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Some properties of the growth and of the algebraic entropy of group endomorphisms. (English) Zbl 1401.20041
The authors study the growth of group endomorphisms, a generalization of the classical notion of growth of finitely generated groups. The idea of growth in arbitrary groups was discussed in [D. Dikranjan and A. Giordano Bruno, in: Proceedings of the international conference on topology and its applications (ICTA 2011), Islamabad, Pakistan, July 4–10, 2011. Cambridge: Cambridge Scientific Publishers. 133–214 (2012; Zbl 1300.54002)]. The generalizations use the language of algebraic entropy, first given for endomorphisms of torsion abelian groups in [R. L. Adler et al., Trans. Am. Math. Soc. 114, 309–319 (1965; Zbl 0127.13102)]. The authors first show that the addition theorem for algebraic entropy does not hold for endomorphisms of arbitrary groups. They then show that for arbitrary groups \(G\), \(G\) has polynomial growth exactly when \(G\) is locally virtually nilpotent. Then they show that if \(G\) is virtually locally soluble, then \(G\) has either polynomial or exponential growth. They also show that if \(G\) is locally finite and if \(\phi:G\longrightarrow G\) is a group endomorphism of zero entropy, then every element of \(G\) belongs to a finite \(\phi\)-invariant subgroup of \(G\).

MSC:
20F65 Geometric group theory
20E07 Subgroup theorems; subgroup growth
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
20F05 Generators, relations, and presentations of groups
20E25 Local properties of groups
20F50 Periodic groups; locally finite groups
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