GROWTH

These are problems about growth functions of groups and related issues.
 

(G1) (S.I.Adian) Is it true that a finitely presented group has either polynomial or exponential growth? Background

(G2) (R.I.Grigorchuk) (a) Is it true that there is a gap between polynomial rate of growth and the rate of growth of the function e^\sqrt{n} on the scale of growth of finitely generated groups?

(b) Is there a finitely generated group G whose growth function is equivalent to e^\sqrt{n} ?

*(G3) (M.Gromov, J.Lafontaine, P.Pansu) Is there a group of exponential, but not uniformly exponential growth? Background

(G4) (R.I.Grigorchuk) Is it true that every finitely generated infinite simple group has exponential growth?

(G5) (R.I.Grigorchuk) Is it true that every hereditary just infinite group (i.e., a residually finite group whose every subgroup of finite index is just infinite) has either polynomial or exponential growth?

(G6) (R.I.Grigorchuk) Is there a cancellative semigroup of subexponential growth whose quotient group (if it exists) has exponential growth?

(G7) (R.I.Grigorchuk) Let G be a group of subexponential growth, and let f(n) be the number of different elements of length at most n in the group G. Does the limit (when n -> \infty) of the ratio f(n+1)/f(n) always exist?

*(G8) Let [F_r, F_r] be the commutator subgroup of the free group F_r of rank r, and let f(n) be the growth function of [F_r, F_r] in F_r (i.e., the number of different elements of length n in [F_r, F_r]). Find the asymptotics of f(n). Background

(G9) Find the growth rate of the free metabelian group of rank r. Background

(G10) (R.I.Grigorchuk, I.Pak) Is it true that every group of intermediate growth contains two infinite subgroups A and B which commute elementwise, i.e., ab=ba for any a \in A, b \in B ? Background

(G11) (R.I.Grigorchuk, P. de la Harpe) (a) Calculate the growth series of Baumslag-Solitar groups B(p,q) with q>p>1.

(b) Part (a) is probably too hard in general; in fact, it is open even for B(2,3). We single out this case as (allegedly) the most tractable one.

(c) At least, find the asymptotic rates of growth for B(p,q). Background