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Growth tightness of free and amalgamated products. (English) Zbl 1018.20036
The ‘entropy’ of a finitely generated group $$G$$, endowed with a finite generating set $$S$$, is the limit $\text{Ent}(G,S)=\lim_{N\to\infty}N^{-1}\cdot\log\beta_{(G,S)}(N)$ where $$\beta_{(G,S)}(N)$$ denotes the number of elements of $$G$$ which can be represented by words on $$S\cup S^{-1}$$ of length smaller than $$N$$. Accordingly, the ‘algebraic entropy’ of $$G$$ is defined as the infimum $$\text{AlgEnt}(G)=\inf_S\text{Ent}(G,S)$$ where $$S$$ runs over all finite generating sets for $$G$$. Let $$(G,d)$$ be a discrete group endowed with a left-invariant distance. We can consider the exponential growth rate of $$G$$ with respect to $$d$$, that is the invariant $\text{Ent}(G,d)=\liminf_{R\to\infty}R^{-1}\cdot\log\#B_{(G,d)}(e,R)$ where $$B_{(G,d)}(e,R)$$ is the ball of radius $$R$$ centred at the identity $$e$$ (here the author always assumes that $$d$$ has the property that balls of finite radius are finite sets). Clearly, $$\text{Ent}(G,S)=\text{Ent}(G,d_S)$$ if $$d_S$$ denotes the word metric of $$(G,S)$$. When $$H$$ is a subgroup of $$G$$, the quotient metric shall be given to the left coset space $$G/H$$, that is the $$G$$-invariant distance $d/H(gH,g'H)=\inf_{h,h'\in H}d(gh,gh')=d(H,g^{-1}g'H).$ It is said that $$(G,d)$$ is ‘growth tight’ if for every infinite normal subgroup $$H$$ of $$G$$ one has $$\text{Ent}(G,d)>\text{Ent}(G/H,d/H)$$. The term growth tightness first appeared in work of R. Grigorchuk and P. de la Harpe [J. Dyn. Control Syst. 3, No. 1, 51-89 (1997; Zbl 0949.20033)], with respect to word metrics of finitely generated groups (here in the author’s definition is a slight modification of that given in the latter reference, where the authors require that the same inequality holds for every nontrivial normal subgroup $$H$$ of $$G$$. However, notice that these definitions are equivalent for free products of nontrivial groups, as any finite normal subgroup of $$G_1*G_2$$ is necessarily trivial [see P. Scott, T. Wall, Homological group theory, Proc. Symp., Durham 1977, Lond. Math. Soc. Lect. Note Ser. 36, 137-203 (1979; Zbl 0423.20023)].)
A distance $$d$$ on a group is called ‘quasi-algebraic’ if $$G$$ is a finite index subgroup of a finitely generated group $$\widehat G$$, and $$d$$ is the restriction to $$G$$ of a word metric of $$\widehat G$$.
$$G=G_1*_FG_2$$ is called a ‘nontrivial’ amalgamated (or free) product when $$G_1\not=F\not=G_2$$ and $$G_1$$ and $$G_2$$ are nontrivial groups.
The main results of the paper under review are as follows:
Theorem 1.3. Every nontrivial free product $$G=G_1*G_2$$, different from the infinite dihedral group $$\mathbb{Z}_2*\mathbb{Z}_2$$, is growth tight with respect to any quasi-algebraic distance.
As direct consequence of Theorem 1.3, the author states: Theorem 3.1. Let $$\widehat G$$ be a finitely generated group which contains a nontrivial free product $$G=G_1*G_2\not=\mathbb{Z}_2*\mathbb{Z}_2$$ as a subgroup of finite index. Then $$\widehat G$$ is growth tight with respect to any word metric.
Theorem 1.4. Let $$G=G_1*_FG_2$$ be a finitely generated group of exponential growth, which is a nontrivial amalgamated product of residually finite groups $$G_i$$ over a finite subgroup $$F$$. Then $$G$$ is growth tight with respect to any word metric.
The construction of Theorem 1.4 is similar to that used in Lemma 7.4 of P. Scott, T. Wall [loc. cit.].
In the first reference quoted above Grigorchuk and de la Harpe have asked: Do there exist finitely generated groups $$G$$ such that $$\text{Ent}(G,S)>\text{AlgEnt}(G)$$ for every $$S$$? In this case the author says, shortly, that “the minimal growth of $$G$$ is not achieved”.
Here the author answers this question positively: the author exhibits a large class of groups of uniform exponential growth whose minimal growth is not achieved. Namely:
Corollary 1.6. Every nontrivial free product whose minimal growth is achieved is Hopfian. In particular, if $$G$$ is the free product of a non-Hopfian group with any nontrivial group, the minimal growth of $$G$$ is not achieved.

##### MSC:
 20F69 Asymptotic properties of groups 20F05 Generators, relations, and presentations of groups 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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##### References:
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