Timeline for Is it possible for a finitely generated group to have polynomial growth rate with leading coefficient less than 1?
Current License: CC BY-SA 4.0
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| Apr 29, 2021 at 21:48 | comment | added | YCor | In the abelian case of rank $d$ a lower bound is given by $\mathbf{Z}^d$ with standard generating subset. For the nilpotent case case I guess there's a lower bound as well (say, for given degree of growth, or for given Hirsch lengths in the subquotient of the lower central series) for the leading coefficient, but it's not clear. Probably the right approach is to study carefully the Breuillard-LeDonne paper. | |
| Apr 29, 2021 at 21:01 | comment | added | Ronnie Pavlov | Fairly embarrassing answer: you can trivially get leading coefficients less than 1. For instance, if $G = \mathbb{Z}^4$ with standard generators, then $c_n(G)$ is the size of the $n$-ball in $\mathbb{Z}^4$ under the $L_1$-norm, which is asymptotically $\frac{2^4}{4!} n^4 = \frac{2}{3} n^4$. I guess my real question is then: what are the minimum possible leading coefficients in terms of degree $d$? Are they given by $\mathbb{Z}^d$ (in which case they would be $\frac{2^d}{d!}$)? | |
| Apr 29, 2021 at 20:58 | comment | added | Ronnie Pavlov | @Sophie: this is cool, although I also don't know enough to understand exactly what that volume is. | |
| Apr 29, 2021 at 20:57 | comment | added | Ronnie Pavlov | @Ycor: thanks! I guess my question is about the value this converges to (this is what I called the "leading coefficient.") | |
| Apr 23, 2021 at 15:53 | history | edited | Ronnie Pavlov | CC BY-SA 4.0 |
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| Apr 22, 2021 at 19:33 | comment | added | Sophie M | When $G$ is nilpotent and torsion-free, Breuillard-Le Donne characterize the limit as the volume of the unit ball of some space that would make sense if one knew some things that I don't. See the paragraph after Corollary 8.1 on page 15 in "On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry", PNAS 110 (48) 2013 pp. 19220-19226, arxiv:1204.1613. | |
| Apr 22, 2021 at 16:49 | comment | added | YCor | More generally the degree of polynomial growth is an integer $D$ which is computable (Gromov + Wolf). In addition, Pansu proved that $c_n/n^D$ converges. | |
| Apr 22, 2021 at 16:48 | comment | added | YCor | An infinite locally finite graph always has a geodesic ray from any point. Hence if $G$ is infinite, the growth rate is $\ge n$ w.r.t. any generating subset. | |
| Apr 22, 2021 at 16:44 | history | asked | Ronnie Pavlov | CC BY-SA 4.0 |