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In 2019 Anna Erschler and Tianyi Zheng gave a very sharp estimate of the growth of Grigorchuk's first group. Although it was one of the first example of a group of intermediate growth (finitely generated group whose growth is neither polynomial nor exponential), how fast it grows was not really known. In Grigorchuk's original paper, the exponent $\alpha$ in $\mathrm{exp}(Cn^\alpha)$ was only known to lie somewhere between 0.5 and 0.991…. Quite a few papers made improvements on these bounds. For example, Bartholdi brought the upper bound down to 0.7675… and Leonov brought the upper bound to 0.504…, but until then it remained unknown.

EDIT: if $b_n$ is the cardinality of the ball of radius $n$ in Grigorchuk's first group, Erschler and Zheng proved that $$ \alpha := \lim_{n \to \infty} \frac{ \log \log b_n}{\log n} = \frac{\log 2}{\log \lambda_0} \approx 0.7674 $$ where $\lambda_0$ is the positive real root of the polynomial $x^3-x^2-2x-4$. Note that the group may still grow somehow faster or slower than $\mathrm{exp}{(C n^\alpha)}$, but they identified the dominating term in the growth. Also, since changing the generating set is a bi-Lipschitz map, this is the only part of the growth function that is guaranteed to be independent of the generating set.

In 2019 Anna Erschler and Tianyi Zheng gave a very sharp estimate of the growth of Grigorchuk's first group. Although it was one of the first example of a group of intermediate growth (finitely generated group whose growth is neither polynomial nor exponential), how fast it grows was not really known. In Grigorchuk's original paper, the exponent $\alpha$ in $\mathrm{exp}(Cn^\alpha)$ was only known to lie somewhere between 0.5 and 0.991…. Quite a few papers made improvements on these bounds. For example, Bartholdi brought the upper bound down to 0.7675… and Leonov brought the lower bound to 0.504…, but until then it remained unknown.

EDIT: if $b_n$ is the cardinality of the ball of radius $n$ in Grigorchuk's first group, Erschler and Zheng proved that $$ \alpha := \lim_{n \to \infty} \frac{ \log \log b_n}{\log n} = \frac{\log 2}{\log \lambda_0} \approx 0.7674 $$ where $\lambda_0$ is the positive real root of the polynomial $x^3-x^2-2x-4$. Note that the group may still grow somehow faster or slower than $\mathrm{exp}{(C n^\alpha)}$, but they identified the dominating term in the growth. Also, since changing the generating set is a bi-Lipschitz map, this is the only part of the growth function that is guaranteed to be independent of the generating set.

In 2019 Anna Erschler and Tianyi Zheng gave a very sharp estimate of the growth of Grigorchuk's first group. Although it was one of the first example of a group of intermediate growth (finitely generated group whose growth is neither polynomial nor exponential), how fast it grows was not really known. In Grigorchuk's original paper, the exponent $\alpha$ in $\mathrm{exp}(Cn^\alpha)$ was only known to lie somewhere between 0.5 and 0.991…. Quite a few papers made improvements on the upper bound e.g. Bartholdi brought it down to 0.7675… and Leonov up to 0.504… but until then it remained unknown.

EDIT: if $b_n$ is the cardinality of the ball of radius $n$ in Grigorchuk's first group, Erschler and Zheng proved that $$ \alpha := \lim_{n \to \infty} \frac{ \log \log b_n}{\log n} = \frac{\log 2}{\log \lambda_0} \approx 0.7674 $$ where $\lambda_0$ is the positive real root of the polynomial $x^3-x^2-2x-4$. Note that the group may still grow somehow faster or slower than $\mathrm{exp}{(C n^\alpha)}$, but they identified the dominating term in the growth. Also, since changing the generating set is a bi-Lipschitz map, this is the only part of the growth function that is guaranteed to be independent of the generating set.

In 2019 Anna Erschler and Tianyi Zheng gave a very sharp estimate of the growth of Grigorchuk's first group. Although it was one of the first example of a group of intermediate growth (finitely generated group whose growth is neither polynomial nor exponential), how fast it grows was not really known. In Grigorchuk's original paper, the exponent $\alpha$ in $\mathrm{exp}(Cn^\alpha)$ was only known to lie somewhere between 0.5 and 0.991…. Quite a few papers made improvements on these bounds. For example, Bartholdi brought the upper bound down to 0.7675… and Leonov brought the upper bound to 0.504…, but until then it remained unknown.

EDIT: if $b_n$ is the cardinality of the ball of radius $n$ in Grigorchuk's first group, Erschler and Zheng proved that $$ \alpha := \lim_{n \to \infty} \frac{ \log \log b_n}{\log n} = \frac{\log 2}{\log \lambda_0} \approx 0.7674 $$ where $\lambda_0$ is the positive real root of the polynomial $x^3-x^2-2x-4$. Note that the group may still grow somehow faster or slower than $\mathrm{exp}{(C n^\alpha)}$, but they identified the dominating term in the growth. Also, since changing the generating set is a bi-Lipschitz map, this is the only part of the growth function that is guaranteed to be independent of the generating set.

In 2019 Anna Erschler and Tianyi Zheng gave a very sharp estimate of the growth of Grigorchuk's first group. Although it was one of the first example of a group of intermediate growth (finitely generated group whose growth is neither polynomial nor exponential), how fast it grows was not really known. In Grigorchuk's original paper, the exponent $\alpha$ in $\mathrm{exp}(Cn^\alpha)$ was only known to lie somewhere between 0.5 and 0.991... Quite a few papers made improvements on the upper bound e.g. Bartholdi brought it down to 0.7675... and Leonov up to 0.504... but until then it remained unknown.

EDIT: if $b_n$ is the cardinality of the ball of radius $n$ in Grigorchuk's first group, Erschler and Zheng proved that $$ \alpha := \lim_{n \to \infty} \frac{ \log \log b_n}{\log n} = \frac{\log 2}{\log \lambda_0} \approx 0.7674 $$ where $\lambda_0$ is the positive real root of the polynomial $x^3-x^2-2x-4$. Note that the group may still grow somehow faster or slower than $\mathrm{exp}{(C n^\alpha)}$, but they identified the dominating term in the growth. Also, since changing the generating set is a bi-Lipschitz map, this is the only part of the growth function that is guaranteed to be independent of the generating set.

In 2019 Anna Erschler and Tianyi Zheng gave a very sharp estimate of the growth of Grigorchuk's first group. Although it was one of the first example of a group of intermediate growth (finitely generated group whose growth is neither polynomial nor exponential), how fast it grows was not really known. In Grigorchuk's original paper, the exponent $\alpha$ in $\mathrm{exp}(Cn^\alpha)$ was only known to lie somewhere between 0.5 and 0.991…. Quite a few papers made improvements on the upper bound e.g. Bartholdi brought it down to 0.7675… and Leonov up to 0.504… but until then it remained unknown.

EDIT: if $b_n$ is the cardinality of the ball of radius $n$ in Grigorchuk's first group, Erschler and Zheng proved that $$ \alpha := \lim_{n \to \infty} \frac{ \log \log b_n}{\log n} = \frac{\log 2}{\log \lambda_0} \approx 0.7674 $$ where $\lambda_0$ is the positive real root of the polynomial $x^3-x^2-2x-4$. Note that the group may still grow somehow faster or slower than $\mathrm{exp}{(C n^\alpha)}$, but they identified the dominating term in the growth. Also, since changing the generating set is a bi-Lipschitz map, this is the only part of the growth function that is guaranteed to be independent of the generating set.