Coulon, Rémi Growth of periodic quotients of hyperbolic groups. (English) Zbl 1353.20025 Algebr. Geom. Topol. 13, No. 6, 3111-3133 (2013). Summary: Let \(G\) be a non-elementary torsion-free hyperbolic group. We prove that the exponential growth rate of the periodic quotient \(G/G^n\) tends to the one of \(G\) as \(n\) odd approaches infinity. Moreover, we provide an estimate for the rate at which the convergence is taking place. MSC: 20F65 Geometric group theory 20F50 Periodic groups; locally finite groups 20F67 Hyperbolic groups and nonpositively curved groups 20F69 Asymptotic properties of groups Keywords:periodic groups; exponential growth; hyperbolic groups PDF BibTeX XML Cite \textit{R. Coulon}, Algebr. Geom. Topol. 13, No. 6, 3111--3133 (2013; Zbl 1353.20025) Full Text: DOI arXiv References: [1] S I Adian, The Burnside problem and identities in groups, Ergeb. Math. Grenzgeb. 95, Springer (1979) [2] G N Arzhantseva, I G Lysenok, Growth tightness for word hyperbolic groups, Math. Z. 241 (2002) 597 · Zbl 1045.20035 · doi:10.1007/s00209-002-0434-6 [3] V S Atabekyan, Uniform nonamenability of subgroups of free Burnside groups of odd period, Mat. Zametki 85 (2009) 516 · Zbl 1213.20036 · doi:10.1134/S0001434609030213 [4] W Burnside, On an unsettled question in the theory of discontinuous groups, Quart. J. Pure Appl. Math. 33 (1902) 230 · JFM 33.0149.01 [5] J W Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984) 123 · Zbl 0606.57003 · doi:10.1007/BF00146825 [6] M Coornaert, Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993) 241 · Zbl 0797.20029 · doi:10.2140/pjm.1993.159.241 [7] M Coornaert, T Delzant, A Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics 1441, Springer (1990) · Zbl 0727.20018 · doi:10.1007/BFb0084913 [8] R Coulon, Detecting trivial elements of Burnside groups · arxiv:1211.4267 [9] T Delzant, Sous-groupes distingués et quotients des groupes hyperboliques, Duke Math. J. 83 (1996) 661 · Zbl 0852.20032 · doi:10.1215/S0012-7094-96-08321-0 [10] T Delzant, M Gromov, Courbure mésoscopique et théorie de la toute petite simplification, J. Topol. 1 (2008) 804 · Zbl 1197.20035 · doi:10.1112/jtopol/jtn023 [11] É Ghys, P de la Harpe, Editors, Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser (1990) · Zbl 0731.20025 [12] M Gromov, Hyperbolic groups, Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 · Zbl 0634.20015 · doi:10.1007/978-1-4613-9586-7_3 [13] M Hall Jr., Solution of the Burnside problem of exponent 6, Proc. Nat. Acad. Sci. U.S.A. 43 (1957) 751 · Zbl 0079.03003 · doi:10.1073/pnas.43.8.751 [14] S V Ivanov, The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput. 4 (1994) · Zbl 0822.20044 · doi:10.1142/S0218196794000026 [15] M Koubi, Croissance uniforme dans les groupes hyperboliques, Ann. Inst. Fourier (Grenoble) 48 (1998) 1441 · Zbl 0914.20033 · doi:10.5802/aif.1661 · numdam:AIF_1998__48_5_1441_0 · eudml:75325 [16] F Levi, B L van der Waerden, Über eine besondere Klasse von Gruppen, Abh. Math. Sem. Univ. Hamburg 9 (1933) 154 · Zbl 0005.38507 · doi:10.1007/BF02940639 [17] I G Lysënok, Infinite Burnside groups of even period, Izv. Ross. Akad. Nauk Ser. Mat. 60 (1996) 3 · Zbl 0926.20023 · doi:10.1070/IM1996v060n03ABEH000077 [18] S P Novikov, Adams operators and fixed points, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968) 1245 · Zbl 0193.51901 [19] A Y Ol’shanskiĭ, The Novikov-Adyan theorem, Mat. Sb. 118(160) (1982) 203, 287 [20] A Y Ol’shanskiĭ, Periodic quotient groups of hyperbolic groups, Mat. Sb. 182 (1991) 543 [21] D V Osin, Uniform non-amenability of free Burnside groups, Arch. Math. \((\)Basel\()\) 88 (2007) 403 · Zbl 1173.43002 · doi:10.1007/s00013-006-2002-5 [22] I N Sanov, Solution of Burnside’s problem for exponent 4, Leningrad State Univ. Annals [Uchenye Zapiski] Math. Ser. 10 (1940) 166 · Zbl 0061.02506 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.