Growth of periodic quotients of hyperbolic groups. (English) Zbl 1353.20025

Coulon, Rémi

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

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