## Groups of oscillating intermediate growth.(English)Zbl 1283.20027

For a finitely generated group $$\Gamma$$, its growth function $$\gamma_\Gamma(n)$$ is given by the number of elements in $$\Gamma$$ with word length $$\leq n$$. Answering a famous question of J. Milnor [Am. Math. Mon. 75, 685-686 (1968)], R. I. Grigorchuk constructed the first example of a group of intermediate growth [Sov. Math., Dokl. 28, 23-26 (1983); translation from Dokl. Akad. Nauk SSSR 271, 30-33 (1983; Zbl 0547.20025)], i.e. with $$\gamma_\Gamma(n)$$ growing strictly faster than any polynomial and strictly slower than the exponential function. During the following thirty years the study of groups of intermediate growth turned into a prominent subject. The present paper is an important new contribution to this field.
The main result can be roughly described as follows. Let $$f_1$$, $$f_2$$, $$g_1$$, $$g_2$$ be monotone increasing subexponential integer functions that satisfy $f_1\succeq f_2\succeq g_1\succeq g_2=\gamma_{G_\omega},\tag{*}$ where $$G_\omega$$ is a Grigorchuk group of intermediate growth with the usual set of generators. Then under some further technical conditions strengthening (*), there exists a finitely generated group $$\Gamma$$ with growth function $$h(n)=\gamma_\Gamma(n)$$ such that $$g_2(n)<h(n)<f_1(n)$$ and $$h(n)$$ takes values in the intervals $$[g_2(n),g_1(n)]$$ and $$[f_2(n),f_1(n)]$$ infinitely often. – The proof of this theorem is based on the beautiful idea of approximating the Grigorchuk group by an infinite sequence of finite groups $$G_N=\text{PSL}_N(\mathbb Z_N)$$ and then using expanding properties of $$G_N$$ to analyze the growth of the group $$\Gamma=\otimes G_N$$ (we refer to the paper for the definition of the product that is used here).
Other important results about the growth functions of groups of intermediate growth were obtained recently by L. Bartholdi and A. Erschler [Invent. Math. 189, No. 2, 431-455 (2012; Zbl 1286.20025)] and J. Brieussel [preprint arXiv 1107.1632].

### MSC:

 20F05 Generators, relations, and presentations of groups 20F69 Asymptotic properties of groups 20E08 Groups acting on trees

### Citations:

Zbl 0547.20025; Zbl 1286.20025
Full Text:

### References:

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