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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.

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
Full Text: DOI Numdam EuDML
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