\begingroup$ 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. $\endgroup$
    – user35370
    Commented May 25, 2017 at 23:02
  • $\begingroup$ @Paul Plummer: Yes, of course , the first question has affirmative answer for all groups. $\endgroup$
    – Misha
    Commented May 25, 2017 at 23:50
  • $\begingroup$ As for Q2 "for all" presentations is very unlikely since it fails for some virtually abelian groups and they embed in mapping class groups. $\endgroup$
    – Misha
    Commented May 25, 2017 at 23:52
  • $\begingroup$ @PaulPlummer You're right. Thanks :) $\endgroup$
    – Mehdi Yazdi
    Commented May 26, 2017 at 0:59
  • $\begingroup$ @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? $\endgroup$
    – Mehdi Yazdi
    Commented May 26, 2017 at 1:10

2 Answers 2

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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.

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  • $\begingroup$ 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). $\endgroup$
    – Misha
    Commented May 25, 2017 at 23:01
  • $\begingroup$ 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? $\endgroup$
    – Mehdi Yazdi
    Commented May 26, 2017 at 1:15
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    $\begingroup$ @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). $\endgroup$
    – Misha
    Commented May 26, 2017 at 14:12
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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.)

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

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