Forum Problem Lists Definitions FAQ

19.31 (2018)

Open

Let $\omega(G, S)$ be the exponential growth rate of a group $G$ with a finite generating set $S$ (see 14.7). Let $\text{MCG}(\Sigma_g)$ be the mapping class group of the orientable surface $\Sigma_g$ of genus $g$. Is there a constant $C > 1$ such that $\omega(\text{MCG}(\Sigma_g), S) \geqslant C$ for every $g > 0$ and every finite generating set $S$ of $\text{MCG}(\Sigma_g)$?

Contributor: K. Fujiwara

Progress

*No, there is no such constant that works for all genera; but if the genus $g$ is fixed, then such a constant $C = C(g) > 1$ does exist (J. Mangahas, Geom. Funct. Anal., 19, no. 5 (2010), 1468–1480).

Comments

All comments are the responsibility of the user. Comments appearing on this page are not verified for correctness. Please keep posts mathematical and on topic. If you want to submit a proof (or a partial proof), please use the dedicated proof submission form rather than posting it in the comments.
Order by newest first or oldest first.

No comments yet. Be the first to comment.

Proof claims

Proof claims are the responsibility of the submitter. Appearance here does not mean the claim has been checked for mathematical correctness. Moderators only screen for spam, abuse, and obviously low-effort submissions.

No proof claims yet.