Abstract
We introduce a model for random groups in varieties of n-periodic groups as n-periodic quotients of triangular random groups. We show that for an explicit dcrit ∈ (1/3, 1/2), for densities d ∈ (1/3, dcrit) and for n large enough, the model produces infinite n-periodic groups. As an application, we obtain, for every fixed large enough n, for every p ∈ (1, ∞) an infinite n-periodic group with fixed points for all isometric actions on Lp-spaces. Our main contribution is to show that certain random triangular groups are uniformly acylindrically hyperbolic.
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References
S. Antoniuk, T. Łuczak and J. świątkowski, Random triangular groups at density 1/3, Compositio Mathematica 151 (2015), 167–178.
M. R. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften, Vol. 319, Springer, Berlin, 1999.
W. Burnside, On an unsettled question in the theory of discontinuous groups, The Quarterly Journal of Pure and Applied Mathematics 3 (1902), 230–238.
M. Cordes, M. Duchin, Y. Duong, M.-C. Ho and A. P. Sanchez, Random nilpotent groups I, International Mathematics Research Notices 2018 (2018), 1921–1953.
M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, Vol. 1441, Springer, Berlin, 1990.
R. Coulon and D. Gruber, Small cancellation theory over Burnside groups, Advances in Mathematics 353 (2019), 722–775.
R. B. Coulon, Partial periodic quotients of groups acting on a hyperbolic space, Université de Grenoble. Annales de l’Institut Fourier 66 (2016), 1773–1857.
R. Coulon, Infinite periodic groups of even exponents, https://arxiv.org/abs/1810.08372.
T. Delzant and M. Gromov, Courbure mésoscopique et théorie de la toute petite simplification, Journal of Topology 1 (2008), 804–836.
C. Druţu and J. M. Mackay, Random groups, random graphs and eigenvalues of p-Laplacians, Advances in Mathematics 341 (2019), 188–254.
N. M. Dunfield and W. P. Thurston, Finite covers of random 3-manifolds, Inventiones Mathematicae 166 (2006), 457–521.
M. Gromov, Hyperbolic groups, in Essays in Group Theory, Mathematical Sciences Research Institute Publications, Vol. 8, Springer, New York, 1987, pp. 75–263.
M. Gromov, Asymptotic invariants of infinite groups, in Geometric Group Theory, Vol. 2 (Sussex, 1991), London Mathematical Society Lecture Note Series, vol. 182, Cambridge University Press, Cambridge, 1993, pp. 1–295.
M. Gromov, Spaces and questions, Geometric and Functional Analysis Special Volume (2000), 118–161.
M. Gromov, Random walk in random groups, Geometric and Functional Analysis 13 (2003), 73–146.
M. Hall, Jr., Solution of the Burnside problem for exponent six, Illinois Journal of Mathematics 2 (1958), 764–786.
S. V. Ivanov, The free Burnside groups of sufficiently large exponents, International Journal of Algebra and Computation 4 (1994), 1–308.
M. Kotowski and M. Kotowski, Random groups and property (T): Żuk’s theorem revisited, Journal of the London Mathematical Society 88 (2013), 396–416.
I. G. Lysenok, Infinite Burnside groups of even period, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 60 (1996), 3–224.
J. M. Mackay and P. Przytycki, Balanced walls for random groups, Michigan Mathematical Journal 64 (2015), 397–419.
J. P. McCammond and D. T. Wise, Fans and ladders in small cancellation theory, Proceedings of the London Mathematical Society 84 (2002), 599–644.
P. S. Novikov and S. I. Adjan, Infinite periodic groups. I, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 32 (1968), 212–244.
P. S. Novikov and S. I. Adjan, Infinite periodic groups. II, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 32 (1968), 251–524.
P. S. Novikov and S. I. Adjan, Infinite periodic groups. III, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 32 (1968), 709–731.
T. Odrzygóźdź, Nonplanar isoperimetric inequality for random groups, https://arxiv.org/abs/2104.13903.
Y. Ollivier, A January 2005 Invitation to Random Groups, Ensaios Matematicos, Vol. 10, Sociedade Brasileira de Matemótica, Rio de Janeiro, 2005.
Y. Ollivier, Some small cancellation properties of random groups, International Journal of Algebra and Computation 17 (2007), 37–51.
D. Osajda, Group cubization, Duke Mathematical Journal 167 (2018), 1049–1055.
I. N. Sanov, Solution of Burnside’s problem for exponent 4, Leningrad State University Annals. Mathematical Series 10 (1940), 166–170.
A. Żuk, Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal. 13 (2003), no. 3, 643–670.
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The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge for support and hospitality during the programme “Non-positive curvature, group actions and cohomology” where work on this paper was undertaken, supported by EPSRC grant EP/K032208/1. The research of the second author was also supported in part by EPSRC grant EP/P010245/1.
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Gruber, D., Mackay, J.M. Random triangular Burnside groups. Isr. J. Math. 244, 75–94 (2021). https://doi.org/10.1007/s11856-021-2170-9
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DOI: https://doi.org/10.1007/s11856-021-2170-9