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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
##### Keywords:
algebraic entropy; group endomorphism; growth function
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##### References:
 [1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. 114 (1965), 309-319. · Zbl 0127.13102 [2] D. Dikranjan and A. Giordano Bruno, The Pinsker subgroup of an algebraic flow, J. Pure Appl. Algebra 216 (2012), no. 2, 364-376. · Zbl 1247.37014 [3] D. Dikranjan and A. Giordano Bruno, Topological entropy and algebraic entropy for group endomorphisms, Proceedings of the International Conference on Topology and Its Applications (ICTA 2011), Cambridge Scientific Publishers, Cambridge (2012), 133-214. · Zbl 1300.54002 [4] D. Dikranjan and A. Giordano Bruno, Entropy on abelian groups, Adv. Math. 298 (2016), 612-653. · Zbl 1368.37015 [5] D. Dikranjan, B. Goldsmith, L. Salce and P. Zanardo, Algebraic entropy for abelian groups, Trans. Amer. Math. Soc. 361 (2009), 3401-3434. · Zbl 1176.20057 [6] M. Fekete, Über die Verteilung der Wurzeln bei gewisser algebraichen Gleichungen mit ganzzahlingen Koeffizienten, Math. Z. 17 (1923), 228-249. · JFM 49.0047.01 [7] A. Giordano Bruno and P. Spiga, Milnor-Wolf theorem for the growth of endomorphisms of locally virtually soluble groups, in preparation. [8] R. I. Grigorchuk, On Milnor’s problem of group growth (in Russian), Dokl. Akad. Nauk SSSR 271 (1983), 31-33; translation in Sov. Math. Dokl. 28 (1983), 23-26. · Zbl 0547.20025 [9] M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. Inst. Hautes Études Sci. 53 (1981), 53-73. · Zbl 0474.20018 [10] J. Milnor, Growth in finitely generated solvable groups, J. Differential Geom. 2 (1968), 447-449. · Zbl 0176.29803 [11] J. Milnor, Problem 5603, Amer. Math. Monthly 75 (1968), 685-686. [12] J. Peters, Entropy on discrete abelian groups, Adv. Math. 33 (1979), 1-13. · Zbl 0421.28019 [13] M. D. Weiss, Algebraic and other entropies of group endomorphisms, Math. Systems Theory 8 (1974/75), no. 3, 243-248. · Zbl 0298.28014 [14] J. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geom. 2 (1968), 424-446. · Zbl 0207.51803
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